Answer to Question #93970 in Quantum Mechanics for Bornface

Question

Consider a particle of mass m confined in an infinite 1-D potential well of width a;
V (x)=0 for 0 less or equal to x and x less than or equal to a,
V(x)=infinite otherwise
Find the eigenstates of the Hamiltonian and the corresponding eigen energies Using three possibilities considering the energy E,
E>0
E<0
E=0

Answer 1

The eigenvalues of the Hamiltonian are:

$$H|\psi\rangle=E|\psi\rangle$$

for every eigenvector.

Write the Hamiltonian:

$$H=-\frac{\hbar^2}{2m}\frac{\text{d}^2}{\text{d}x^2},$$

substitute this to the first equation:

$$-\frac{\hbar^2}{2m}\frac{\text{d}^2}{\text{d}x^2}|\psi\rangle=E|\psi\rangle,$$

the solution of this equation gives the set of eigenstates:

$$|\psi\rangle=C\space\text{sin}\Big(\frac{\pi n x}{a}\Big).$$

The corresponding energies (eigenvalues):

$$E=\frac{n^2\pi^2\hbar^2}{2ma}.$$