Quantum double well — tunnelling, doublets and instantons

Double well — tunnelling and doublets

Two minima separated by a barrier: the low-lying eigenstates organize into near-degenerate doublets, with a splitting exponentially small in the barrier height — the model of quantum tunnelling (ammonia inversion, conformational transitions, double-well qubits). There is no closed form: the spectrum is computed by finite differences in the Python backend (via gw2py); the interface shows in 3D the complex wave function and the coherent oscillation of the density between the two wells.

Two degenerate wells separated by a barrier of height \(B\): states come in near-degenerate pairs (symmetric and antisymmetric) whose splitting \(\Delta\) is exponentially small in the barrier. It is the paradigm of quantum tunnelling.

Double-well potential

\[ V(x)=B\left(\frac{x^2}{x_0^2}-1\right)^2, \]

two degenerate minima at \(x=\pm x_0\) (depth 0) separated by a barrier of height \(B\) at \(x=0\).

Tunnelling doublets

The low-lying eigenstates come in near-degenerate pairs: a symmetric combination \(\psi_+\) and an antisymmetric one \(\psi_-\), separated by

\[ \Delta = E_- - E_+ \sim \hbar\omega\,e^{-S_0/\hbar}, \]

with \(S_0\) the instanton action (an imaginary-time trajectory between the two wells). The barrier controls tunnelling: the higher it is, the smaller \(\Delta\).

Starting from a state localized in one well (superposition \(\tfrac{1}{\sqrt2}(\psi_++\psi_-)\)), the density \(|\Psi|^2\) oscillates between the two wells with period \(T=2\pi/\Delta\): coherent tunnelling.

Where it appears

Inversion of the ammonia molecule (NH\(_3\)), conformational transitions, and double-well qubits: the physics is always the doublet and its splitting.

State to represent

State type Level \(n_1\)

Level \(n_2\)

The \((0,1)\) doublet shows coherent tunnelling between the two wells.

Potential parameters

Barrier height \(B\) = 6

Higher barriers ⇒ ever tighter doublets (tunnelling suppressed).

Half-separation of the wells \(x_0\) = 1.6

Dimensionless units (\(\hbar=m=1\)). The spectrum is computed by finite differences in the Python backend.

Confluent Heun

The quartic double well belongs to the confluent Heun class — the same as the Rabi model. As for the anharmonic oscillator, there is no general closed form: the splitting must be computed (numerically, or via instantons in the high-barrier regime).

Instantons

The exponential splitting \(\Delta\sim e^{-S_0}\) is the non-perturbative result par excellence: invisible at every order of perturbation theory, it emerges from the classical imaginary-time trajectories (instantons). It is the bridge between the double well and field theory.

Computational evidence

The finite-difference solver resolves the near-degenerate doublets thanks to re-orthogonalization (orthonormality \(\sim10^{-15}\) even when \(\Delta\ll1\)). Increasing \(B\), \(\Delta=E_1-E_0\) decreases, as expected.

References

S. Coleman, Aspects of Symmetry, Cambridge Univ. Press, 1985 (ch. «The uses of instantons»).
C. M. Bender, T. T. Wu, Phys. Rev. D 7, 1620 (1973). doi.
NIST DLMF, ch. 31 (Heun functions). dlmf.nist.gov/31.
F. Cooper, A. Khare, U. Sukhatme, Phys. Rep. 251, 267 (1995). doi.

WebNIR · CNR-IFAC | demo interface — numerical work is provided by the Python backend (gw2py).

Keywords: double well, tunnelling, doublets, instantons, confluent Heun, ammonia inversion, finite differences, Schrödinger, quantum mechanics