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Quantum Chemistry

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Complex numbers

Complex numbers

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Complex numbers are expressed as a combination of a real and imaginary part, in the form . The variables and are both real integers but is the real component and is imaginary due to the presence of the imaginary unit . It is defined as 2 and therefore .

Working with the square root of negative numbers was impossible until the introduction of this concept and since its inception it has found numerous industrial and theoretical applications.

For example, electrical engineers use it to calculate impedance, the resistance of an alternating current circuit as it varies with frequency, which an important analysis tool. The relation goes as follows, where , is resistance and is reactance in electrical engineering.

In this case resistance is real while reactance is an imaginary component.

In quantum chemistry the probability distribution of a quantum mechanical (QM) particle, is obtained by multiplying the wavefunction, by it's complex conjugate *.

Hydrogen Electron Orbitals as Predicted by Quantum Mechanics
The Probability Distribution of a QM particle

2*

This is an expression of the likelihood of a quantum mechanical being at a certain point in space over time. In combination with the Schrödinger equation it has allowed us to map the density of electrons around the nucleus. These distributions form highly complex shapes which can be depicted and understandable to the human mind. The knowledge of molecules at the atomic level which this equation has brought to mankind is immense.

Euler's Relation

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When working with imaginary numbers there are several shortcuts to simplify the task. An example of a simple yet useful relation is ·22.

In 1748 Leonhard Euler published his first work. It contained a relation which showed the fundamental relationship between the imaginary exponential, , and the combination of a cosine curve and an imaginary sine curve.

Illustration of Euler's Relation
Euler's Relation


It demonstrated the point at which algebra and geometry unified and changed mathematics and all disciplines descended from it forever. It was groundbreaking, so much so that Richard Feynman, a renowned physicist called it "one of the most remarkable, almost astounding, formulas in all of mathematics"[1]

Sample Question

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Use Euler's relation to simplify the following expression, which is relevant to the phase factor in quantum wavefunctions:

Express your answer in the form a+bi, where a and b are real numbers.

Answer

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Both halves of the expression can be written in trigonometric form using Euler's formula.

They can then be plugged back into the expression.

Special Right Triangle for 45 degree angle (π/4)
Special Right Triangle for 30 (π/6) and 60 (π/3) degree angles

These sine and cosines have real integer answers. They can be found using a calculator or by using shortcuts such as the special right angle triangles and "SOHCAHTOA" which relates trigonometric expressions to the opposite, adjacent and hypotheneuse of triangles.

The two parentheses can then be multiplied by one another. Note here that when is multiplied by itself it equals -1.

The last step is collecting like terms.

Which can also be expressed with decimals.

  1. Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew L. (2011). The Feynman lectures on physics (in Eng) (New millennium ed.). New York: Basic Books. ISBN 978-0-465-02414-8. OCLC 671704374.{{cite book}}: CS1 maint: unrecognized language (link)


Integration by parts

Example Problem:

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Evaluate the triple integral shown below in spherical polar coordinates using the given boundary conditions:

Solution:

The triple integral can be rewritten as an iterated integral under the boundary conditions as follows:

=

The integral can then be evaluated using Fubini's theorem to separate the multivariable integrals into single variable integrals that can be solved accordingly as follows:

=

= =

Multiplying these results together then gives the final solution:

(2) =

Therefore,

=


Probability and statistics

Probability distributions describe the likelihood of a variable taking on a given range of values. The probability (P) of an arbitrary event happening can be calculated using the following formula, where M represents the number of times the event occurred over N number of trials.

Definition of Probability

The value of P can range between 0 (event is impossible) and 1 (event is guaranteed). Suppose an event has a range of possible outcomes. By repeatedly calculating the frequency of each individual outcome, one can derive a probability distribution over all outcomes, describing the likelihood of any possible outcome occurring in a single trial.

Several key properties characterize these distributions. The probability of any event or outcome must be a non-negative value, as negative probabilities are physically meaningless. Second, the total probability must be normalized, meaning the sum of probabilities for all possible outcomes must equal one. Additionally, probability distributions can either be discrete or continuous. Discrete distributions apply when a variable can take on a limited number of distinct values, allowing the probability of a specific outcome to be determined. In contrast, continuous distributions apply for variables that can assume any value within a range, such as the position of an electron in space. This is common in quantum mechanics, where probabilities are associated with continuous variables, like the x-axis. In such cases, calculating the probability of finding a particle at an exact point (e.g., x = 0.5000) is practically meaningless, as the probability at any single point is effectively zero. Instead, it is more useful to consider the probability of locating the particle within a small interval along the x-axis. The probability (P) can be calculated by integrating the probability distribution (P(x)) over an interval .

Probability of Finding a Value within an Interval

In quantum mechanics, probability and statistics play an essential role in interpreting and predicting the behavior of particles. The system is defined by a wavefunction, 𝜓, which is used to derive various properties of the system, such as position, momentum and energy.  The probability distribution for a quantum mechanical particle is the square of the wavefunction. The probability distribution can then be used to determine the probability of a particle being in a certain interval or the average value of that property, via integration.

The probability distribution of a quantum mechanical particle in 3D

The simplest quantum system that can be described is called “particle in a 1D Box”. The particle has a probability of existing anywhere along a 1D box ( when ) but can never exist outside of the box ( when or ). The probability of the particle existing within the bounds of the box is one, since the total probability of it being located somewhere in the allowed region is 100%. Using probability distributions, one can determine the likelihood of a particle being in any specific range along the x-axis.

Example

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Calculate the probability of the particle being located between and in the first quantum state (n=1) for a particle in a 1D box.

Solution: The wavefunction for a particle in a 1D box is,

The probability density is the square of the wavefunction:

For the interval , the integral becomes,

Substituting ,

Therefore, the probability of finding a particle in the first half of the 1D box is , or .


Multivariable differentiation

Functions dependent on multiple variables require a different approach to differentiation than single variable functions. The partial derivative of a multivariable function is the derivative with respect to one of its variables, with all other variables treated as constants.

The notation convention is like single variable differentiation as the function is written on top of the variable it is differentiated with respect to. A cursive d is used instead of d, and the variables held constant are indicated outside the parentheses in a subscript. For example, the partial derivative of the function with respect to , holding constant, is:

Partial derivatives show how the function changes as one variable is changing if all others do not change. To determine the change in the function if all variables are changing, partial derivatives where one variable changing is determined for each variable of the function separately.

In two dimensions:

In three dimensions:

The second partial derivative of a function can be determined from partial differentiation with respect to one variable, and then partial differentiation again with respect to the other variable, the one held constant in the last partial derivative. Second partial derivatives with different sequence of differentiation may not always be equal. In the following two-dimensional example, they are equal:

Alongside the full notation, it is also common notation to write the order of differentiation as a superscript on the function, then below it the variables of partial differentiation. This does not account for order in which the function is differentiated.

Useful rules in calculus can be applied to partial derivatives, such as the chain rule:

Partial derivatives are used in the derivation of the time-independent Schrödinger equation, angular momentum, the model for the hydrogen atom, and appears in operators such as the Hamiltonian. It is also used extensively in thermodynamics.

Example

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The derivative of this wavefunction with respect to :

Differentiating with respect to while holding constant means is treated as a constant in differentiation, so only terms with are differentiated:

is a constant independent of so it can be moved out of the differential:

To differentiate the term, the chain rule is used:

Once again, constants can be removed from differentiation:


The derivative of the wavefunction with respect to :

Differentiating with respect to while holding constant means is treated as a constant in differentiation, so only terms with are differentiated:

To differentiate the term, the chain rule is used:

Once again, constants can be removed from differentiation:


Implicit differentiation

Provide a brief description of implicit differentiation.

Example

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Differentiate the function with respect to x

Where


Operator algebra

Overview

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Operator algebra is a subset of functional analysis and involves the use of continuous linear operators on a topological vector space. Operators are mathematical objects that, when applied to functions, generate a corresponding function or value. A key concept in operator algebra is eigenvalues and eigenfunctions. An eigenfunction is a function that, when acted upon by an operator, yields a scalar multiple of itself, with the scalar being the eigenvalue.

