Delta Potential - Bound and Continuum States

  • #1
1,734
13

Homework Statement




I am studying my lecturer's notes and in this part he uses a delta potential to illustrate a simple example of Fermi's golden rule, that the rate of excitation is ##\propto t##.


2s8jkn5.png


Homework Equations





The Attempt at a Solution



I've managed to get the bound states, by solving boundary conditions. They turn out to be ##\frac{\sqrt{Wm}}{\hbar}exp(-\frac {mW|x|}{\hbar^2})##.

How are the continuum states derived? And why do only odd parity states contribute, when there is a perturbation ##-Fxe^{-i\omega t}##?
 

Answers and Replies

  • #2
TSny
Homework Helper
Gold Member
13,159
3,458
The continuum states can be derived is a similar manner to the bound state. You can assume the solutions are either even or odd functions. Start by thinking about the general form of the solutions for E>0 for the regions x>0 and x<0.

Fermi's golden rule contains a matrix element of the perturbation function between the initial and final states. Consider the overall parity of the integrand that expresses the matrix element.
 
  • #3
1,734
13
The continuum states can be derived is a similar manner to the bound state. You can assume the solutions are either even or odd functions. Start by thinking about the general form of the solutions for E>0 for the regions x>0 and x<0.

Fermi's golden rule contains a matrix element of the perturbation function between the initial and final states. Consider the overall parity of the integrand that expresses the matrix element.

The matrix element in the integral is: ##|<p|V|i>|^2 = \left(\int \psi_p^* V \psi_i\right)^2##

If ##\psi_p## is odd, multiplied by ##V## which is odd, together they form an even function so the integral ##\int_{-\infty}^{\infty} = 2\int_0^{\infty}## which gives a non-zero value.
 
  • #4
TSny
Homework Helper
Gold Member
13,159
3,458
That's right. So the odd ##\psi_p## will generally have a nonzero matrix element. Similarly, you can show that the even ##\psi_p## must yield zero for the matrix element.
 

Related Threads on Delta Potential - Bound and Continuum States

Replies
19
Views
279
Replies
1
Views
1K
Replies
8
Views
6K
Replies
1
Views
6K
  • Last Post
Replies
8
Views
2K
Replies
0
Views
825
Replies
4
Views
684
  • Last Post
Replies
1
Views
1K
Replies
2
Views
2K
Top