Two Electrons in a Well

The infinite square well is the simplest place to watch the exclusion principle organize a system of electrons. We already know its single-particle spectrum; identical-particle physics tells us how to populate it.

Single-particle levels

For a 1D infinite well of width $L$ , the single-particle energy levels are

E_n = \frac{n^2 \pi^2 \hbar^2}{2 m L^2} = n^2 E_1, \qquad n = 1, 2, 3, \ldots,

with spatial wavefunctions $\psi_n(x) = \sqrt{2/L}\,\sin(n\pi x/L)$ . We will measure energies in units of the ground level $E_1$ , so the rungs of the ladder sit at $1, 4, 9, 16, \ldots$

Two non-interacting electrons

Ignore the Coulomb repulsion between the electrons for now, so the only physics is the well plus the exclusion principle. Each spatial level $n$ can hold two electrons because the electron's spin gives two distinct states for the same orbital: $|\psi_n\rangle\otimes|\!\uparrow\rangle$ and $|\psi_n\rangle\otimes|\!\downarrow\rangle$ . The full single-particle state — orbital and spin — is what must be distinct for two fermions, and here we have two distinct spin labels available.

The ground state

To minimize total energy, drop both electrons into the lowest available levels. The first electron goes into $n=1$ with spin up. The second electron may also go into $n=1$ , taking spin down — this is allowed because its spin label differs. Neither electron is forced up to $n=2$ :

E_\text{ground} = E_1 + E_1 = 2E_1.

The overall two-electron state is the product of a symmetric spatial part $\psi_1(x_1)\psi_1(x_2)$ and the antisymmetric spin singlet $\tfrac{1}{\sqrt2}(|\!\uparrow\downarrow\rangle - |\!\downarrow\uparrow\rangle)$ . Symmetric $\times$ antisymmetric is antisymmetric overall, exactly as a two-fermion state must be.

The first excited state

The third unit of structure shows up when we excite the system. Promote one electron to $n=2$ :

E_\text{first excited} = E_1 + E_2 = 1 + 4 = 5E_1.

Now the two electrons sit in different spatial orbitals, $\psi_1$ and $\psi_2$ . The spatial part may be either symmetric or antisymmetric, and the spin part must take the opposite symmetry to keep the total antisymmetric. So this level splits into a spin-singlet (symmetric spatial) and a spin-triplet (antisymmetric spatial) — a fourfold set of states that ordinary Coulomb repulsion would later split in energy.

Try it

Compute the total ground-state energy of two non-interacting electrons in the infinite well, in units of $E_1$ . Remember that both electrons fit in $n=1$ thanks to their opposite spins.

Run your code to see the quantum state.