Quantum Mechanics Time Evolution - Hamiltonian

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) {>} \; \rightarrow \; | \psi(t) {>} %}
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) {>} = \; U(t,t_0) | \psi(t_0) {>} %}

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){>} \; - \; |\psi(t){>}}{\delta t} = -i H |\psi(t){>} %}
Utilizing the equation
{% \psi(x,t) = {<} x | \psi(t) {>} %}
the above equation becomes the Schrodinger Equation

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