HW5 - V 2 , show that T ≤ T 1 , T ≤ T 2 . Offer a...

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PHY4221 Quantum Mechanics I Fall term of 2004 Problem Set No.5 (Due on November 24, 2004) 1. If we define the probability particle density ρ and probability current density J as follows: ρ ( x,t ) = | ψ ( x,t ) | 2 J ( x,t ) = ¯ h 2 mi ( ψ * ( x,t ) ∂x ψ ( x,t ) - ψ ( x,t ) ∂x ψ * ( x,t ) ) , then show that they satisfy the continuity equation: ∂ρ ∂t + ∂J ∂x = 0 . [Note that the above definitions and continuity equation can be generalized to the 3-D case in a straightforward manner.] 2. Show that the reflection coefficients for the two 1-D potential barriers: (a) V ( x ) = ( V 0 , x > 0 0 , x < 0 (b) V ( x ) = ( 0 , x > 0 V 0 , x < 0 are equal. [Note that the incident particle of energy E > V 0 is going from x → -∞ to x → ∞ .] 3. (a) Calculate the transmission coefficient T for the 1-D double potential step: V ( x ) = 0 , x < 0 V 1 , 0 < x < a V 2 , x > a . [Note that the incident particle of energy E > V 2 is going from x → -∞ to x → ∞ .] (b) If we call T 1 the transmission coefficient appropriate to the single potential step V 1 , and T 2 that appropriate to the single potential step
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Unformatted text preview: V 2 , show that T ≤ T 1 , T ≤ T 2 . Offer a physical explanation for these inequalitites. (c) What are the three sets of conditions under which T is maximized? What do these conditions correspond to physically? (d) A student argues that T is the product T 1 T 2 on the following grounds. The particle current that penetrates the V 1 barrier is T 1 J inc . This current is incident on the V 2 barrier so that T 2 ( T 1 J inc ) is the current transmit-ted through the second barrier. What is the incorrect assumption in his argument? 4. Calculate the transmission coefficient T for the 1-D potential barrier: V ( x ) =      , x <-a V ,-a < x < a-αV , x > a . [Note that the incident particle of energy E < V is going from x → -∞ to x → ∞ .] —— 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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HW5 - V 2 , show that T ≤ T 1 , T ≤ T 2 . Offer a...

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