Question

Eigenvalue of the total energy operator partical box

Answer 1

As a simple example, we will solve the 1D Particle in a Box problem. That is a particle confined to a region 0<x<a

We can do this with the (unphysical) potential which is zero within those limits and +$\infin$ outside the limits.

$V(x)= \begin{cases} 0 &\text 0<x<a \\ \infin &\text{elsewhere} \end{cases}$

Because of the infinite potential, this problem has very unusual boundary conditions. (Normally we will require continuity of the wave function and its first derivative.) The wave function must be zero at x=0 and x=a since it must be continuous and it is zero in the region of infinite potential. The first derivative does not need to be continuous at the boundary (unlike other problems), because of the infinite discontinuity in the potential.

The time-independent Schrödinger equation (also called the energy eigenvalue equation) is

$Hu_j=E_ju_j$

with the Hamiltonian (inside the box) $H=\frac{h^2}{2m}\frac{d^2}{dx^2}$

Our solutions will have $u_j=0$ outside the box.

The solution inside the box could be written as $u_j=e^{ikx}$ where k can be positive or negative.