HW4 - would be found outside the well ( x > a )? 3. (a)...

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PHY4221 Quantum Mechanics I Fall term of 2004 Problem Set No.4 (Due on November 13, 2004) 1. Use the Heisenberg’s equation of motion to show that d dt h xp i = 2 h T i - h x dV dx i , where T and V are the kinetic energy operator and potential energy operator associated with the Hamiltonian H = T + V , respectively. Note that the expec- tation values are evaluated with respect to an arbitrary state. In a stationary state the left side is zero (why?) so 2 h T i = -h x dV dx i . This is called the virial theorem. Use it to prove that h T i = h V i for stationary states of the harmonic oscillator. 2. A particle of mass m is in the potential V ( x ) = 0 , x > a - 32¯ h 2 ma 2 , 0 x a , x < 0 . (a) How many bound states are there? (b) In the highest-energy bound state, what is the probability that the particle
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Unformatted text preview: would be found outside the well ( x > a )? 3. (a) Show that Ψ( x,t ) = ± mω π ¯ h ¶ 1 / 4 exp (-mω 2¯ h " x 2 + a 2 2 ‡ 1 + e-i 2 ωt · + i ¯ ht m-2 axe-iωt #) satisfies the time-dependent Schr¨odinger equation for the harmonic oscil-lator potential. Here a is any real constant with the dimensions of length. (b) Find | Ψ ( x,t ) | 2 , and describe the motion of the wave packet. (c) Compute h x i and h p i , and check that Ehrenfest’s theorem is satisfied: m d dt h x i = h p i and d dt h p i =-h dV dx i . —— End ——...
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This note was uploaded on 12/10/2011 for the course PHYS 4221 taught by Professor Cflo during the Spring '11 term at CUHK.

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