Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2012 / Paper 1 / 10 / ii

Here $p=w=1$ and $q=x^2$ on the whole real line. The quantum harmonic oscillator energy form is $Q[\psi ]={\int }_{\mathbb{R}}(|{\psi }^{\prime }{|}^2+x^2|\psi {|}^2)dx$. Admissible trials have a square-integrable weak derivative and finite potential energy; they need not have a classical second derivative everywhere.

Put $C_a=\sqrt{15/(16a^5)}$, with $a>0$. Direct integration confirms the normalization:The kinetic energy and potential energy terms in the dimensionless Rayleigh quotient areThus ${\lambda }_0\le 5/(2a^2)+a^2/7$. Differentiating with respect to $a$ gives $-5/a^3+2a/7=0$, so the unique minimizing width is $a=(35/2{)}^{1/4}$. The two energy contributions are equal there, and the compact parabolic trial bound for the quantum harmonic oscillator isThe trial is continuous and zero at $\pm a$, so its weak first derivative is $-2C_{a}x$ inside and zero outside. The jumps of that derivative do not add delta terms to the first derivative, and its squared norm is finite. The energy-form calculation is therefore valid even though the trial does not belong to the operator domain of the harmonic oscillator. Applying the second-derivative operator naively and omitting its endpoint distributions would be incorrect.

For any nonzero constant $C$, differentiating ${\psi }_0=C{e}^{-x^2/2}$ gives ${\psi }_0^{\prime \prime }=(x^2-1){\psi }_0$. HenceIt is indeed the lowest eigenvalue, since the factorization $H=(-d/dx+x)(d/dx+x)+1$ makes $⟨\psi ,H\psi ⟩\ge \Vert \psi {\Vert }^2$, with equality for this Gaussian. The variational bound exceeds the exact value by $\sqrt{10/7}-1\simeq 0.195229$, or about $19.52\%$.

The PDF's prefactor $1/\sqrt{2\pi }$ does not normalize this wavefunction: its squared norm is $1/(2\sqrt{\pi })$. The normalized ground state has prefactor ${\pi }^{-1/4}$. This normalization error does not affect the eigenvalue, because the equation is homogeneous in the wavefunction.