Double-Well Potential

A double-well potential is a one-dimensional potential with two separated minima. The symmetric double well is the canonical first encounter with tunneling splitting: low-energy states localized in the left and right wells mix through the barrier, producing even and odd energy eigenstates separated by a small energy difference.

A standard smooth example is

$$V(x)=\lambda(x^2-a^2)^2, \qquad \lambda\gt0.$$

It has minima at $x=\pm a$ and a barrier of height $\lambda a^4$ at $x=0$. This page explains the qualitative and elementary two-state structure. Double-Well Tunneling develops the graduate physical picture, while Tunneling Splittings compares quantitative Herring, WKB, and instanton estimates.

Double-Well Splitting is the graduate worked calculation for the quartic model, with a two-state reduction, WKB action, instanton saddle, and numerical benchmark.

Helpful background. Finite Square Well to match oscillating interiors to decaying tails.

Two minima connected by tunneling

The double well combines three ideas from earlier wave mechanics:

• each individual well has approximately discrete bound states;
• the barrier allows exponentially small leakage between wells;
• parity symmetry organizes the exact eigenstates when $V(x)=V(-x)$.

The result is not two independent copies of a single well. If the barrier is finite, the two wells communicate quantum mechanically.

The exactly solvable Double Delta Potential is the clean algebraic model of this same splitting mechanism.

Localized Well States

Near either minimum, a smooth double well is approximately harmonic. For the quartic example,

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

so the local oscillator frequency is

$$\omega_0 =\sqrt{\frac{8\lambda a^2}{m}}.$$

If the barrier were impenetrable, the left and right local ground states would be separate states, denoted schematically by

$$\lvert L\rangle, \qquad \lvert R\rangle.$$

They would have nearly the same local energy $E_0$. For a finite barrier, these localized states are useful approximations, but they are not exact stationary states of the symmetric Hamiltonian.

Parity Eigenstates

When

$$V(x)=V(-x),$$

the Hamiltonian commutes with parity. Exact energy eigenstates can be chosen even or odd:

$$\psi_+(x)=\psi_+(-x), \qquad \psi_-(x)=-\psi_-(-x).$$

For a high barrier, the lowest pair is 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 ground state is lower in energy because it has no node between the wells, while the odd state has a node at the center.

Two-State Effective Hamiltonian

Use an orthonormal localized basis. One precise choice is $|L\rangle=(|+\rangle+|-\rangle)/\sqrt2$ and $|R\rangle=(|+\rangle-|-\rangle)/\sqrt2$ formed from the exact lowest even and odd states. These combinations are concentrated in opposite wells when the barrier is high. A two-state approximation for more general initial states requires negligible occupation of higher levels; $\hbar\omega_0\ll\lambda a^4$ is a useful deep-well condition.

In this basis, the lowest doublet is captured by

$$H_{\mathrm{eff}} = \begin{pmatrix} E_0 & -K \\ -K & E_0 \end{pmatrix}, \qquad K\gt0.$$

The eigenstates are the even and odd combinations above. The eigenvalues are

$$E_+ =E_0-K, \qquad E_- =E_0+K.$$

The splitting is

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

The coupling $K$ is controlled by barrier penetration. It becomes smaller when the barrier is higher, wider, or when the mass is larger.

Tunneling Oscillation

A localized state is a superposition of stationary states:

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

Under time evolution,

$$\lvert \psi(t)\rangle =e^{-i(E_++E_-)t/(2\hbar)} \left[ \cos\left(\frac{\Delta E\,t}{2\hbar}\right) \lvert L\rangle +i\sin\left(\frac{\Delta E\,t}{2\hbar}\right) \lvert R\rangle \right],$$

up to an overall phase convention for $\lvert R\rangle$. Therefore

$$P_R(t) =\sin^2\left(\frac{\Delta E\,t}{2\hbar}\right).$$

Here $P_R$ is the probability of projection onto $|R\rangle$. It approximates the probability of being on the right half-line when the localized states have negligible opposite-well tails. The first complete transfer within this two-state subspace occurs at

$$t_{\mathrm{transfer}} =\frac{\pi\hbar}{\Delta E}.$$

Small splitting means slow tunneling oscillation. This is why exponentially small energy differences can still have observable dynamical consequences.

