Phy5646/Another example
(Submitted by Team 1)
This example was taken from "Theory and Problems of Quantum Physics", SCHAUM'S OUTLINE SERIES, p. 190-192.
Problem: Consider a one dimensional harmonic oscillator embedded in a uniform electric field. The field can be considered as a small perturbation and depends on time according to
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon(t) = \frac{A}{\sqrt{\pi} \tau} exp [-(\frac{t}{\tau})^2] }
where A is constant. If the oscillator was in ground state until the field was turned on at t=0, compute in the first approximation, the probability of its excitation as a result of the action of the perturbation.
Solution:
The probability of a transition from the state n to the state k is given by
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_{nk} = \frac{1}{\hbar^2} |\int^\infty_\infty <k|V|n> e^{i t (E^{(0)}_k - E^{(0)}_n )/\hbar} dt| }
Let e, m and w denote the charge, mass and natural frequency of the oscillator, respectively, where x denotes its deviation from its equilibrium position. In the case of an uniform field, the perturbation is given by
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(x,t) = -e x \epsilon(t) - x }