Overview
The Schrodinger equation in three dimensions is derived by using the three dimensional momentum operator, and the three dimensional potential in the Schrodinger equation given here:
{% \hat{H} = - \frac{\hat{p}^2}{2m} + V(x,t) %}
The three dimensional momentum operator can be written as
{% \hat{p}^2 = (p_x^2 + p_y^2 + p_z^2) %}
where for instance
{% p_x = -i\hbar \frac{\partial}{\partial x} %}
{% i\hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m} \nabla^2 \psi + V\psi %}