QM of a particle in a 3D infinite square well

A particle in the infinite square well potential

𝑉 (𝐫) = {0 𝐫 \in [- 𝑎, 𝑎] 3 \infty e l s e w h e r e

has stationary states $𝜑 𝑛 𝑥, 𝑛 𝑦, 𝑛 𝑧 (𝑥, 𝑦, 𝑧)$ that are (tensor) products of stationary states $𝜓 𝑛 (𝑥)$ analogous 1D potential, i.e.

𝜑 𝑛 𝑥, 𝑛 𝑦, 𝑛 𝑧 (𝑥, 𝑦, 𝑧) = 𝜓 𝑛 𝑥 (𝑥) 𝜓 𝑛 𝑦 (𝑦) 𝜓 𝑛 𝑧 (𝑧)

with energies

𝐸 𝑛 𝑥, 𝑛 𝑦, 𝑛 𝑧 = 𝜋 2 ℏ 2 8 𝑚 𝑎 (𝑛 2 𝑥 + 𝑛 2 𝑦 + 𝑛 2 𝑧)

Proof by separation of variables

Inside $[- 𝑎, 𝑎] 3$ the TISE reads

- ℏ 2 𝑚 \nabla 2 𝜓 = 𝐸 𝜓

we look for solutions of the form

𝜓 (𝐫) = 𝑋 (𝑥) 𝑌 (𝑦) 𝑍 (𝑧)

for which the TISE becomes

- ℏ 2 𝑚 (𝑋 ″ 𝑌 𝑍 + 𝑌 ″ 𝑋 𝑍 + 𝑍 ″ 𝑋 𝑌) = 𝐸 𝑋 𝑌 𝑍

hence

𝑋 ″ 𝑋 + 𝑌 ″ 𝑌 + 𝑍 ″ 𝑍 = - 2 𝑚 𝐸 ℏ 2

since each of the terms are functions of $𝑥$ , $𝑦$ , and $𝑧$ respectively, the only way the LHS can equal the constant RHS is if each of the terms equals a constant, i.e.

𝑋 ″ = - 𝑘 2 𝑗 𝑋 𝑌 ″ = - 𝑘 2 𝑗 𝑌 𝑍 ″ = - 𝑘 2 𝑗 𝑍

Once boundary conditions are applied, the general solutions for $𝑋$ , $𝑌$ , and $𝑍$ are thus precisely those for QM of a particle in a 1D infinite square well. Let $𝜑 𝑛 (𝑥)$ denote solutions for the 1D case. We thus have

𝜓 𝑛 𝑥, 𝑛 𝑦, 𝑛 𝑧 = 𝜑 𝑛 𝑥 (𝑥) 𝜑 𝑛 𝑦 (𝑦) 𝜑 𝑛 𝑧 (𝑧)

which is already normalized.

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