Time evolution

In Schrödinger’s formulation of quantum mechanics, a state vector is parameterized by time, and time evolution can be written as the application of a unitary operator

\begin{align*} \ket{\psi,t} = U(t,t_0) \ket{\psi,t_0}. \end{align*}

Since $U$ is unitary, normalization is preserved

\begin{align*} \braket{\psi,t|\psi,t} = \braket{\psi,t_0 | U\adj(t,t_0)U(t,t_0)|\psi,t_0} = \braket{\psi,t_0|\psi,t_0}. \end{align*}

We also require that $U(t_0,t_0) = \hat 1$ , that time evolution composes $U(t_2,t_1)U(t_1,t_0) = U(t_2,t_0)$ , and that $U$ is unique and independent of state. Since $U$ is unitary, its hermitian conjugate is its inverse, so the conjugate reverses the time translation $U\adj(t,t_0) = U(t_0,t)$ .

Now consider how $U$ relates to the time derivative of the state.

\begin{align*} \frac{d }{d t} \ket{\psi,t} &= \frac{\partial }{\partial t} U(t,t_0) \ket{\psi,t_0} \\ &= \frac{\partial U(t,t_0)}{\partial t} U(t_0,t) \ket{\psi,t} \\ &= \frac{\partial U(t,t_0)}{\partial t} U\adj(t,t_0) \ket{\psi,t}. \end{align*}

Now write

\begin{align*} \frac{\partial U(t,t_0)}{\partial t} U\adj(t,t_0) =: -\frac{i}{\hbar} H(t,t_0). \end{align*}

This yields something similar to the Schrödinger equation, with the notable difference that $H$ is parameterized by both $t$ and $t_0$ .

\begin{align*} -i\hbar \frac{d }{d t} \ket{\psi,t} = H(t,t_0) \ket{\psi,t} \end{align*}

First show that $H$ is hermitian, observing that $H\adj = -i\hbar U\partial_t U\adj$ .

\begin{align*} 0 &= \partial_t \hat 1 = \partial_t (U(t,t_0) U\adj(t,t_)) = (\partial_t U) U\adj + U\partial_t U\adj \\ &= -\frac{i}{\hbar} H + \frac{i}{\hbar} H\adj. \end{align*}

Then show that $H$ is $t_0$ independent by introducing an arbitary time $t_1$ and showing that $H(t,t_0) = H(t,t_1)$ .

\begin{align*} H(t,t_0) &= i\hbar (\partial_t U(t,t_0)) \hat 1 U\adj(t,t_0) \\ &= i\hbar \underbrace{(\partial_t U(t,t_0)) U(t_0,t_1)}_{\partial_t(U(t,t_0)U(t_0,t_1))} \underbrace{U(t_1,t_0) U\adj(t,t_0)}_{U\adj(t,t_1)} \\ &= i\hbar (\partial_t U(t,t_1)) U\adj(t,t_1) \\ &= H(t,t_1) = H(t). \end{align*}

We thus arrive at the Schrödinger equation for the hermitian Hamiltonian $\hat H(t)$ .

\begin{align*} i\hbar \frac{d }{d t} \ket{\psi,t} = \hat H(t) \ket{\psi,t}. \end{align*}

The only thing we assumed about $H$ is that it is hermitian. We see that any hermitian operator generates unitary time evolution! The question becomes which operators correspond to physical systems. Following classical intuition we extend the Hamiltonian from classical mechanics to quantum mechanics.