Work out an approximation to the energy splitting between...

Name: Problem 8, Chapter 8: Quantum mechanics
Uploaded: 2021-11-30T20:24:28-08:00
Duration: 4 min 19 s
Channel: Mahnoor Amin
Description: Work out an approximation to the energy splitting between the second and third excited levels of the double oscillator defined in Section 8.5, assuming the distance between the wells to be very large compared with the classical amplitude of the zero-point vibrations.

Work out an approximation to the energy splitting between the second and third excited levels of the double oscillator defined in Section 8.5, assuming the distance between the wells to be very large compared with the classical amplitude of the zero-point vibrations.

Key Concepts

Quantum Tunneling

Quantum tunneling is the phenomenon where a particle can pass through a potential barrier even if its energy is less than the height of the barrier. In a double-well potential, tunneling allows the particle to transition between the two wells, leading to a mixing of the otherwise degenerate energy states and resulting in a splitting of these levels.

Energy Level Splitting

Energy level splitting in a double-well system arises due to the overlap of the localized wave functions in each well caused by tunneling. When the wells are weakly coupled (for example, when they are far apart compared to the scale of zero-point vibrations), the symmetric and antisymmetric combinations of the localized states have slightly different energies, hence the splitting between otherwise degenerate levels.

WKB Approximation

The Wentzel-Kramers-Brillouin (WKB) approximation is a semi-classical method used to estimate tunneling probabilities and energy splittings in quantum systems. In the context of a double-well potential, the WKB method helps to evaluate the exponentially small overlap between the states in the separate wells when the barrier is high or the wells are widely separated.

Zero-Point Vibrations

Zero-point vibrations refer to the inherent quantum fluctuations in a system's lowest energy state. These small oscillations around the minimum of a potential well set the scale for the spatial extent of a particle's wave function. In the double-well context, the magnitude of zero-point vibrations relative to the distance between the wells determines the degree of wave function overlap and thus influences the tunneling amplitude and energy splitting.

Double-Well Potential

A double-well potential is a system in which the potential energy has two nearly degenerate minima separated by a barrier. This setup is frequently used to model systems where a particle can be localized in either of the two wells, providing the framework for studying phenomena such as quantum tunneling and the resulting energy level splitting.

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