= Solution
A tall, broad central <potential barrier> has $\phi\gg1$. Let $\sigma=+1$ denote even <parity> and $\sigma=-1$ odd <parity>, and write $\theta=(n+\tfrac12)\pi+\delta_\sigma$, where $n=0,1,\ldots$. Since $\tan[(n+\tfrac12)\pi+\delta]=-\cot\delta$, the condition in part (d) gives
$$
\delta_\sigma=-\sigma\arctan\left(\frac12e^{-\phi}\right)=-\frac\sigma2e^{-\phi}+O(e^{-3\phi}).
$$
Thus the leading correction to the isolated-well <Bohr-Sommerfeld quantization condition> is
$$
\boxed{\int_{x_1}^{x_2}p(x)\,dx=\pi\hbar\left(n+\frac12\right)-\sigma\frac\hbar2\exp\left[-\frac1\hbar\int_{-x_1}^{x_1}\chi(x)\,dx\right]+O(\hbar e^{-3\phi}).}
$$
This is an implicit condition, with all <action integrals> evaluated at the corrected <energy>. To convert it into an explicit leading <energy> correction, let $E_n^{(0)}$ solve $\theta(E_n^{(0)})=(n+\tfrac12)\pi$ and define the full classical <oscillation period> in one well by $\mathcal T(E)=2m\int_{x_1}^{x_2}dx/p(x)$. Differentiation of the <action integral> has no endpoint terms because $p(x_1)=p(x_2)=0$, and gives
$$
\theta'(E)=\frac m\hbar\int_{x_1}^{x_2}\frac{dx}{p(x)}=\frac{\mathcal T(E)}{2\hbar}>0.
$$
Consequently, to leading exponentially small order within the <WKB approximation>,
$$
\boxed{E_{n,\sigma}\simeq E_n^{(0)}-\sigma\frac{\hbar}{\mathcal T(E_n^{(0)})}e^{-\phi(E_n^{(0)})},\qquad E_{n,-}-E_{n,+}\simeq\frac{2\hbar}{\mathcal T(E_n^{(0)})}e^{-\phi(E_n^{(0)})}.}
$$
The even member lies below its odd partner. For an approximately <harmonic oscillator> well, $\mathcal T\simeq2\pi/\omega$, giving the familiar <double-well tunneling splitting> $\Delta E\simeq(\hbar\omega/\pi)e^{-\phi}$. The explicit <energy> formula retains only its leading exponential correction; varying $\phi$ and $\theta'$ with the shifted <energy> also contributes at higher orders.
The <ground state> has an even, everywhere same-sign <wavefunction> with a peak in each well and a small nonzero value beneath the central <potential barrier>. Its derivative vanishes at zero. The first <excited state> is odd, has opposite-sign lobes in the two wells, and has exactly one interior <node of a wavefunction>, at zero. Both <wavefunctions> decay outside the wells. In terms of localized normalized <wavefunctions> $\varphi_L,\varphi_R$, with overlap $s=\langle\varphi_L|\varphi_R\rangle$, their qualitative shapes are the <lowest parity states of a symmetric double well>:
$$
\psi_{\rm ground}\simeq\frac{\varphi_L+\varphi_R}{\sqrt{2(1+s)}},\qquad \psi_{\rm first}\simeq\frac{\varphi_R-\varphi_L}{\sqrt{2(1-s)}}.
$$
The overall sign of the odd <wavefunction> is arbitrary. The sketch uses localized Gaussian profiles to show the <parity> and the central <node of a wavefunction>, rather than claiming exact <energy eigenfunctions> for an unspecified <potential energy>.
\Image[/past-exam-of-the-further-quantum-theory-course-of-the-university-of-oxford/2023-parity-wavefunctions.png]
{title=Qualitative even ground-state and odd first-excited-state wavefunctions of a symmetric double well}
Back to article page
