Instantons in Quantum Mechanics, Double Well Potential and Quantum Decay Rate

Proseminar

Supervisor: Dr. Maciej Janusz-Koch, ETH Zürich

Summary: We showed how to calculate the energy splitting between the two classical states of a symmetric double well potential using the instanton method, and evaluated the formula for quantum decay rate at zero temperature.

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Introduction:

Quantum Physics predicts that even if the energy of the particle is smaller that the potential well’s height, the particle will have a non-zero probability to cross the other side of the barrier. This process is known as quantum tunnelling. An important consequence of quantum tunnelling is that the ground states mix causing a shift in the energy levels. The in

stanton method can be used to calculate the transition probability of a particle tunnelling through a potential barrier as well as these energy shifts. This report treats the case of a symmetric double well potential, for which the instanton method provides solutions to the the equations of motion in the semiclassical approximation on a Euclidean spacetime. We first show how to calculate the energy splitting between the two classical states of a double well potential. Subsequently, we evaluate the formula for quantum decay rate, at zero temperature.

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Discussion

We considered a symmetric double well potential (s. image)

with two classical ground states with potential

where x̃ = x − a.

In the first part we have provided a method for calculating the energy splitting for the symmetric double well potential in the semiclassical approximation, namely the instanton method. We have provided the mathematical tools in order to understand some basic knowledge of the Euclidean theory and the Feynman path integral formalism, which we used in the instanton method. As a result, we have obtained the formula for the energy splitting

which is non-perturbative and cannot be thus obtained with standard perturbation theory. In the second part of the report, we have treated quantum mechanical tunnelling from a false vacuum (local minimum) to a true vacuum (ground state) in the semiclassical approximation. We have determined the formula for the tunnelling rate

which depends on the bounce action S_0, and which more precisely tells us that the probability of remaining in the false vacuum will decay with time.

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(Full Project Report available upon request)