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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, Ψ ( x , t = 0) = Ae- ax 2 , with A and "a" both real and positive constants. a) Normalize your Ψ (x,t=0) and then calculate the Fourier transform φ(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 Ψ (x,t) Answer: Ψ ( x , t ) = 2 a p 1/ 4 e- ax 2 /(1+( i W t )) 1+ i W t , with Ω 2 a h / m c) Find the probability density Ψ ( 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?...
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- Fall '08
- mechanics, Old HW, Gaussian Wave Packets, time dependent wavefunction, Problem 6.2Gaussian wave