In quantum mechanics, operator algebra forms the mathematical basis for describing physical observables in quantum systems. Operators are generally applied to wave functions and can be used to determine physical properties like position, momentum, and energy.

The most common operator is the Hamiltonian operator, as it represents the total energy of a quantum system as described by a specific wavefunction.

The Hamiltonian operator is made up of the kinetic and potential energy operators.

Defining the kinetic energy operator in 1D, shows that operators are not always in the form of physical quantities and can also represent simple mathematical operations such as a derivative.

In this case, the second partial derivative is defined as the Laplacian operator .

Example Problem & Solution

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Problem

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There is a known commutator between the position and momentum operators that is used to prove the Heisenberg uncertainty principle.

,

Provide a detailed derivation using operator algebra for this expression by applying the position and momentum commutator to an arbitrary wavefunction where:

and  

Solution

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The commutator of two operators and  is defined as:

Substituting the position and momentum operators:

Apply the commutator to an arbitrary wavefunction :

Expand and

Substitute into the initial commutator expression and simplify:

Since the commutator acts on the wavefunction to give we know that:


Example 1

Find ⟨x⟩, ⟨x2⟩, ⟨px⟩ and ⟨px2⟩ for a quantum harmonic oscillator in the ground state, then determine the uncertainty on the position and momentum. Is the product of the uncertainty on position and momentum consistent with the Heisenberg's Uncertainty Principle?

Heisenberg's Uncertainty Principle

The wavefunction of a quantum harmonic oscillator in the ground state is:

Using this wavefunction the average position and the average of the square of the position can be calculated.


The average position:

use



The average square of the position:

use

use and



The uncertainty on the position:


The average momentum:

use


The average square of the momentum:

use

use

use and



The uncertainty on the momentum:


The product of the uncertainty on the position and the uncertainty on the momentum is:


This is equal to , therefore, a quantum harmonic oscillator in the ground state is consistent with the Heisenberg Uncertainty Principle.


Example 2

Question

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Starting with the wave equation of a 1D box:

The classical probability of a particle existing in the region of the box [0, 1/L] is,

Where is the length of the 1D box, is the principle quantum number, and is the position of the particle in a 1D box.

Derive an equation that calculates the probability of the particle being in the [0, 1/L] for the quantum mechanical particle in a 1D box. Is this result compatible with the correspondence principle?

Solution

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Example 3

Question 3

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Calculate the number of nodes for a particle in a 1D box when n=2 and n=3 when the length of the box is L=5, and give the x-intercept of the node(s).

Wavefunction for a particle in a 1D-Box

Answer

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Part 1

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nnodes=n-1

nnodes=2-1=1

We're looking for one x-intercept.





Use the form where





is the x-intercept

x-intercepts are 0, , 5

0 and 5 are the edges of the box and is the only node.

Part 2

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nnodes=n-1

nnodes=3-1=2

We're looking for two x-intercepts.







x-intercepts are

x-intercepts = 0,

0 and 5 are the edges of the box, and are the nodes.


Example 4

Question

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An electron in a 1D box emits a photon as the electron transitions to a lower energy level. If the length of the 1D box is equal to 1.0 cm, and the quantum number transition is , what is the electromagnetic radiation frequency of the emitted photon?

Solution:

The energy level of a particle in a 1D box at a specific quantum number () is,

Where is equal to Planck's constant (6.62607015 x 10-34 Js), is equal to the quantum number ( = 1, 2, 3, ...), is equal to the mass of the particle, and is equal to the length of the 1D box. For an electron, the mass is equal to 9.10938356 x 10-31 kg.

Since (assuming the mass and box length are constant), the energy level increases by a factor of 4 as the quantum number increases by a factor of 2. Therefore, if a particle in a 1D box undergoes an energy level transition, there is a difference between the initial and final quantum number energy levels. The energy level difference () of a particle in a 1D box that has undergone an energy level transition is,

Where is equal to the final quantum number, and is equal to the initial quantum number.

If , is a positive value; photon absorbed.

If , is a negative value; photon emitted.

Therefore, the energy level difference of an electron in a 1D box with a length of 1.0 cm, which has undergone a transition is,

The energy level difference for the electron which underwent a transition is equal to -3.01 × 10-33 J. Since , 3.01 × 10-33 J was emitted from the electron. If the electron underwent a transition, the electron would absorb the same amount of energy that was emitted from the transition which was 3.01 × 10-33 J.

Therefore,

The energy of a photon has a specific frequency of electromagnetic (EM) radiation, and the energy is directly proportional to the frequency. The energy of the photon is equal to,

Where is equal to Planck's constant (6.62607015 x 10-34 Js), and is equal to EM radiation frequency.

Rearranging this equation allows for the calculation of the photon EM radiation frequency,

The calculated photon energy was equal to 3.01 × 10-33 J, therefore the EM radiation frequency of the emitted photon from the transition of an electron in a 1D box with a length of 1.0 cm is,


Example 5

For a particle (assume the particle is an electron with 1 quantum number.) in a 1D box of length 5 cm, the equation of energy levels of a particle in a 1D box is,

a. If the length of the 1D box is increased to 10 cm, what is the change in the energy level of this particle in the box?

b. If the length of the 1D box is decreased to 2 cm, what is the change in the energy level of this particle in the box?

c. Explain the effect of length changes on the energy levels of particles in a 1D box.

Solutions

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The energy levels of a particle in 1D box:

where h is Planck's constant equal to

Because this particle is an electron with 1 quantum number:

m is the mass of this particle equal to

n is the quantum number of the particle equal to 1

L is the length of this 1D box equal to

for this question,

a. for part a, the initial length of this 1D box: , the final length of this 1D box:

b. for part b, the initial length of this 1D box: , the final length of this 1D box:

c. Based on the calculations in part a and b, the energy level of the particles in the 1D box is decreased by the increase of this 1D box length, and the energy level of the particles in the 1D box is increased by the decrease of this 1D box length. Therefore, the energy level of the particles in the 1D box is negatively related to the length of this 1D box.


Example 6

Using the following equation calculate the ground state energies of H, He+ and Li2+, , then draw a conclusion on the effect of increasing atomic number on the energy of the ground state.


Solution:

Using the Schrödinger equation (ĤΨ=EΨ) the wave functions of the hydrogen atom have been solved. From the Schrödinger equation, the energy of the quantized states of the hydrogen atom can also be determine, as energy is the eigenvalue of the Schrödinger equation. The equation for energy for each energy level in the hydrogen atom is as follows,

Energy of Energy Levels

In the equation stated above Z is the nuclear charge, me (9.10938356 x 10-31 kg) is the mass of the electron, e (1.60217662 x 10-19 C) is the elementary charge, ε0 (8.8541878128 x 10-12 Fm-1 ) is the vacuum permittivity, ħ is the reduced Planck's constant and n is the principal quantum number.

The wave functions of the hydrogen atom can be extended to any atom with only one electron. Any atom with only one electron will have the electron in the same energy level as the hydrogen atom as electrons must be placed in the lower energy levels before higher ones. The only thing that will differ between hydrogen and other atoms with one electron is the attraction between the nucleus and the electron. As other atoms such as He+ and Li2+ have a greater number of protons in their nucleus resulting in a greater Coulombic attraction of the electron to the nucleus affecting the orbital size. As a result the energy of the ground state will differ between one electron systems with different nuclear charges.

This can be demonstrated by calculating the ground state energies for H, He+ and Li2+ and seeing how the energies differ. Starting with calculating the ground state of hydrogen where Z=1 and n=1,

Then for He+ where Z=2 and n=1,

Finally for Li2+ where Z=3 and n=1,

The ground state energy for the hydrogen atom is equal to -0.5 Hartree, for He+ it is -2 Hartree and for Li2+ it is -4.5 Hartree. The energies are increasing in a non-linear fashion which can be attributed to the Z2 term in the numerator. This means as Z increases more energy will be required to ionize a one electron atom.


Example 7

Example Problem 7

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Using calculus, derive the most probable radius of an electron 1s orbital.

