Past exam of the physics A course of the University of Cambridge / 2010 / Paper 2 / Section B / B9

With angular-coordinate measure $d\varphi$, the normalized one-particle orbitals are $u_l(\varphi )=e^{il\varphi }/\sqrt{2\pi }$. Single-valuedness requires integer $l$. Their products obeyIf arc-length measure is used instead, each one-particle wavefunction normalization includes $R^{-1/2}$; the physical energy is unchanged. For the following parts impose antisymmetry on the full spatial-and-spin state of the two fermions.

Both particles can occupy $l=0$ with opposite spins, giving a unique singlet state at $E=0$, quantum degeneracy 1. At the next distinct energy ${ϵ}_R$, one orbital is $l=0$ and the other is $l=+1$ or $-1$. For each orbital choice there is one symmetric spatial state paired with the singlet state, and one antisymmetric spatial state paired with three triplet state spin projections. HenceSwapping particle labels adds no independent state after the required exchange symmetry is imposed.

The lowest singlet state isFor the lowest triplet states take $\lambda =\pm 1$ and $m_s=-1,0,1$:orthogonality of $u_0$ and $u_{\lambda }$ verifies wavefunction normalization of the antisymmetric spatial factor. At this same energy there are also two excited singlet states, obtained by replacing its minus sign with a plus sign and the spin factor by $|0,0⟩$. Together they reproduce the eight states counted in part a.

At coincidence ${\varphi }_1={\varphi }_2$, the triplet state spatial factor vanishes, while the lowest singlet state has finite, uniform pair probability. Consequently a weak short-range repulsion raises the singlet state energy more strongly: antisymmetry suppresses close encounters in the triplet state. In the contact-interaction limit the triplet state contribution is exactly zero, whereas the singlet state shift is positive. A finite-range interaction gives a nonzero but suppressed triplet state shift. This explains a relative energy increase; it does not by itself establish a crossing or justify using unchanged noninteracting wavefunctions for arbitrarily strong repulsion.