Hamiltonian (quantum Mechanics) - Hamilton's Equations

Hamilton's Equations

Hamilton's equations in classical Hamiltonian mechanics have a direct analogy in quantum mechanics. Suppose we have a set of basis states, which need not necessarily be eigenstates of the energy. For simplicity, we assume that they are discrete, and that they are orthonormal, i.e.,

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 t, can be expanded in terms of these basis states:

where

The coefficients a_n(t) are 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

$\langle H(t) \rangle \ \stackrel{\mathrm{def}}{=}\ \langle\psi(t)|H|\psi(t)\rangle = \sum_{nn'} a_{n'}^* a_n \langle n'|H|n \rangle$

where the last step was obtained by expanding in terms of the basis states.

Each of the a_n(t)'s 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 a_n(t) and its complex conjugate a_n*(t). With this choice of independent variables, we can calculate the partial derivative

$\frac{\partial \langle H \rangle}{\partial a_{n'}^{*}} = \sum_{n} a_n \langle n'|H|n \rangle = \langle n'|H|\psi\rangle$

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

$\frac{\partial \langle H \rangle}{\partial a_{n'}^{*}} = i \hbar \frac{\partial a_{n'}}{\partial t}$

Similarly, one can show that

$\frac{\partial \langle H \rangle}{\partial a_n} = - i \hbar \frac{\partial a_{n}^{*}}{\partial t}$

If we define "conjugate momentum" variables π_n by

then the above equations become

which is precisely the form of Hamilton's equations, with the s as the generalized coordinates, the s as the conjugate momenta, and taking the place of the classical Hamiltonian.

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