Solution

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To determine the equation for the most probable radius of an electron in an 1s orbital we must calculate the value of r at the maximum point on the probability distribution. This is done by setting the derivative of radial probability to zero and solving for radius.

The general formula for the radial probability is.

[1]

After inserting the wavefunction of a 1s orbital we can simplify the radial probability equation as follows.

Radial Component of Wave Function of 1s Orbital

[2]

The derivative of the simplified probability function must then be calculated.

The constants can be excluded from the rest of the equation by moving them to the front of the derivative.

We see that the function is composed of the product of two smaller functions of the radius, multiplied by constants. Therefore we can apply the derivative product rule to solve for the equation of the derivative of the probability function.

Product Rule of Derivatives


Simplifying and moving the constants now will be beneficial to simplifying the final equation


After setting the derivative of radial probability to zero as seen above it can be seen that the only way that the equation can be equal to zero is if the polynomial part of the formula is equal to zero. This is the case because and the first part of the equation is constant which means it cannot be zero. Therefore...

The equation for the exact value of the most probable radius in a 1s orbital is then calculated using the quadradic equation.

Quadratic Equation

Therefore let , , and


The most probable radius cannot be a negative distance or zero meaning the numerator must be negative so that it is cancelled out by the negative value of the denominator. This means that the plus or minus operator in the numerator must be minus.


In conclusion the most probable radius for an electron in an 1s orbital is where that radius is proportional to the bohr radius and inversely proportional to the nuclear charge of the nucleus. This means that for an atom with one proton like hydrogen [H] the most probable radius is while for the helium ion [He+] which also only has one electron in the 1s orbital the most probable radius is .


Example 8

Write a question and its solution that quantitatively demonstrates the Heisenberg uncertainty principle holds for the J=0 state of a quantum rigid rotator

Background Information

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Rigid rotor model

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Rigid rotor model for the example molecule HCl

In quantum chemistry, the rigid rotor model is used to describe the rotations in molecules, such as HCl. The assumptions used in the rigid rotor model is that the rotating molecules is rigid, and the changes in bond length that naturally occur in a molecule (such as vibrational) is insignificant compared to the bond length re, and thus, negligible. [3]

The total energy of the system is the summation of the potential energy and the kinetic energy. The potential energy of a rigid rotor is 0 given the assumption that the rigid rotor bond length is constant. As such, the total energy of the system is equal to KE, which is equal to the angular momentum. [3] The energy level (EJ) of a linear rigid rotor model (such as HCl) is given by the equation:

[3]

In which I is the inertia, based on the reduced mass of the diatomic and bond length, and J is the quantum energy level.

Rigid rotor model for reduced mass

The rigid rotor model is 3 dimensional, and for ease of calculations, rather than 2 sets of masses that are used (m1 and m2), 1 reduced mass is used instead (μ). [3]


And in a spherical model of the rigid rotor, there are 2 variables that are used to determine position, the angels θ and ϕ, given that r is the constant bond length re.


Thus, the position of the reduced mass is given by the wave function:

Heisenberg Uncertainty Principle

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The Heisenberg uncertainty principle states that the exact position and momentum of a particle cannot be determines at one given point, and the more precisely either is determined, the less certain the other would be.[4]

However, even thou the exact position and momentum cannot be calculated at a given moment, they can be related, and that relation is: [4]

In which  is the reduced Planck constant,   is the uncertainty of position, and  is the uncertainty of momentum.

However, in the rigid rotor model, this equation does not work. The Heisenberg inequality is recalculated: [4]

Example

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Question: using the rigid rotor model at J=0, what is the moment of angular momentum and position? Does the rigid rotor follow Heisenberg’s principle at J=0?

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In the ground state zero-point, J = 0, and thus, the energy level of the rigid rotor is 0.

Given that the energy is 0, the angular momentum is known (Lz = 0).


The probability density of the position can be obtained using the wavefunction:

Average of position = 0, as such, the variance is calculated:

at m=0 J=0


if is plotted on a sphere

Where m = integers between J and – J.

|m| = number of longitudinal node

J = number of latitudinal nodes

At ground state, J = 0, |m| = 0. Thus, probability of position is equally spread across the sphere.


at J=0,

thus

is satisfied

Thus, this shows that at ground state, J=0, the rigid rotor model follows the Heisenberg uncertainty.

References

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[3] Anderson, J.M. Introduction to Quantum Chemistry, 1969, W.A. Benjamin, Inc, pg.91-100.

[4] Fayngold & Fayngold, Quantum Mechanics and Quantum Information, pp 384-388.

  1. Engel, Thomas, and Philip Reid. “Chapter 20: The Hydrogen Atom.” Physical Chemistry, 3rd ed., Pearson, Upper Saddle River, 2018, p. 478.
  2. Engel, Thomas, and Philip Reid. “Chapter 20: The Hydrogen Atom.” Physical Chemistry, 3rd ed., Pearson, Upper Saddle River, 2018, p. 468.
  3. a b c d e Anderson, J.M. Introduction to Quantum Chemistry, 1969, W.A. Benjamin, Inc, pg.91-100.
  4. a b c d Fayngold & Fayngold, Quantum Mechanics and Quantum Information, pp 384-388.


Example 9

Write a question and its solution that calculates the locations of the nodes of an electron in a 2s orbital.

The Question

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The question is to find the location of the radial node in a 2s electron for a hydrogen atom. To find the node one can start by analyzing, generally, how many nodes one should expect to see in a 2s electron system. There are two equations that give the number of nodes present in an orbital, the angular node equation and radial node equations:

1. Radial Nodes:

2. Angular Nodes:

Therefore ℓ must be determined, and based on the table 1 data one can determine ℓ is equal to 0. And the n is equal to the which is 2, this comes from the number before the orbital type which tells you the principle quantum number.

Table 1: Orbitals and Quantum Number
Orbital Angular

Momentum

Quantum

Number (ℓ)

s 0
p 1
d 2
f 3

So how many nodes are there?

First analyze the number of angular nodes:

Therefore, the number of angular nodes is 0.

Radial nodes:

Therefore there is one radial node present in a 2s orbital, resulting in the question becoming where is the location of that radial node?

The Wavefunction

Now the next main step is to determine what wavefunction describes the wavefunction this scenario of the 2s electron. The wavefunction can be found online which is

In the equation the is the wavefunction for the 2s electron, 𝒓 is the radius, and is the Bohr radius.

Based on the equation we can then solve for the position of the electron. The best way to do this is to find where is the equation going to be equal to zero and what term that contains the position causes this. The first part is a constant thus won't change with the radius, the position, of the electron. So the only place that will change with the position of the electron are the 𝒓 terms. With the 𝒓 term can be any number and the term won't be zero unless the 𝒓 is approaching infinity, while the can potentially be equal to zero since it has 2 subtracted by the position term. Therefore one can set this term equal to zero and solve for 𝒓.

Therefore, we get the solution to the position of the radial node which is 𝒓 , so when 𝒓 the probability of the electron being there is 0 all around the nucleus creating a node. The has a length of 52.9 pm which means the node is 105.8 pm in radius away from the nucleus.



In Conclusion

In conclusion the approach to the problem of finding nodes for an electron in an orbital boils down to first finding the number of theoretical nodes, then determining the wavefunction, analyzing the wavefunction's variables and solving for 0. After this is all done you will have the solution to the radial node locations. Further problems can be solved as well, because they are follow up questions that are made easier to solve after finding the nodes, like the position of the electron in it's most probable state. This problems follows the solution of arranging the P= then finding the derivative of the wavefunction and then simplifying it. The final step is to find the zero points, where the 𝒓 is equal to zero which will give the most probable locations of the electron. The practical applications of finding the nodal locations can help with understanding how orbitals work which can help with making molecular orbital diagrams and SALCs that can be used to determine the way atoms and molecules bond. Other applications include understanding the energy levels of the bonds and orbitals to predict possible interactions between molecules and atoms, for research purposes and chemical engineering.


Reference

1) Branson, J. The Radial Wavefunction Solutions. https://quantummechanics.ucsd.edu/ph130a/130_notes/node233.html (accessed November 16, 2021).


