Unformatted text preview: PHYS851 Quantum Mechanics I, Fall 2008 HOMEWORK ASSIGNMENT 8: 1. [20pts] Evolution of a gaussian wavepacket: Consider the time dependent Gaussian wavefunction ψ (x, t) = x|ψ (t) = √ π σ+ it mσ
−1/2 exp ip0 x− p0 t 2m exp −(x − p0 t/m)2 . 2σ 2 (1 + i t/(mσ 2 )) Show by direct diﬀerentiation that this satisﬁes Schrodinger’s equation for a free particle: i
2 d2 d ψ (x, t) = − ψ (x, t) dt 2m dx2 Then compute |ψ (x, t)|2 and show that it is a normalized gaussian |ψ (x, t)|2 = √
(x−x0 (t)) 1 − σ 2 (t ) e πσ (t) 2 What are the center, x0 (t), and width, σ (t), as functions of time? 2. Consider a baseball with a mass of .14 kg. Assume that it is localized to a size of 10−7 m. Assume that a pitcher throws a fast ball and it is clocked at 96 miles per hour. What does the Heisenberg uncertainty relation tell us that the fundamental quantum limit to the accuracy of this measurement is? In other words, v=(96 mph) ±∆v , where v is the baseball velocity. Compute the quantum lower limit to the uncertainty ∆v in units of miles per hour. Now consider an electron localized also to 10−7 m, also traveling at 96 mph, what is the uncertainty ∆v in this case?
1 3. [20 pts]Consider a particle in a harmonic oscillator potential V (X ) = 2 kX 2 . What are the equations of motion for X and P ? How do they compare with the classical equations for a particle in a harmonic potential? What is the most general solution? What are the equations of motion for X 2 , X P , and P 2 ? Describe how you might go about solving them. 4. [20pts]Redo the previous problem but with a small cubic term added to the potential, i.e. V (X ) = 1 1 2 3 2 kX + 3 bX . Can you form a closed set of equations to solve in this case? ...
View Full Document