We have derived approximate expressions for the change in...

Name: Problem 1, Chapter 12: The Physics of Quantum Mechanics
Uploaded: 2022-01-17T05:54:58-08:00
Duration: 1 min 19 s
Channel: Amita Prajapat
Description: We have derived approximate expressions for the change in the energies of stationary states when an electric or magnetic field is applied. Discuss whether the derivation of these results implicitly assumed the validity of the adiabatic principle.

We have derived approximate expressions for the change in the energies of stationary states when an electric or magnetic field is applied. Discuss whether the derivation of these results implicitly assumed the validity of the adiabatic principle.

Key Concepts

Stationary States

Stationary states are the eigenstates of the unperturbed Hamiltonian which have well-defined energies that do not change with time. They form the basis for analyzing how a system responds to external perturbations, such as electric or magnetic fields, and are central to understanding energy level shifts in quantum mechanics.

Perturbation Theory

Perturbation theory is a framework used to calculate small corrections to the energy levels and states of a quantum system when it is subject to an external or internal disturbance. In this context, it provides approximate expressions for energy changes when fields are applied, under the assumption that the perturbation is sufficiently weak so that the system's behavior is close to that predicted by the unperturbed system.

Adiabatic Principle

The adiabatic principle (or adiabatic theorem) states that if a quantum system’s Hamiltonian is altered slowly enough, the system remains in its instantaneous eigenstate, with its quantum numbers preserved. This principle underpins many approaches that assume a gradual change in external conditions, ensuring that the system’s transition probabilities between different states are negligible.

Implicit Adiabatic Assumptions

When deriving approximate expressions for energy shifts under the influence of electric or magnetic fields, it is often assumed implicitly that the external fields are turned on slowly compared to the natural timescales of the system. This slow variation ensures that the system can adjust adiabatically, justifying the use of stationary state perturbation theory even as the Hamiltonian changes.

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