By Dmitry Ivanov


Example 10

Question

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Use Planck's radiation law to find the surface temperature of a star with a maximum intensity of EM radiation is emitted at 504 nm. Report your answer in Kelvin.

Getting the correct function

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A blackbody material is defined by its ability to absorb ALL radiation that falls onto it.[1] When the blackbody is at constant temperature, the distribution of its emission frequency can be determined by assuming its only direct relation is to temperature.[1] Furthermore, the frequency of the electromagnetic (EM) radiation can also be measured in units of wavenumber .

Planck's approach to defining was deriving for a closed form Harmonic Oscillator was based on Boltzmann's distribution.[2][3] The resulting form of Planck's law can then be applied to the question now that all parameters are known except for the parameter in question. The model below shows the radiation intensity distribution (i.e., area of the curve) of EM with respect to frequency (in Hertz; Hz) and temperature (in Kelvins; K). Planck's law for a blackbody material predicts the behaviour of its quantitative properties (such as frequency, wavelength, and temperature) when the environment are in extreme conditions.[3]

Planck's Law of BlackBody Radiation

Solving for the temperature

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The maximum value of the independent variable of any function can be found by equating the function's derivative to zero. Only is unknown and is with respect to the independent variable, . Thus, Planck's law will be derived with respect to .

First, for simplicity, let .

The first term consists only of non-zero constants so that leaves:

Subbing back in gives:

At max intensity, . Subbing in all known values and constants (in SI units) then rearranging to solve for T will determine the surface temperature of the Sun under these specific conditions.

Known Values[3][4]
Variable Value Units

Therefore, the surface temperature of the star with a maximum blackbody radiation emission wavelength of 504 nm is just above 5709 Kelvins.

References

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  1. a b Quantum Hypothesis Used for Blackbody Radiation Law https://chem.libretexts.org/@go/page/210776 (accessed Nov 20, 2021).
  2. "Rotations and Vibrations of Polyatomic Molecules", Molecular Physics, Weinheim, Germany: Wiley-VCH Verlag GmbH, pp. 203–236, retrieved 2021-11-20
  3. a b c "1.1: Blackbody Radiation Cannot Be Explained Classically". Chemistry LibreTexts. 2020-03-18. Retrieved 2021-11-20.
  4. "Boltzmann constant | Value, Dimensions, Symbol, & Facts | Britannica". www.britannica.com. Retrieved 2021-11-21.


Example 11

Example 11

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Use the 1D particle in a box model to estimate the wavelength of light required to excite an electron from a pi to pi* MO in ethene.


Solution:

The energy levels of a particle in a 1D box with a specific quantum number , are as follows.

In this equation represents Planck's constant, is the mass of the particle, and is the length of the box.

The pi electron in the double bond between the carbon atoms in ethene can be approximated to the particle in a 1D box model. This means that the mass of the particle in this question will be the mass of an electron, and the length of the box corresponds to the bond length between the carbon atoms in the molecule ethene.

Additionally, the energy equation above needs to be transformed into a equation for since the electron is moving from one energy level to another.

The change in energy between the pi and pi* MO in ethene can now be calculated knowing that the bond length between doubly bonded carbon atoms is 133pm and the mass of an electron is 9.1093856x10-31 kg. Moving from the ground state n=1 to an excited state of n=2 :

Now that the energy required to excite the electron to the pi* orbital is known, the wavelength of light can be calculated through the following equation, where c is the speed of light in a vacuum and is the wavelength of light.

The equation can then be re-arranged to solve for the wavelength of light.

By plugging in the known constants and the value for that has been calculated above, the wavelength can be found.

Therefore the wavelength of light required to excite an electron from a pi to pi* molecular orbital in ethene is 19nm.


Example 12

Write a question and its solution that shows the specific selection rule for a quantum rigid rotor.


Consider an N2 molecule with a bond length of 1.09 Å.

(a) Calculate the energy at the quantum number 3 using the specific selection rule for a rigid rotor.

Solution: This bond length is given (1.09 Å), and the reduced mass (μ) and the inertia (I) must be calculated to determine the energy at the angular momentum quantum number . The bond length and reduced mass must also be changed to SI units.

[1]

Now, calculating the moment of inertia ():

[2]

Since the molecule is linear (N2), the energy level () can be calculated with the linear rigid rotor equation:

[3]


(b) Calculate the quantum number if the transition energy is 2.4176 × 10-22J. In reference to part (a), does this value adhere to the specific selection rule? Why or why not?


The linear rigid rotor equation must be rearranged into linear form to solve for .

To solve this relationship, the quadratic formula must be utilized:

Rigid rotor quantum numbers cannot be negative, ∴ .

This transition adheres to the specific selection rule for a quantum rigid rotor because the change in rotational quantum number is .




(c) Calculate the quantum number if the transition energy is 1.2073 × 10-21J. In reference to part (a), does this value adhere to the specific selection rule? Why or why not?

The linear rigid rotor equation must be rearranged into linear form to solve for .

To solve this relationship, the quadratic formula must be utilized:

Rigid rotor quantum numbers cannot be negative, ∴ .This transition does not adhere to the specific selection rule for a quantum rigid rotor, because the change in rotational quantum number is not within the . Compared to part (a), it has a change of .

References

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  1. https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Exercises%3A_Physical_and_Theoretical_Chemistry/Exercises%3A_Aktins_et_al./12.E%3A_Rotational_and_Vibrational_Spectra_(Exercises)
  2. https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University_Physics_(OpenStax)/Book%3A_University_Physics_I_-_Mechanics_Sound_Oscillations_and_Waves_(OpenStax)/10%3A_Fixed-Axis_Rotation__Introduction/10.06%3A_Calculating_Moments_of_Inertia
  3. Anderson, J.M. Introduction to Quantum Chemistry, 1969, W.A. Benjamin, Inc, pg.91-100.


Example 13

Write a question and its solution that shows the specific selection rule for a quantum harmonic oscillator

Calculate the energy for the vibrational transitions and . If they have the same energy gaps comment on why?

The energy of the vibrational transitions from from and have the same energy gap . This is because of the specific selection rule for the quantum harmonic oscillator. The rule states that the molecule is only allowed to move up or down one vibrational energy level for the transition to occur. If the molecule diverges from the rule then it is considered an overtone, and these are unlikely to occur.


Example 14

Show using calculus the most probable position of a quantum harmonic oscillator in the ground state (n=0)

Question:


What is the most probable position of a quantum harmonic oscillator at the ground state? Calculate this using the probability density equation to find the most probable position at n=0.

Probability distribution



Solution:

The Hermite polynomial at n=0 is:

The normalization factor at n=0 is:

α is a constant and is equal to:

The probability distribution at n=0:

The most probable position is when the maximum probability distribution is:

Applying this partial derivative to the probability distribution gives:

The constants can be taken out of the derivative:

The derivative gives:

Since it is equal to zero the constants can be divided out leaving:

Since all of the parts are multiplied they can be divided out leaving:

The point where the probability distribution is at a maximum for the ground state of n=0 for the quantum harmonic oscillator is 0.


Example 15

Write an example question showing the determination of the bond length of CO using microwave spectroscopy

Example 15:

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Deriving the Required Equations

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When a photon is absorbed by a polar diatomic molecule, such as carbon monoxide, the molecule can be excited rotationally. The energy levels of these excited states are quantized to be evenly spaced energetically. This is illustrated on a microwave spectrum where the interval between the absorption peaks, correlating to the energy transitions, is constant. As long as we know the masses of the elements that comprise the diatomic molecule, we can use a microwave spectrum to determine the bond length. The distance between each rotational absorption line is defined as twice the rotational constant, which can be measured via the following equation:

[1]

h = plank's constant = 6.626 ᛫10−34 J ᛫ s

c = speed of light = 2.998᛫108 m ᛫ s-1

I = moment of inertia


The energy required to rotate a molecule around its axis is the moment of inertia . It can be calculated as the sum of the products of the masses of the component atoms and their distance from the axis of rotation squared:

[2]

Working it out for an heterogeneous diatomic molecule:


The distance from the atom to the center of mass cannot easily be measured; however, by setting the origin at the center of mass, an equation can be derive for the two values that uses the bond length as a variable:

Substituting these equations into the moment of inertia equation:


This equation can be simplified further if we imagine the rigid rotor as a single particle rotating around a fixed point a bond length away. The mass of this particle is the reduced mass of the two atoms that make up the diatomic molecule:

[3]

Simplifying the previous moment of inertia equation, we get:

Solving for Bond Length

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From here we have everything we need to be able to determine the bond length of a polar diatomic molecule such as carbon monoxide.

