Hamiltonian operator

The Hamiltonian operator gives us the energy of a wavefunction. Generally the Hamiltonian is

\HH = \frac{\pp^2}{2m} + V

where $\pp$ is the momentum operator and $V$ is the potential.

In position space the Hamiltonian is

\HH = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V.

The eigenfunctions $u_n(x)$ of the Hamiltonian are the eigenstates of the system, the eigenvalues are their respective energies. A wave equation can be constructed a superposition of eigenstates and their time-dependent functions (see Schrödinger equation).

\Psi(x,t) = \sum_{n=1}^\infty C_n u_n(x) e^{-iE_nt/\hbar}.

The coefficients $C_n$ are given by the projection of the wavefunction onto the eigenstate $C_n = \braket{u_n|\psi}$ .

Since the Hamiltonian is hermitian, it is a generator of time evolution. This Hamiltonian generates the time evolution that we observe physically. It’s not obvious why this is the case, but we can think of the classical Hamiltonian (generator of time translation classicaly) as a special case of this operator.

For some time translation operator $U(t,t_0)$ , the Hamiltonian is $H(t) = i\hbar (\partial_t U)U\adj$ . We can find $U$ in a few special cases.

For the simplest case, assume $H(t) = H$ is time independent. Then it can be shown that

\begin{align*} U(t,t_0) = \tilde U e^{-i Ht/\hbar} = U(0,t_0) e^{-iHt/\hbar}. \end{align*}

Or assume $[H(t),H(t')] = 0$ . Then guess

\begin{align*} U(t,t_0) &= \exp \left[ -\frac{i}{\hbar} \int_{t_0}^t dt'\, H(t') \right] =: e^{G(t,t_0)} \\ \frac{d }{d t} G(t,t_0) &= - \frac{i}{\hbar} H(t) \\ [G'(t,t_0), G(t,t_0)] &= \frac{1}{\hbar^2} \left[H(t), \int_{t_0}^t dt'\, H(t') \right] \\ &= -\frac{1}{\hbar^2} \int_{t_0}^t dt'\, [H(t),H(t')] = 0. \end{align*}