Question

Consider two electrons, both in the ground state of an in?nite potential well. Write the wave function ?(x1,x2) for this system.

Assume the electrons are non-interacting

Answer 1

Assuming that the electrons are non-interacting, the collective two-electron wavefunction will be:

where A is the normalization constant

from the condition that this wavefunction should entirely be contained within the potential well and that both the states (left side term and the right side term) are equally probable, the value of this constant A will be:

therefore,

for the Anti-symmetric case: [electrons]

If n = m, then the wavefunction becomes zero which is not possible because it must be non-zero for two existing electrons inside the well. Thus, n cannot be equal to m which is Pauli's Exclusion Principle.

This implies that if n = 1, smallest value m can have is 2 [since n,m cannot be zero in an infinite potential well].

therefore, the ground state wavefunction will be:

likewise, m can also be 1 and then n will be 2.