First, we must solve for the moment of inertia using the rotational constant:


As explained earlier, the rotational constant can be determined by measuring the distance between the rotational absorption lines and halfling it. In the case of the rotational constant is m-1 [4]. Plugging this value in, we can determine the moment of inertia:

kg ᛫ m2


Now that we know the moment of inertia, we can rearrange the equation we derived earlier in order to determine the bond length:


The exact atomic mass of is 12.011 amu and is 15.9994 amu [5]. As such, the reduced mass is calculated to be:

amu

amu ᛫


Plugging in the reduced mass back into our equation, we can finally solve for the bond length of a carbon monoxide molecule:

References

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  1. https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Exercises%3A_Physical_and_Theoretical_Chemistry/Exercises%3A_Aktins_et_al./12.E%3A_Rotational_and_Vibrational_Spectra_(Exercises)
  2. https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University_Physics_(OpenStax)/Book%3A_University_Physics_I_-_Mechanics_Sound_Oscillations_and_Waves_(OpenStax)/10%3A_Fixed-Axis_Rotation__Introduction/10.06%3A_Calculating_Moments_of_Inertia
  3. https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Exercises%3A_Physical_and_Theoretical_Chemistry/Exercises%3A_Aktins_et_al./12.E%3A_Rotational_and_Vibrational_Spectra_(Exercises)
  4. https://webbook.nist.gov/cgi/cbook.cgi?ID=C630080&Mask=1000
  5. https://www.angelo.edu/faculty/kboudrea/periodic/structure_mass.htm


Example 16

Write an example question showing the calculation of the frequency of EM radiation emitted when a HCl molecule transitions from the J=1–>0 rotational state.


Example 16: Giving the bond length re=1.27Å, find the frequency of the EM radiation emitted when a HCl molecule transitions from the J=1→0 rotational state

Given the reduced mass (μ) and inertia (I), the energy can be found and subsequently be used to find frequency of the transition.

Note that for rotational states for J=1→0, ΔE is not required as the transition states are in the ground state.

Inertia can be found by using reduced mass of the molecule and their bond length(re = 1.27Å):

where reduced mass can be calculated by the following:

, where and

Convert to SI units:

, thus

Note that the bond length must also be in SI units,

1Å= 1×10-10m

Bond length = 1.27×10-10m


Inertia can now be calculated:


After calculating the inertia, energy can be found:

and , where h is the Planck's constant.

Frequency of the transmission can therefore be found:


Example 17

The force constant of HCl is determined computationally to be 480 N/m. Given this information find the frequency of EM radiation required to excite the HCl molecule from its ground state to its first excited state.

Solution

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Given the bond constant (k) of HCl , we use the relationship between fundamental frequency and bond constant to find the bond constant.

where is the bond constant.

is the reduced mass of HCl.

To find the reduced mass of HCl the masses of H and Cl are multiplied and divided by the sum of the masses.

For HCl the reduced mass is calculated as

convert to the SI unit of Kg

To find the fundamental frequency

After finding the fundamental frequency, the Energy at different quantum levels can be found by

For the ground state i.e.

For the first excited state i.e.

The difference in energy between the two states is

and Energy is defined as Planck's constant multiplied by frequency


Example 18

Write an example question showing the determination of the force constant of CO using IR spectroscopy.

Question

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Using IR spectroscopy, calculate the force constant of carbon monoxide (CO) and report the value in kN/m.

Solution

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To solve this question, the IR spectrum of CO is required. The website, webbook.nist.gov/chemistry/ is a database that is available to the public for access to IR spectra. If you are unfamiliar on acquiring an IR spectrum, follow these steps: with the nist website open, go to Search Option -> Name -> Enter: Carbon Monoxide -> Select: IR Spectrum -> Change Transmittance to Absorbance -> Finally, take a screen shot.

The CO IR spectrum earlier acquired is a low resolution rotational-vibrational spectrum. In IR spectrum's, the fundamental frequency can be determined by utilizing the P-branch, Q-branch, and R-branch. Remember that the Q-branch is pure vibrational, which is forbidden, therefore the peak does not exist and is located between the P-branch and R-branch.

Using the website, apps.automeris.io/wpd/ will allow for an accurate collection of data points from the image of the CO IR spectrum taken earlier. For low resolution spectrums, locate the R-branch and P-branch, use the peak maxima as the data point. If you are unfamiliar on acquiring the data points from a image, follow these steps: Open the website -> Load Image -> Choose File -> Open -> 2D(X-Y) Plot -> Align Axes -> Plot Known -> Complete -> Insert Known Values -> Ok -> Add Points -> View Data.


Equation 1: Using the data points from the image, the fundamental frequency can be determined by the following relationship.

Where,

is the fundamental frequency

is the highest wavenumber

is the lowest wavenumber


Equation 2: The following equation is used to relate the fundamental frequency to the force constant.

Where,

is the fundamental frequency

is the speed of light

is the force constant

is the reduced mass


Equation 3: The reduced mass equation is required for relating the fundamental frequency to the force constant. Remember to use exact masses, and since the isotopes were not specified, that means we assume the most abundant version is to be used.

For oxygen:

For carbon:

Where,

is the reduced mass

is the exact mass 1

is the exact mass 2


Equation 4: Rearrange equation 2 to solve for the force constant.


The fundamental frequency of CO from IR spectra is determined using equation 1:


The reduced mass of CO is calculated using equation 3:

The force constant of CO is calculated using equation 4:

Answer: Using IR spectroscopy, the force constant of carbon monoxide was determined to be 1.87 kN/m.


Example 19

==Write an example question showing the calculation of the frequency of EM radiation required to excite an H atom from ground to n=2 electronic state.==
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The energy of the hydrogen atom is calculated using the following equation:

'''Energy of Hydrogen Atom'''

To calculate the change in energy required to go from the ground state to n=2, the following equation is used:

δE =((9.10938x10)−31kg(1.60218x10−19c)432π28.854x1012c2S2/kgm2)2(1.054x10−24J.s)2)54


Example 20

An electronic transition of H atom from an energy level to the ground state is observed with a corresponding wavelength of 102.5 nm. Determine the initial state of the electron from which transition has occurred.

The Rydberg's phenomenological equation will be used to solve the energy transition problem

Rydberg's phenomenological equation for hydrogen

where the is Rydberg constant for hydrogen and is equal to 109737 . is the final energy level, is the initial energy level for hydrogen transition. Both and are integers and .


The wavelength is given as 103nm

Concert to SI units:


The H atom's transition is from unknown energy level to the ground state.

So we can know the final state is the ground state which means


The now can be calculated:


Because the energy level always

Therefore, the initial state of the electron from which transition has occurred is energy level 3.


Example 21

Write a question and its solution that shows the calculation of the vibrational force constant of H19F. Show what the force constant would have to be for HF to absorb visible red light instead of IR light.

If H19F has a vibrational fundamental wavenumber of 4141.3 cm, What is the bond force constant? What would the force constant need to be to absorb visible red light instead of IR light? H has a n exact mass of 1.00782 u, and 19F has an exact mass of 18.998403 u.

Solution:

First, calculate the reduced mass for H19F.

Now, solve for the force constant:

Therefore, the bond force constant is 967 N m-1.

Red light is around 700 nm, or 700×10-9 m. Converted to wavenumber, that is 14285.71 cm-1. Substituting this in,

Therefore, the bond force constant would have to be 11506 N m-1 to absorb visible light.


