Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 75 / 3

A Euclidean configuration-space transition kernel is obtained by slicing $\tau$ into $M$ steps $\delta =\tau /M$, inserting position resolutions and using the free Gaussian kernel. With initial $q_i=\pm a$ and final $q_f=a$,The normalization means the limit of $(m/(2\pi \hslash \delta ){)}^{M/2}\int {\prod }_{j=1}^{M-1}d{q}_j$ with exponent $-{\sum }_j[m(q_{j+1}-q_j{)}^2/(2\delta )+\delta V(q_j)]/\hslash$, in a consistent time-slice prescription. This fixes the endpoint normalization that an informal continuum symbol alone leaves unspecified. The Wick rotation converts oscillatory real-time weight into the positive-potential Euclidean weight; classical extrema of this action organize the semiclassical approximation.

For a trajectory connecting the degenerate minima in infinite Euclidean time, the first integral gives $m{\dot{q}}^2/2=V(q)$. The quartic double-well instanton and its reverse areTheir action isThe instanton fluctuation prefactor $K$ has dimensions of inverse time. It incorporates nonzero Gaussian fluctuation eigenvalues and the translation zero-mode Jacobian; a conventional determinant expression iswith common endpoint regularization and the prime removing the translation mode. Only its positive quantum tunnelling rate $\kappa =K{e}^{-S_0/\hslash }$ is needed here.

In the dilute instanton gas, well-separated crossings alternate in direction. Integrating $n$ ordered centers gives ${\tau }^n/n!$. A path returning to the same well has $n$ an even number, and one connecting opposite wells has $n$ an odd number. The common leading harmonic endpoint factor follows from the supplied harmonic oscillator transition kernel:Expanding these functions gives the sums over even numbers and odd numbers, with the position-kernel normalization ${\mathcal{A}}_0$ that its abbreviated formula suppresses. Validity requires $S_0/\hslash \gg 1$, $\omega \tau \gg 1$ and $\kappa \ll \omega$, so typical crossing separation greatly exceeds an instanton width. Local anharmonic corrections replace the harmonic well energy by its perturbatively corrected value; the leading formula does not claim uniformly negligible relative error at arbitrarily large $\tau$.

The two exponents identify the double-well tunneling splitting:The superposition with an even spatial wavefunction is the lower state. Coherent quantum tunnelling removes the classical degeneracy, and the exponentially small splitting sets the long quantum tunnelling timescale.

For the new periodic potential, let $\kappa$ be the single-neighbour hopping rate and $\Delta ϵ=\hslash \kappa$. It is determined by the barrier between adjacent minima; an unspecified periodic potential does not determine it from the preceding quartic potential's numerical action. The minimum spacing is now $a$.

A path with $r$ right hops and $s$ left hops has $r-s=n-m$. Summing the ordered-center weights and their direction choices givesInsert the Fourier representation of a Kronecker delta. The two exponential series sum independently, yielding the dilute-hopping lattice propagatorThe two Fourier-sign choices are equivalent by $\theta \mapsto -\theta$. Equivalently the integral is the modified Bessel function $I_{n-m}(2\kappa \tau )$.

The original PDF prints $e^{+\omega \tau /2}$ here. Its sign is inconsistent with both the previous double-well result and the requested positive on-site oscillator energy. The correct factor is $e^{-\omega \tau /2}$: a positive oscillator ground-state energy must decay under $e^{-H\tau /\hslash }$. The other factors and their derivation remain as displayed. The energy $\Delta ϵ$ is the positive nearest-neighbour quantum tunnelling matrix-element magnitude, exponentially small relative to the local well scale in the semiclassical regime.

The low-energy localized-well basis gives the tight-binding modelA Bloch state with coefficients $e^{ipna}$ is an eigenstate because the two neighbouring coefficients add to $2\cos (pa)e^{ipna}$. ThusHere $p$ is a Bloch wavenumber; physical momentum is $\hslash p$, and writing $p$ for physical momentum would require $pa/\hslash$. Spectral evolution of these eigenstates is exactly the Fourier integral in the preceding solution. The negative hopping lowers the symmetric $p=0$ state, and coherent quantum tunnelling spreads a formerly degenerate family of localized levels into a band of width $4\Delta ϵ$. Using the printed positive Euclidean prefactor instead would give on-site energy $-\hslash \omega /2$, showing its contradiction directly.