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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 ——...
View Full Document

## This note was uploaded on 12/10/2011 for the course PHYS 4221 taught by Professor Cflo during the Spring '11 term at CUHK.

### Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online