Example 22

Make a table showing all the absorption transitions for the first 5 rotational states of HCl. Include a column that shows which are allowed and which are forbidden

The rotational energy for any molecule is all kinetic and comes from the angular momentum of the molecule. To get the rotational energy of any one molecule the moment of inertia first needs to be derived (inertia units:(kg·m2). For a diatomic molecule, the inertia is dependent on the equilibrium bond length and the mass of the two atoms in the form of reduced mass. Inertia for a diatomic:

Moment of Inertia (Diatomic)

,


Where is the moment of inertia, is the reduced mass, and are the masses of the two atoms, and is the equilibrium distance between the two atoms.

The rotational energy levels of a molecule is related to the reduced Planck's constant, the inertia of the molecule, and the J quantum number.

Energy Levels for Rigid Rotor

Where  is the energy of the energy state of the molecule,  is the inertia,  is the reduced planck's constant, and J is the second quantum number of the rigid rotor.  

The Energy of a photon is related to the wavelength, the speed of light and Planck's constant. The energy between two rotational energy states of a molecule is equal to the energy of the photon absorbed to get to the higher energy state. Therefore the wavelength of a photon can be determined from the difference in rotational energy states of a molecule.

Energy of a Photon

Where is the energy of the photon, is Planck's constant, is the speed of light and is the wavelength of the photon.

Table 1. First five rotational energy states of HCl and the associated energy. and wavelengths of light associated with them. .

Rotational Absorption

Transition States

J → J'

Allowed/Forbidden Energy of Transitions (J) Wavelength of Light Absorbed (μm)
0 → 1 Allowed 4.75 418.5
0 → 2 Forbidden 1.42 139.5
0 → 3 Forbidden 2.85 69.8
0 → 4 Forbidden 4.75 41.9
0 → 4+n Forbidden
1 → 2 Allowed 0.95 209.3
1 → 3 Forbidden 2.37 83.7
1 → 4 Forbidden 4.27 46.5
1 → 4+n Forbidden
2 → 3 Allowed 1.42 139.5
2 → 4 Forbidden 3.32 59.8
2 → 4+n Forbidden
3 → 4 Allowed 1.90 104.6
3 → 4+n Forbidden
4 → 5 Allowed 2.37 83.7
4 → 5+n Forbidden

Example 22

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What is the wavelength of light in μm needed to excite an HCl molecule in its second rotational excited state to its third rotational excited state if it has an equilibrium bond length of 1.200 Å?

Solution:

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Divide by Avogadro constant and convert to SI units:



Convert Units:


Example 23

Create a table showing all the possible absorption transitions between the first 5 vibrational states of HCl. Include columns for the energies and frequencies. The last column should indicate if the transition is allowed or forbidden.

The simplified diatomic quantum harmonic oscillator models the vibration of a chemical bond as the displacement of a reduced mass on a spring fixed to an origin point. The displacement of the reduced mass is measured with respect to the equilibrium bond length (x=0).

Solution

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The vibrational frequency of a linear diatomic molecule depends on its reduced mass and vibrational spring constant (k). The spring constant for 1H35Cl was measured and reported as 515.825 N/m.[1]

The reduced mass was calculated as follows:

The reduced mass was converted to SI units as follows:

The fundamental vibrational frequency was calculated as follows:

The fundamental vibrational frequency was converted to wavenumbers as follows:

Vibrational energy levels for the quantum harmonic oscillator (where h is Planck's constant, and ω is frequency).

The energy for a given vibrational state can be calculated as follows:

For example, the energy of the vibrational ground state:

The transition energy between the ground state and the first excited vibrational state was calculated as follows:


According to the specific selection rule, vibrational transitions are only allowed between adjacent energy levels (where ). Consequently, due to the quantization of energy levels, vibrational transitions allowed by the specific selection rule are equivalent in energy and frequency, such that . Conversely, overtone transitions are those for which . As such, overtone frequencies can be calculated as integer multiples of the fundamental frequency. For example, the vibrational frequency for the transition between the ground state and the second excited vibrational state was calculated as follows:

Similarly, the wavenumber for the transition between the ground state and the third excited vibrational state was calculated as follows:

Finally, the energy for the overtone transition between the first and fourth excited vibrational states was calculated as follows:

However, because , this transition is forbidden .

Frequency, wavenumber, and energy values for transitions between the first five vibrational energy levels of 1H35Cl.
Transition Frequency (s-1) Wavenumber (cm-1) Energy (J) Allowed/Forbidden
1←0 8.96237·1013 2989.52 5.93853·10-20 Allowed
2←0 1.79247·1014 5979.05 2.37541·10-19 Forbidden
2←1 8.96237·1013 2989.52 5.93853·10-20 Allowed
3←0 2.68871·1014 8968.57 5.34468·10-19 Forbidden
3←1 1.79247·1014 5979.05 2.37541·10-19 Forbidden
3←2 8.96237·1013 2989.52 5.93853·10-20 Allowed
4←0 3.58495·1014 11958.1 1.18771·10-18 Forbidden
4←1 2.68871·1014 8968.57 5.34468·10-19 Forbidden
4←2 1.79247·1014 5979.05 2.37541·10-19 Forbidden
4←3 8.96237·1013 2989.52 5.93853·10-20 Allowed
5←0 4.48119·1014 14947.6 1.48463·10-18 Forbidden
5←1 3.58495·1014 11958.1 1.18771·10-18 Forbidden
5←2 2.68871·1014 8968.57 5.34468·10-19 Forbidden
5←3 1.79247·1014 5979.05 2.37541·10-19 Forbidden
5←4 8.96237·1013 2989.52 5.93853·10-20 Allowed

References:


Example 24

Write a question and its solution showing the probability of a particle being located in the bottom left quadrant of the ground state of the particle in a square.

Background Information

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Particle In a Square (2D)

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The particle in a square model is where a particle in confined to a restricted

region inside a 2D box. For this model we consider a particle which is free to move

anywhere inside the x,y dimensions but cannot exist outside of the 2D box where it would

have infinite potential energy.

The conditions for the potential energy are meaning that the particle spends 100% of its time within the walls of the box.

Since the particle in a square is a 2D model the wave function must take into account the x and y direction ;

The wave function for a particle in a square

The states the quantum system is in are defined by the quantum numbers

In quantum energy levels only discrete energy levels are possible. In this model the particle cannot be motionless meaning the lowest possible energy level must be non-zero (n=1). Since the particle is always moving even in the ground state the particle has a a non- zero energy level. Therefore the ground state of a particle in a box is . In quantum mechanical systems there is a presence of nodes where in states above the ground state for the 2D model there are nodal lines that divide the square. As the quantum numbers increase so does the amount of nodal lines as . The sign of the wavefunction changes between the nodes and has a value of zero on the nodes.

Probability Distributions

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The probability (P) of a particle being found in an interval can be found by integrating the probability distribution (P(x))

The probability of finding a particle in the range , is given by;

The probability distribution of a a particle is the wave function squared.

For one particle in 2D (particle in a box):

,where * is the complex conjugate of the wave-function

Therefore the probability of finding a particle over a range can be re-written as:

Example

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Question: For a particle in a square that is length L, what is the probability of finding the particle in the bottom left quadrant if in the ground state?

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Solution
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If the particle is in the bottom left quadrant it is in the interval;

Therefore the probability density is:

The wave-function for a particle in a square is:

In the ground state , therefore the probability distribution can be written as:

Then solving the integral,

From Table of Integrals


Since is zero the whole term goes to zero

Therefore the the probability of finding a particle in the bottom left quadrant of a square in the ground state is 0.25 of 25%.


Example 25

Introduction

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Visual representation for the particle in a one-dimensional box model.

The particle in a one-dimensional box model describes the movement of a particle in one dimension. It is confined to a "box" where it can only move horizontally along the x-axis within the confines of the box. The particle cannot escape the box due to the areas outside of the box having an infinite potential energy, thus trapping the particle inside an infinitely deep well. If the leftmost boundary of the box is position 0 and the rightmost position is position L, the potential energy of the particle inside the box is zero between position 0 and L. This is denoted mathematically as where is potential energy as a function of position. Outside the box, the potential energy is for and . The problem below proposes a situation where the potential energy is zero everywhere along the x-axis, meaning the particle is no longer bound by areas of infinite potential energy like in the 1D box model. Rather, across the interval .