Barrier Estimate

At a rough semiclassical level, the splitting has the form

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

where $S$ is an action through the classically forbidden barrier region. In a WKB estimate,

$$S \sim \int_{x_1}^{x_2} \sqrt{2m(V(x)-E_0)}\,dx,$$

with $x_1$ and $x_2$ the turning points inside the barrier. This formula is a scaling guide here, not a complete derivation. The derivation and prefactors are part of Barrier Penetration and Tunneling and the graduate double-well tunneling page.

Asymmetric Wells

Real double wells are often not exactly symmetric. A useful two-state Hamiltonian is

$$H_{\mathrm{eff}} = \begin{pmatrix} E_L & -K \\ -K & E_R \end{pmatrix}.$$

The energy splitting becomes

$$E_{\mathrm{high}}-E_{\mathrm{low}} =2\sqrt{ K^2+\left(\frac{E_L-E_R}{2}\right)^2 }.$$

If $\lvert E_L-E_R\rvert\gg K$, the eigenstates are mostly localized in one well. Small tunneling coupling cannot overcome a large energy bias. This is the basic reason why controllable two-state systems often tune a bias parameter through an avoided crossing.

Physical Examples

Double-well language appears in many places:

• ammonia inversion, where the nitrogen position relative to the hydrogen plane has two equivalent configurations;
• tunneling splittings in molecular conformations;
• flux and charge qubits, where two macroscopic circuit configurations can form an effective double well;
• symmetric coupled wells in semiconductor and cold-atom settings.

These examples differ in microscopic detail. The shared structure is a pair of nearly degenerate localized states mixed by tunneling.

Common Mistakes

• Treating $\lvert L\rangle$ and $\lvert R\rangle$ as exact stationary states in a symmetric finite double well.
• Forgetting that exact symmetric-well eigenstates have definite parity.
• Confusing the splitting $\Delta E$ with the barrier height.
• Assuming tunneling requires energy above the barrier.
• Applying infinite-wall boundary conditions at the central barrier.
• Ignoring asymmetry; even a small bias can suppress left-right oscillations if it is large compared with $K$.
• Using a WKB exponential estimate outside its semiclassical regime.

Exercises

1. Diagonalize the symmetric two-state Hamiltonian.

Solution

For

$$H_{\mathrm{eff}} = \begin{pmatrix} E_0 & -K \\ -K & E_0 \end{pmatrix},$$

the symmetric vector $(1,1)/\sqrt2$ has eigenvalue

$$E_+=E_0-K,$$

and the antisymmetric vector $(1,-1)/\sqrt2$ has eigenvalue

$$E_-=E_0+K.$$

Thus the splitting is $\Delta E=2K$.

2. A doublet has splitting $\Delta E$. If the particle begins in $\lvert L\rangle$, when is the first maximum probability to find it in $\lvert R\rangle$?

Solution

The right-well probability is

$$P_R(t) =\sin^2\left(\frac{\Delta E\,t}{2\hbar}\right).$$

The first maximum occurs when

$$\frac{\Delta E\,t}{2\hbar}=\frac{\pi}{2}.$$

Therefore

$$t=\frac{\pi\hbar}{\Delta E}.$$

3. Explain qualitatively why increasing the particle mass reduces the tunneling splitting.

Solution

In the forbidden region, the WKB decay rate contains

$$\sqrt{2m(V(x)-E)}.$$

A larger mass increases the decay rate under the barrier, reducing the overlap between left and right localized states. Since the splitting is controlled by this overlap, $\Delta E$ becomes smaller.

References

• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.