Overview
The time evolution of a state in quantum mechanics describes how a state changes from an initial time {% t_0 %} to the time {% t %}.
{% | \psi(t_0) \rangle \; \rightarrow \; | \psi(t) \rangle %}
In order to investigate the time evolution of a quantum state, define the following operator which maps
the state at one time, to another time.
{% | \psi(t) \rangle = \; U(t,t_0) | \psi(t_0) \rangle %}
Properties of the Time Evolution Operator
- Composition - {% U(t_2,t_0) = U(t_2,t_1)U(t_1,t_0) %}
- {% U(t,t) = 1 %}
and
{% lim _{\Delta t \rightarrow 0} U(t+\Delta t, t) = 1 %}
Hamiltonian
Given the assumption that the limit as {% \delta t %} goes to zero is 1, we can can assume that
{% U(t+\delta t,t) = 1 - iH \delta t %}
where {% H %} is here some (as yet unknown) operator. It is named the Hamiltonian.
{% \frac{|\psi(t+ \delta t)\rangle \; - \; |\psi(t)\rangle}{\delta t} = -i H |\psi(t)\rangle %}
Utilizing the equation
{% \psi(x,t) = \langle x | \psi(t) \rangle %}
the above equation becomes the
Schrodinger Equation