Example Problem and Solution

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Problem

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Suppose an electron is free to move along the x-axis and its potential energy everywhere is zero. Derive a wavefunction for this system.

Solution

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Schrödinger Equation and Definition of the Hamiltonian

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First the Hamiltonian operator must be defined in order to solve the Schrödinger equation for this system.

Time-independent Schrödinger equation

The Hamiltonian is simply the total energy of the system. It is expressed as the sum of the total kinetic energy () and total potential energy () of the system. Therefore the Hamiltonian is defined as,

However, the problem states that the potential energy everywhere is zero, so the Hamiltonian is simply equal to the total kinetic energy of the system.

Definition of the Kinetic Energy Operator and Substitution into the Schrödinger Equation

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In one dimension, the kinetic energy operator is defined as,

So the Hamiltonian becomes,

Where is the mass of an electron and is the reduced Planck's constant. This can then be substituted into the Schrödinger equation to give,

The above is simplified by multiplying both sides of the equation by and then dividing both sides by .

Finding a General Solution and Solving the Differential Equation

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is a linear homogenous second order differential equation where , so,

Since the potential energy everywhere is zero, the electron can exist anywhere on the x-axis. The position of the particle exists across the interval , therefore the probability of finding the electron at any position is infinitely small and approaches zero.

Thus it can be assumed that for both terms of the general solution due to the absence of any boundary conditions. Putting into the general solution yields,

Therefore, the wavefunction collapses and is zero if the potential energy is zero everywhere along the x-axis, due to the probability of finding the electron at any position being infinitely small.


Example 26

The square of the angular momentum of a hydrogen atom is measured to be .  What are the possible values of the z-component of the orbital angular momentum, , that could be measured for this atom?

Solution

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The eigenvalues of the square of the angular momentum operator (2) for a quantum mechanical system, such as an electron in a hydrogen atom, are given by:



where are the spherical harmonics, which are eigenfunctions of 2, is the orbital quantum number, and is the magnetic quantum number. The given value for = 20 ħ2 , so we set up the equation:



Dividing by and simplifying, we get:



In this quadratic equation, can be factored to get:



Since must be a non-negative integer. The magnetic quantum number can take on any integer value from to , thus for , can be:



The z-component of the angular momentum, , is quantized in units of and given by:


.


Therefore, the possible values of the z-component of the orbital angular momentum, , that could be measured for the atom with a given are .


Example 27

Question

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Find the wavenumber (in cm-1) of the photon required to excite a CO molecule from its ground vibrational state to its first excited vibrational state, given that the force constant for the CO bond is k=1860 N.m-1.

Solution

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The frequency of the photon required for this transition corresponds to the fundamental vibrational frequency of the molecule, . To determine the wavenumber of this transition, the first step is to calculate the reduced mass of the molecule. The reduced mass () for a diatomic molecule is given by the following equation:

The obtained atomic masses of Carbon and Oxygen are as below:

C= 12.011u, O =15.999u

Plugging these values into the reduced mass equation gives:

The unit conversion from atomic mass units (u) to kilograms (kg) is as follows:

Now, we can calculate the frequency of the photon using the following equation:

Plugging in the values for the reduced mass and force constant gives:

Finally, we need to covert the frequency of the photon () in s-1 to the wavenumber (ṽ) in cm-1:

By substituting the known values, the wavenumber of the photon can be determined as follows::


Example 28

Question

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For an electron in a 3D cube with a length of 6.00 Å, what is:

(a) the energy of its state?
(b) the degeneracy of this energy state?
(c) the total number of nodal planes for a particle in the , , and ?

Solution

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(a) The mass of an electron must be known to determine the energy of the particle. To determine the energy state of an electron in a 3D box, the equation that relates the energy levels of a particle in a cube must be used with every variable in its SI units. The mass of an electron is known to be . Planck's constant is known to be . The length of the box is since one angstrom is known to be . The energy of the electron in the cube can be calculated as follows,

(b) Degeneracy is defined as an energy state with an identical energy to that of a particle with a different set of quantum numbers. In this question, the energy of degenerate states must equal the following,

To determine the degeneracy of a system, the simplest method is to use a matrix to ensure a systematic process is followed. In this case, the sum of the square of all quantum numbers must equal 50. Therefore, the matrix is as follows,

4 5 3 4² + 5² + 3² = 50
4 3 5 4² + 3² + 5² = 50
5 3 4 5² + 3² + 4² = 50
5 4 3 5² + 4² + 3² = 50
3 4 5 3² + 4² + 5² = 50
3 5 4 3² + 5² + 4² = 50

This results in the identification of 6 different sets of quantum numbers, and thus, this energy state is said to be 6-fold degenerate.

(c) To determine the number of nodal planes, each axis must be considered individually. Mathematically, to calculate the number of nodal planes associated with the x-axis, the following equation is used,

Therefore, in this example, the number of nodal planes in x-axis is,

Similarly, for the y-axis and the z-axis, the number of nodal planes is,

Therefore, the total number of nodal planes in this system is 9. This was determined by summing the number of nodal planes associated with each individual axis.


Example 29

Quantum Mechanical Harmonic Oscillator Wavefunction

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A system undergoing harmonic motion around an equilibrium is known as a harmonic oscillator.

In quantum chemistry, the harmonic oscillator refers to a simplified model often used to describe how a diatomic molecule vibrates. This is because it behaves like two masses on a spring with a potential energy that depends on the displacement from the equilibrium, but the energy levels are quantized and equally spaced. The potential energy is non-zero and can theoretically range from .

The wavefunction for the quantum harmonic oscillator is given by the Hermite polynomials multiplied by the Gaussian function. The general form of the wavefunction is:

With:

The first four Hermite polynomials are:

Normalization of a Wavefunction

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The probability of finding the particle in any state is given by the square of the wavefunction. Therefore normalizing a wavefunction in quantum mechanics means ensuring that the total probability of finding a particle in all possible positions is equal to 1. A normalized wavefunction is one that satisfies the normalization condition:

Normalization is important because it ensures that the probability of finding the particle somewhere in space is 100%.

Example

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Show the derivation of the normalization factor of the v=1 state of the harmonic oscillator beginning from the unnormalized wavefunction.

Sub in v=1

Check

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Prove that is normalized (using notation from class)

, where


Example 30

Energy in the particle in a 1D box model

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Energy in quantum mechanical systems is quantized, unlike in classically mechanical systems, where the energy can take any value. The energy of a quantum mechanical system may only hold specific discrete values, which are known as energy levels. Consider the model for the 1D particle in a box. In this model, the possible discrete energy levels are given by the relation,

Energy levels of a particle in a 1D box

The ground state of any quantum mechanical system is the state with the lowest possible energy (otherwise known as the zero-point energy of the system). Defined in the equation for the energy levels of a particle in a 1D box, the quantum number can hold any integer value equal to or greater than one and as such, the ground state of this model occurs when . All energy levels above the ground state are referred to as excited states. There are an infinite number of these states and for the particle in a 1D box, are defined by the quantum number , like the ground state. The first excited state is the energy level one above the ground state, in which , the second excited state is the energy level one above the first excited state, where , and so forth.

Particle transitions between states

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A particle in a 1D box can transition between energy levels by the means of three different processes, whose probability can be measured by Einstein coefficients. The transitional processes involve electromagnetic radiation in the form of photons. They are the following:

  • Spontaneous emission: The spontaneous decay of a particle from a higher energy state to a lower energy state, emitting a photon of the transition energy.
  • Stimulated emission: The decay of a particle from a higher energy state to a lower energy state stimulated by the presence of a photon of the same energy as the transition.
  • Photon absorption: The promotion of a particle from a lower energy state to a higher energy state caused by the absorption of a photon of the transition energy.

The energy of a photon can be determined by the relationship between a photon's frequency of oscillation and energy, known as the Planck relation,

Planck's relation

In quantum chemistry, it is often more convenient to express frequency in terms of wavenumber (), which has units of compared to . There is a simple conversion as frequency and wavenumber are related to the speed of light by,

As such, the Planck relation can be rewritten as,

Planck's relation (wavenumber)

This relation allows the calculation of the energy of a photon involved in one of the transitions between states, as the energy of the transition will be equal to the energy of the photon, whether it is the emitted photon in spontaneous emission, the stimulating photon in stimulated emission or the absorbed photon in the absorption transition.

