p137bp6sol

# p137bp6sol - Physics 137B Fall 2007 Moore Problem Set 6...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Physics 137B, Fall 2007, Moore Problem Set 6 Solutions 1. For t < 0 the particle is in the stationary state ψ ( t,x ) = r 2 L e- iE t/ ¯ h sin πx L . Since this is a box potential, (i.e. an infinite square well), the wavefunction vanishes outside of the interval 0 ≤ x ≤ L . At t = 0 the Hamiltonian changes abruptly to the form of a shifted harmonic oscillator, H = p 2 2 m + 1 2 k x- L 2 2 . The ground state and first excited state of this Hamiltonian are φ ( x ) = mω π ¯ h 1 / 4 e- mω ( x- L/ 2) 2 / 2¯ h , φ 1 ( x ) = mω π ¯ h 1 / 4 2 mω ¯ h 1 / 2 x- L 2 e- mω ( x- L/ 2) 2 / 2¯ h . To find the probability of observing the particle in either of these states, we need to calculate the overlap with the wavefunction at t = 0. (a) For the ground state we have d ≡ h φ | ψ (0) i = Z ∞-∞ dxφ * ( x ) ψ (0 ,x ) 1 = r 2 L mω π ¯ h 1 / 4 Z L dx sin πx L e- mω ( x- L/ 2) 2 / 2¯ h = r 2 L mω π ¯ h 1 / 4 L π Z π/ 2- π/ 2 du sin( u + π 2 ) e- mωL 2 u 2 / 2 π 2 ¯ h ( u = π L ( x- L 2 ) ) = r 2 L mω π ¯ h 1 / 4 L π Z π/ 2- π/ 2 du cos ue- mωL 2 u 2 / 2 π 2 ¯ h ....
View Full Document

## This note was uploaded on 08/01/2008 for the course PHYSICS 137B taught by Professor Moore during the Fall '07 term at Berkeley.

### Page1 / 5

p137bp6sol - Physics 137B Fall 2007 Moore Problem Set 6...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online