Principles of Quantum Mechanics · Problem 21.15

Path Integrals—II — R. Shankar

Physics — Quantum Mechanics · path-integrals-instantons

Problem

Consider a double-well potential

V(x) = \lambda(x^2 - a^2)^2

. (a) Find the instanton solution in imaginary time that connects the two minima

x = \pm a

. (b) Compute the classical action

S_E

of the instanton. (c) Show that the instanton contribution to the tunneling amplitude between the two wells goes as

e^{-S_E/\hbar}

and relate this to the energy splitting between the ground and first excited states.