Double-Well Tunneling

A symmetric double well is the canonical model of tunneling splitting. Classically, a low-energy particle trapped in one well cannot cross the barrier. Quantum mechanically, the two localized well states mix through the barrier, producing even and odd energy eigenstates separated by an exponentially small splitting.

Take a smooth symmetric potential with two minima:

$$V(x)=V(-x), \qquad V(\pm a)=0, \qquad V(0)>0.$$

A standard example is

$$V(x)=\lambda(x^2-a^2)^2, \qquad \lambda>0.$$

This page owns the physical two-well splitting story. Tunneling Splittings develops the quantitative dictionary among the effective two-state coupling, Herring formula, WKB action, instanton sum, and exact spectral doublet. The Euclidean saddle method is developed in Instantons in Quantum Mechanics.

Double-Well Splitting carries this quartic model through a fixed dimensionless convention, an analytic instanton, WKB quadrature, and numerical even–odd gaps. The general physical discussion remains here.

Localized Well States

Near either minimum, the potential is approximately harmonic. For the quartic double well,

$$V''(a)=8\lambda a^2,$$

so the local oscillator frequency is

$$\omega = \sqrt{\frac{8\lambda a^2}{m}}.$$

If the wells were completely isolated, there would be a left-localized ground state $\lvert L\rangle$ and a right-localized ground state $\lvert R\rangle$ with nearly equal local energy

$$E_{\mathrm{loc}} \approx \frac12\hbar\omega.$$

The barrier is finite, so these states are not exact eigenstates.

Even and Odd Eigenstates

Because the Hamiltonian has parity symmetry, exact energy eigenstates can be chosen with definite parity. The two lowest states are approximately

$$\lvert +\rangle = \frac{1}{\sqrt2} \left( \lvert L\rangle+\lvert R\rangle \right),$$

and

$$\lvert -\rangle = \frac{1}{\sqrt2} \left( \lvert L\rangle-\lvert R\rangle \right).$$

The even state is usually lower because it has no node at the center, while the odd state has a node.

Two-State Effective Hamiltonian

In the localized basis, the low-energy Hamiltonian has the schematic form

$$H_{\mathrm{eff}} = \begin{pmatrix} E_{\mathrm{loc}} & -K \\ -K & E_{\mathrm{loc}} \end{pmatrix}, \qquad K>0.$$

Its eigenvalues are

$$E_\pm = E_{\mathrm{loc}}\mp K.$$

The splitting is

$$\Delta E = E_- - E_+ = 2K.$$

The off-diagonal matrix element $K$ is controlled by barrier penetration and is exponentially small when the barrier is high or wide.

WKB Estimate

Let $E_{\mathrm{loc}}$ lie below the barrier, and let $x_1$ and $x_2$ be the turning points that bound the forbidden barrier region between the wells. Define

$$\kappa(x) = \sqrt{2m(V(x)-E_{\mathrm{loc}})}.$$

The splitting has the leading exponential form

$$\Delta E \sim A \exp\left[ - \frac{1}{\hbar} \int_{x_1}^{x_2} \kappa(x)\,dx \right].$$

The prefactor $A$ depends on the well frequency, matching conventions, and corrections near the turning points. The exponent is the most robust part of the estimate.

The important lesson is not the exact prefactor but the nonanalytic dependence:

$$\Delta E \propto e^{-S/\hbar}.$$

No finite power series in $\hbar$ around one isolated well can reproduce this splitting.

Instanton Estimate

In Euclidean time, tunneling is described semiclassically by a finite-action path connecting one minimum to the other. For a one-dimensional double well with minima normalized to $V=0$, the instanton action is

$$S_0 = \int_{-a}^{a} dx\, \sqrt{2mV(x)}.$$

For the quartic example,

$$S_0 = \frac{4}{3} a^3 \sqrt{2m\lambda}.$$

The splitting has the instanton form

$$\Delta E \sim A_{\mathrm{inst}} e^{-S_0/\hbar},$$

with a prefactor determined by fluctuations around the instanton. Computing that prefactor carefully requires treating a zero mode associated with the instanton’s center in Euclidean time.

Numerical Comparison

A numerical one-dimensional diagonalization gives a direct check:

1. choose a grid or basis that resolves both wells and the barrier;
2. diagonalize the Hamiltonian;
3. identify the two lowest parity eigenstates;
4. compute $\Delta E=E_1-E_0$;
5. compare the logarithm of the splitting to the WKB or instanton action.

The logarithm is the cleanest comparison because the exponent controls the dominant dependence. Prefactors require more precise numerics and more careful asymptotics.

Double-Well Instanton Numerical Check carries out this comparison with parity-resolved basis diagonalization, an independent finite-difference discretization, retained WKB quadratures, and explicit basis and roundoff diagnostics.

Common Mistakes

• Treating $\lvert L\rangle$ and $\lvert R\rangle$ as exact energy eigenstates in a symmetric finite barrier.
• Expecting ordinary perturbation theory around one well to see the splitting at finite order.
• Quoting the WKB exponent as a complete formula for the prefactor.
• Forgetting that exact eigenstates of a symmetric double well have definite parity.
• Confusing barrier transmission probability with the energy splitting; they are related semiclassically but not identical observables.

Exercises

1. Diagonalize the two-state effective Hamiltonian and find the splitting.

Solution

For

$$H_{\mathrm{eff}} = \begin{pmatrix} E_{\mathrm{loc}} & -K \\ -K & E_{\mathrm{loc}} \end{pmatrix},$$

the symmetric and antisymmetric eigenvectors have eigenvalues

$$E_+=E_{\mathrm{loc}}-K, \qquad E_-=E_{\mathrm{loc}}+K.$$

Therefore

$$\Delta E=E_- - E_+=2K.$$

2. For $V(x)=\lambda(x^2-a^2)^2$, compute $V''(a)$.

Solution

First,

$$V'(x)=4\lambda x(x^2-a^2).$$

Then

$$V''(x)=4\lambda(3x^2-a^2).$$

At $x=a$,

$$V''(a)=8\lambda a^2.$$

3. Why is the splitting called nonperturbative in $\hbar$?

Solution

The leading dependence is of the form

$$\Delta E\propto e^{-S/\hbar}.$$

As $\hbar\to0^+$, this is smaller than any finite power of $\hbar$. A power series in $\hbar$ around one well cannot produce such an exponential term at finite order.

References

• S. Coleman, Aspects of Symmetry, Cambridge University Press, 1985.
• A. Garg, “Tunnel splittings for one-dimensional potential wells revisited,” American Journal of Physics 68, 430-437, 2000.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Butterworth-Heinemann, 1981.
• H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 5th ed., World Scientific, 2009.