Two Electrons in a Box

Two electrons are confined in one dimension to a box of length a. A clever experimentalist has arranged that both electrons have the same spin state. Ignore the Coulomb interaction between electrons.

a) Write the ground state wave function ψ (x₁, x₂) for the two-electron system.

b) What is the probability that both electrons are found in the same half of the box?

SOLUTION

a) If the box is in the region 0 < x < a, then the one-electron orbitals are

(1)

If both electrons are in the spin state α (spin up), then the spin part of the wave function α₁α₂ is symmetric under exchange of coordinates. Therefore, the orbital part has to be antisymmetric, and both particles cannot be in the n = 1 state. Instead, the lowest energy occurs when one electron is in the n = 1 state and the other is in the n = 2 state:

(2)

b) The probability that both are in one half, say the left side, is

(3)

Three integrals must be evaluated:

(4)

(5)

(6)

 (7)

(8)

The result is rather small. Naturally, it is much more favorable to have one particle on each side of the box.