Example

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What is the wavenumber () of electromagnetic radiation emitted when a proton in a one-dimensional box with length 472 nm decays from the fifth excited state to the fourth excited state?

SOLUTION: This transition is a spontaneous emission where the energy of the emitted photon can be determined by the difference between the energy levels of the transition,

As this transition occurs between the fourth and fifth excited states, and will be 5 and 6, respectively. From the energy levels of a particle in a 1D box,

Substituting these values into the above equation gives,

Using the wavenumber variation of Planck's relation, the wavenumber of a photon of this exact energy can be determined and expressed in convenient units of .


Example 31

Write an example problem related to the conversion between wavelength, frequency, energy and wavenumber of electromagnetic radiation. Show each step in detail and explain the conversion.

Example: Carbon monoxide shows a sharp peak in its IR spectrum at 2143 cm-1

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a) Calculate the frequency and wavelength of the molecule's vibration.

b) Calculate the energy of carbon monoxide at its ground state.

c) Calculate the wavelength of a photon that excites an electron from the ground state up 2 levels.

Solutions

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a) The example provides the wavenumber as an IR spectrum in a plot of wavenumber vs intensity. In this example the wavenumber is 2143 cm-1. This can be converted to frequency using the following equation:

where is the frequency, is the wavenumber and is the speed of light. The speed of light is a constant of 2.9979248x1010 cm/s.

This value then must be converted to the SI units for frequency. One Hz is equal to one s-1, so the conversion is simple by adding the corresponding prefix for

Next in order to determine the wavelength of the vibration we can use the equation:

where is the wavelength.

This equation can be rearranged to solve for wavelength:

This should be converted to the SI units of wavelength which is nm.

b) To find the energy of a molecule at its ground state we can use the equation:

where is the energy, is Planck's constant , is the frequency of the ground state, and is the quantum number of the energy level.

We can use this equation since carbon monoxide's vibrations act as a harmonic oscillator. As this is the ground state the quantum number is equal to 0.

c) We can use the same equation used in b) to first solve the change in energy caused by the photon.In this case, the quantum number is equal to 2.

Now that we have found the change in energy we can calculate the wavelength of the photon using the equation:

Rearrange the equation to solve for λ:

Convert the wavelength to nm for proper SI units

Conclusion

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After working thought part a-c you should be able to see how the conversion between wavelength, frequency, energy and wavenumber of electromagnetic radiation occurs.


Example 32

The Wavefunction of a Particle in a One-Dimensional Box

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Despite the wavefunction having very little physical interpretation, the solution to the wavefunction of a particle is beneficial in order to determine the probability of such particle existing in a given area. When deriving the equation for the wavefunction of a particle in a one-dimensional box, we first begin by defining the Hamiltonian operators (i.e., kinetic and potential energy) for the given system, and substituting these values into the Schrodinger equation.

In quantum mechanics, the Schrodinger equation is given as,

The Schrodinger Equation


where is the Hamiltonian operator, is the kinetic energy operator, is the potential energy operator, is the wavefuntion, and is the total energy of the system. The first equation is a simplification of the second equation, since . Mathematically, the wavefunction is an eigenfunction of the Hamiltonian operator with an eigenvalue of .

Defining the Hamiltonian

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For the case of a particle in a one-dimensional box, its potential energy is equal to zero within the constraints of such box (i.e., ), but the potential energy of the particle is infinite outside of these constraints:

Potential Energy Range of Particle in 1D Box

Additionally, the kinetic energy operator in the dimension is given as . Therefore, since inside of the box, the values defined for and can be substituted into the Schrodinger equation, which simplifies to .

Boundary Conditions of the Wavefunction

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The boundary conditions must then be identified for a particle in a one-dimensional box. The constraints of the box have been previously defined as the position of the particle, , being contained within a box of length . This means that the particle cannot exist when or when . The boundary conditions for the value of the wavefunction must therefore be and .

General Solution

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This is a second-order linear homogenous equation with respect to , and has a general solution of

, where

Thus, the solution for the given second-order linear homogenous equation is

To determine the specific solution for this wavefunction, the constants and must be found.

Determining the Unknown Variables

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By substituting the first boundary condition that into the general solution of the wavefunction, the value of may be found.

We now have . The second boundary condition, , may be applied.

The value of A cannot be equal to zero, otherwise the value of the wavefunction would be equal to zero at any position of . So, .

The sine function is periodic, therefore, there are multiple values of at which . The values of x in which this is true are integer multiples of . Thus, . This can be further simplified by noting that the sine function is symmetric, therefore, the negative value of some will have the same wavefunction as its positive value.

Additionally, substituting into our above equation would result in a value of 0 for the wavefunction at every position.

Therefore, .

We can then plug back into our wavefunction equation, which gives

The second property of a wavefunction is that it is normalized, meaning that the integral of the square of the wavefunction (i.e., the probability distribution) over all space must be equal to 1. We can now apply this to the above function to determine the constant value of .

The tabulated solution to this integral gives a constant value for as

Therefore, the finalized specific solution of the wavefunction for a particle in a one-dimensional box is as follows.

, where

Example

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The wavefunction of an electrion moving along a platinum wire with length is given by , where is the position of the electron. Outside the boundaries of this wire, the value of the wavefunction is zero (). Determine the equation of the wavefunction of this electron by calculating , the normalization constant, in terms of . Is this normalized wavefunction a valid wavefunction? Explain why or why not.

As previously mentioned, one of the properties of a wavefunction is that it must be normalized. When normalizing the wavefunction, a normalization constant appears. Therefore, to determine the normalization constant for this wavefunction, we must integrate over all space and set it equal to 1.

In this case, the length of the wire is an arbitrary constant . Thus, . Additionally, the equation for the wavefunction is , which can be substituted in for .

To ensure that this is a valid wavefunction, one of two properties have been satisfied: the function is normalized. We must now ensure that the boundary conditions are valid as well (i.e., and ).

The first boundary condition is valid.

The value of the wavefunction at will never be equal to zero, since the length of the wire is always a nonzero value. Thus, because this wavefunction disobeys the second property of and , it is not considered a valid wavefunction.


Example 33

Derive the wave function of a free electron.

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A free, or unbound, electron is a particle that has no forces acting on it. In other words, it travels in a region of uniform potential. Thus, a free electron travelling in either direction along the x-axis has only kinetic energy, and no potential energy.

since

then,

The wavefunction describes a particle's behaviour as a wave in terms of its position as a function of time. Since electrons move very quickly, a time-averaged distribution of electron density is observed. Therefore, the wave function of a free electron can be derived using Schrödinger's time-independent wave equation.

Recall the definition of the Hamiltonian operator.

This definition can be substituted into the time-independent Schrödinger wave equation.

Recall also the definition of the kinetic energy operator.

Now, the definitions of the kinetic energy and potential energy operators can be substituted into the time-independent Schrödinger wave equation.

This differential equation can be rearranged to isolate for the second derivative on the left-hand side. The result is the Schrödinger equation of a free electron.

Recall the definition of the angular wavenumber.

This definition can be substituted into the Schrödinger equation of a free electron.

The second derivative in the equation above requires solving a second order differential equation for the wavefunction.

This linear second order differential equation can be solved by separating it into two first-order differential equations.

The above holds true if either

or

Now, the first-order differential equations can be solved simultaneously.

To eliminate the derivative, integrate both sides of the equation.

To eliminate the natural logarithm, exponentiate both sides of the equation.

Set the term equal to a constant, .

This is the wave function of a free electron, where the sign represents the direction in which the wave is propagating, since the free electron does not have defined boundary conditions.


  1. Feller, S. E.; Blaich, C. F. Error Estimates for Fitted Parameters: Application to HCl/DCl Vibrational/Rotational Spectroscopy, J. Chem. Educ., 2001, 78 (3), 409. DOI: 10.1021/ed078p409