Time Independent Schrodinger Equation
Overview
Separation of Variables
{% \Psi(x,t)= \psi(x)\phi(t) %}
{% \frac{\partial \Psi}{\partial t} = \psi \frac{d\phi}{dt} %}
{% \frac{\partial \Psi ^2}{\partial x^2} = \frac{d^2 \psi}{dx^2} \phi %}
Then the Schrodinger equation becomes
{% i \hbar \psi \frac{d \phi}{dt} = - \frac{\hbar ^2}{2m} \frac{d^2 \psi}{dx^2} \phi + V \psi \phi %}
after dividing by {% \psi \phi %} it becomes
{% i \hbar \frac{d \phi}{dt} = - \frac{\hbar^2}{2m} \frac{1}{\psi} \frac{d^2 \psi}{dx^2} + V %}
the left side is dependent only on t, which the right is dependent only on x. This implies
that each side is equal to a constant, which we label {% E %}
The equation for {% \phi %} is
{% i \hbar \frac{1}{\phi} \frac{d\phi}{dt} = E %}
{% \frac{d \phi}{dt} = - \frac{iE}{\hbar} \phi %}
The equation for {% \psi %} is
{% -\frac{\hbar^2}{2m}\frac{1}{\psi}\frac{d^2 \psi}{dx^2} + V = E %}
{% - \frac{\hbar^2}{2m} \frac{d^2 \psi}{dx^2} + V \psi = E \psi %}
Operator Formalism
{% \hat{H} \psi = E \psi %}
eigenvector/eigenvalue
