Past exam of the quantum mechanics quantum theory and quantum computers course of the University of Oxford / 2008 / B / 6

Choose an orthonormal eigenbasis $H_0|n⟩=E_n|n⟩$ and the phase convention $⟨0|{\psi }^{\prime }⟩=0$. Comparing coefficients of the eigenvalue equation at first order yieldsProjecting onto $|0⟩$ and the other eigenvectors givesProjection of the next-order equation onto $|0⟩$ givesThese are coefficients of the printed series, so $E^{\prime \prime }$ denotes the quadratic coefficient rather than the second derivative itself. Without a phase convention an arbitrary purely imaginary multiple of ${\psi }_0$ can be added to the first vector correction. Continuous spectrum, if present, requires the corresponding spectral integral; the oscillator has a discrete complete eigenbasis.

Put $K=m{\omega }^2/2$ and let $g(x)=(2\pi {\sigma }^2{)}^{-1/2}{e}^{-x^2/(2{\sigma }^2)}$ denote the ground-state density. Then $H^{\prime }=K(a^2-x^2)1_{[-a,a]}$ and $p={\int }_{-a}^{a}g(x)dx$. Because $g^{\prime }=-xg/{\sigma }^2$, integration by parts yieldsThus the two first-order shifts areThe first is the supplied ground-state expression; the second follows because the excited-state density is $x^{2}g/{\sigma }^2$, so $E_1^{\prime }=K(a^2{I}_2-I_4)/{\sigma }^2$.

Subtracting, and using $K{\sigma }^2=\hslash \omega /4$, givesThe bracket is zero at $a=0$, tends to one at infinity, and has derivative $2a^{2}g(a)/{\sigma }^2\ge 0$. Therefore the first-order predicted level gap isFor every $\lambda >2$ it becomes negative for all sufficiently large $a$, crossing when $p-2ag=2/\lambda$. It is still positive at $a=0$: the stated conclusion must be read with the large-$a$ regime, rather than for every interval width.

For $\lambda >1$ the inner potential curves downward, placing a central barrier between two symmetric minima at $\pm a$. The even unperturbed ground state has its greatest density at the central barrier, while the odd first excited state has a node there and is less penalized. This explains the relative first-order shifts. A negative predicted gap signals failure of that perturbation approximation, not inversion of the exact lowest two states: the true one-dimensional ground state remains nodeless and even, with the odd state above it. At a high barrier the actual low levels instead form a close double-well pair. The apparent level crossing from a finite-interval oscillator perturbation is useful precisely as a diagnostic of the approximation's range.