This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Physics 3220 Quantum Mechanics 1 Fall 2008 Problem Set #6 Due Wednesday, Oct 8 at 2pm Problem 6.1 Gaussian wave packets: part 1 [20 pts] Free particles are often modeled as "wave packets", meaning we start with an initial Gaussian wave function, ( x , t = 0) = Ae ax 2 , with A and "a" both real and positive constants. a) Normalize your (x,t=0) and then calculate the Fourier transform (k) for this wavefunction. Hint: Old HW 2.4a will help with normalization. Griffiths Eq 2.103 will help with details of Fourier transforms. There's a clever trick to do integrals of the form e ( ax 2 + bx ) dx , called "completing the square": let y a x + b 2 a and note that ( ax 2 + bx ) = y 2 b 2 4 a . (Look again at HW 2.4 for the handy Gaussian integral!) b) Now calculate the time dependent wavefunction (x,t) Answer: ( x , t ) = 2 a p 1/ 4 e ax 2 /(1+( i W t )) 1+ i W t , with 2 a h / m c) Find the probability density ( x , t ) 2 . Please simplify your result by expressing it in terms of a new quantity w a 1+ (W t ) 2 d) Find x , and p as functions of time. Problem 6.2 Gaussian wave packet  part 2. [20 pts] a) Using your results from the previous problem, find x 2 , p 2 , and then s x , and s p (all as functions of time) Discuss the Heisenberg uncertainty principle for this problem. Does it work? When is the system closest to the lower limit? Note: Old HW#3.3 might help with the x 2 integral. The p 2 algebra is trickier! I get p 2 = a h 2 , do you?...
View
Full
Document
This note was uploaded on 02/27/2012 for the course PHYSICS 3220 taught by Professor Stevepollock during the Fall '08 term at Colorado.
 Fall '08
 STEVEPOLLOCK
 mechanics

Click to edit the document details