Editing Hamiltonian (quantum mechanics) (section)

==Hamilton's equations==

[[William Rowan Hamilton|Hamilton]]'s equations in classical [[Hamiltonian mechanics]] have a direct analogy in quantum mechanics. Suppose we have a set of basis states <math>\left\{\left| n \right\rangle\right\}</math>, which need not necessarily be eigenstates of the energy. For simplicity, we assume that they are discrete, and that they are orthonormal, i.e.,

<math display="block"> \langle n' | n \rangle = \delta_{nn'}</math>

Note that these basis states are assumed to be independent of time. We will assume that the Hamiltonian is also independent of time.

The instantaneous state of the system at time <math>t</math>, <math>\left| \psi\left(t\right) \right\rangle</math>, can be expanded in terms of these basis states:

<math display="block"> |\psi (t)\rangle = \sum_{n} a_n(t) |n\rangle </math>

where

<math display="block"> a_n(t) = \langle n | \psi(t) \rangle. </math>

The coefficients <math>a_n(t)</math> are [[Complex number|complex]] variables. We can treat them as coordinates which specify the state of the system, like the position and momentum coordinates which specify a classical system. Like classical coordinates, they are generally not constant in time, and their time dependence gives rise to the time dependence of the system as a whole.

The expectation value of the Hamiltonian of this state, which is also the mean energy, is

<math display="block"> \langle H(t) \rangle \mathrel\stackrel{\mathrm{def}}{=} \langle\psi(t)|H|\psi(t)\rangle
= \sum_{nn'} a_{n'}^* a_n \langle n'|H|n \rangle </math>

where the last step was obtained by expanding <math>\left| \psi\left(t\right) \right\rangle</math> in terms of the basis states.

Each <math>a_n(t)</math> actually corresponds to ''two'' independent degrees of freedom, since the variable has a real part and an imaginary part. We now perform the following trick: instead of using the real and imaginary parts as the independent variables, we use <math>a_n(t)</math> and its [[complex conjugate]] <math>a_n^*(t)</math>. With this choice of independent variables, we can calculate the [[partial derivative]]

<math display="block">\frac{\partial \langle H \rangle}{\partial a_{n'}^{*}}
= \sum_{n} a_n \langle n'|H|n \rangle
= \langle n'|H|\psi\rangle
</math>

By applying [[Schrödinger's equation]] and using the orthonormality of the basis states, this further reduces to

<math display="block">\frac{\partial \langle H \rangle}{\partial a_{n'}^{*}}
= i \hbar \frac{\partial a_{n'}}{\partial t} </math>

Similarly, one can show that

<math display="block"> \frac{\partial \langle H \rangle}{\partial a_n}
= - i \hbar \frac{\partial a_{n}^{*}}{\partial t} </math>

If we define "conjugate momentum" variables <math>\pi_n</math> by

<math display="block"> \pi_{n}(t) = i \hbar a_n^*(t) </math>

then the above equations become

<math display="block"> \frac{\partial \langle H \rangle}{\partial \pi_n} = \frac{\partial a_n}{\partial t},\quad \frac{\partial \langle H \rangle}{\partial a_n} = - \frac{\partial \pi_n}{\partial t} </math>

which is precisely the form of Hamilton's equations, with the <math>a_n</math>s as the generalized coordinates, the <math>\pi_n</math>s as the conjugate momenta, and <math>\langle H\rangle</math> taking the place of the classical Hamiltonian.