Classical mechanics is divided into two branches, Lagrangian mechanics and Hamiltonian mechanics. In Lagrangian mechanics, a system having
degrees of freedom
is described by a function
of the degrees of fredom and their temporal derivatives. The function
is called the Lagrangian of the system. The equations of motion of the system are given by Hamilton's principle, stating that the degrees of freedom change in such a way that the integral
![{\displaystyle S[t_{1},t_{2}]=\int _{t_{1}}^{t_{2}}L(q,{\dot {q}},t)\mathrm {d} t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4a0c8c494bcb1cd088da24d1640bf6d4d13cf36)
is at an extremum with respect to the path. This is a problem in variational calculus which we will not discuss here. It's solution is
![{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial L}{\partial {\dot {q_{i}}}}}\right)-{\frac {\partial L}{\partial q_{i}}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/097252c9e8610a744628052850b2952fec72b8f0)
for each individual index
. These equations are called the Euler-Lagrange equations. Thus we obtain
second-order partial differential equations describing the system. This gives us
initial conditions which determine the evolution of the system. However, the Lagrangian formalism is not suited for quantum mechanics. We need the other formalism, Hamiltonian mechanics. The Hamiltonian formalism is based on the following fact. Assume that the degree of freedom
does not appear in
for some
. Then
, so we get
![{\displaystyle {\frac {\partial L}{\partial {\dot {q_{i}}}}}=k_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b8fdcfafc88a80f77ddebcfd6712f53dddff49d)
where
is a constant. In other words, we get a conserved quantity. The hamiltonian formalism is based on replacing
in
by
for all
. We can do this by performing a Legendre transformation on
.
is called the canonical or conjugate momentum associated with
. We define
.
Solving
for
and inserting, we get the Hamiltonian function
. The equations of motion can be found from the Euler-Lagrange equations. We get
,
,
.
These are called Hamilton's equations. We get
first-order equation describing our system, again giving us
initial conditions, as expected.
Define the Poisson bracket as the expression
.
It is readily checked that
![{\displaystyle \{q_{i},p_{j}\}=\delta _{ij}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25d5dac794ae71e0803cd28e64a3ec6073c32730)
where
if
and
otherwise.
We can also obtain a useful expression about the time evolution of arbitrary quatities. Let
be any (differentiable) quantity. We then have
![{\displaystyle {\frac {\mathrm {d} F}{\mathrm {d} t}}={\frac {\partial F}{\partial t}}+\sum _{i=1}^{n}{\frac {\partial F}{\partial q_{i}}}{\dot {q_{i}}}+{\frac {\partial F}{\partial p_{i}}}{\dot {p_{i}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f8b20b58d5025c76684218559a4a4bcf047dbdf)
by the chain rule. Insering for the time derivatives, we get
.
The poisson brackets have an important counterpart in quantum mechanics, and are used as a starting point for the theory.
In quantum mechanics is based on the following postulates:
- 1. To each state of a physical system there corresponds a state vector
in a complex Hilbert space. The state vector has length 1, meaning
, and its time evoltution satisfies the Schrödinger equation
![{\displaystyle i\hbar \partial _{t}|\Psi (t)\rangle ={\hat {H}}|\Psi (t)\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/900a466c5f90454233cf4d2fa3ff14cf16a420cc)
- where
is the Hamiltonian operator of the system. We will get back to determining
for a given system.
- 2. To each physical observable
there corresponds a linear operator
on the Hilbert space. The operators
and
for the generalized coordinates and momenta satisfy
.
- 3. The expectation value of an observable
is
.
- 4. The only possible results when measuring the observable
are the eigenvalues
of
.