The eigenfunctions of a particle in a cubic box with potential V=0 in the region 0≤x≤L,0≤y≤L,0≤z≤LandV=∞0 \leq x \leq L,\quad 0 \leq y \leq L,\quad 0 \leq z \leq L \quad \text{and} \quad V = \infty0≤x≤L,0≤y≤L,0≤z≤LandV=∞ outside are denoted as ψnxnynz\psi_{n_x n_y n_z}ψnxnynzWhich of the following functions is also an eigenfunction of the Hamiltonian?

The eigenfunctions of a particle in a cubic box with potential V=0 in the region $0 \leq x \leq L,\quad 0 \leq y \leq L,\quad 0 \leq z \leq L \quad \text{and} \quad V = \infty$ outside are denoted as $\psi_{n_x n_y n_z}$

Which of the following functions is also an eigenfunction of the Hamiltonian?

$\phi_1=\psi_{123}-\psi_{312}$

$\phi_2=\psi_{111}+\psi_{222}$

$\phi_3=\psi_{121}-\psi_{122}$

$\phi_4=\psi_{212}+\psi_{113}$

Correct option is A

Particle in a box

A particle is confined to a rectangular surface of length L₁ in the x-direction and L₂ in the y-direction; the potential energy is zero everywhere except at the walls, where it is infinite. The wavefunction is now a function of both x and y and the Schrödinger equation is

Writing the wavefunction as a product of functions, one depending only on x and the other only on y:

The first step in the justification of the separability of the wavefunction into the product of two functions X and Y is to note that, because X is independent of y and Y is independent of x, we can write

with the quantum numbers taking the values n₁ =1, 2, . . . and n₂ =1, 2, . . . independently.

We treat a particle in a three-dimensional box in the same way. The wavefunctions have another factor (for the z-dependence), and the energy has an additional term in