# HW5 - δ k 1-k 1 for the E> V stationary states of the...

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Physics 731 Assignment #5, due Wednesday, October 14 1. Sakurai, Problem 2, Chapter 2. 2. Sakurai, Problem 3, Chapter 2. 3. Sakurai, Problem 8, Chapter 2. 4. Verify by explicitly doing the appropriate integration that for the free particle in one dimension, an initial state Gaussian wavepacket of the form ψ ( x, 0) = 1 π 1 / 4 d e ik 0 x - x 2 / (2 d 2 ) (1) evolves into ψ ( x,t ) = 1 π 1 / 4 ± 1 d + i ¯ ht/ ( md ) ² 1 / 2 e ik 0 ( x - ¯ hk 0 t/ (2 m )) e - ( x - ¯ hk 0 t/m ) 2 / (2 d 2 (1+ i ¯ ht/ ( md 2 ))) , (2) and using this result, evaluate | ψ ( x,t ) | 2 . 5. Consider a particle subject to a one-dimensional constant force F . (a) Show that h p | U ( t, 0) | p 0 i = δ ( p - p 0 - Ft ) e i ( p 0 3 - p 3 ) / (6 m ¯ hF ) . (3) (b) Then, show that K ( x,t ; x 0 , 0) = ³ m 2 π ¯ hit ´ 1 / 2 exp µ i ¯ h ± m ( x - x 0 ) 2 2 t + 1 2 Ft ( x + x 0 ) - F 2 t 3 24 m ²¶ . (4) 6. Prove that h k 1 | k 0 1 i =
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Unformatted text preview: δ ( k 1-k 1 ) for the E > V stationary states of the step function potential: ψ k 1 ( x ) = 1 √ 2 π µ θ (-x ) ± e ik 1 x + k 1-k 2 k 1 + k 2 e-ik 1 x ² + θ ( x ) 2 k 1 k 1 + k 2 e ik 2 x ¶ , (5) where k 1 = √ 2 mE/ ¯ h , k 2 = p 2 m ( E-V ) / ¯ h . Hint: Z ∞ e ikx dx = lim α → Z ∞ e ikx-αx dx = lim α → µ α α 2 + k 2 + ik α 2 + k 2 ¶ = πδ ( k ) + i k . (6) 7. Calculate R and T as function of k for the potential V ( x ) = λδ ( x ) , with λ > . 1...
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