- Only Part E Needs Solving The Last Question 1 (72.8 KiB) Viewed 35 times
Only part e needs solving (the last question)
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Only part e needs solving (the last question)
Only part e needs solving (the last question)
3. One-dimensional potential barriers (a) Sketch the one-dimensional potential barrier U() r. 0<<L, 0, otherwise, where I'>0. [2 marks] (b) Consider quantum mechanical particles of energy E incident on the barrier from I= -o and show that the wavefunction may be written as (*) | Aeike + Ae-ikk, I<0, B.eks + B,e-KI, 0<I<L, Celke I>L, where ke2 = 2mE/h2 and K2 = (2m/h?)(T – E). [6 marks) T 2 RE (c) Further, show that the transmission and reflection coefficients are given by C12 (2kk) A (12+)2 sinh"(KL) + (2kk)?' (k+ x2) sinh?(KL) A (k + 1)2 sinh?(L)+(2k: c)? [6 marks) (d) In the case E<T, explain the physical implication of the above result for quantum mechanical particles, and compare to the case of classical particles. Likewise, explain the physical implication in the case E >I and compare to the classical result. [2 marks] (e) The classical result for the transmission and reflection coefficients is recovered from the quantum mechanical result in the limit E»T. It is also recovered for finite specific E>T. Determine the values of E where this occurs in terms of the width L and height l' of the potential. [4 marks)
3. One-dimensional potential barriers (a) Sketch the one-dimensional potential barrier U() r. 0<<L, 0, otherwise, where I'>0. [2 marks] (b) Consider quantum mechanical particles of energy E incident on the barrier from I= -o and show that the wavefunction may be written as (*) | Aeike + Ae-ikk, I<0, B.eks + B,e-KI, 0<I<L, Celke I>L, where ke2 = 2mE/h2 and K2 = (2m/h?)(T – E). [6 marks) T 2 RE (c) Further, show that the transmission and reflection coefficients are given by C12 (2kk) A (12+)2 sinh"(KL) + (2kk)?' (k+ x2) sinh?(KL) A (k + 1)2 sinh?(L)+(2k: c)? [6 marks) (d) In the case E<T, explain the physical implication of the above result for quantum mechanical particles, and compare to the case of classical particles. Likewise, explain the physical implication in the case E >I and compare to the classical result. [2 marks] (e) The classical result for the transmission and reflection coefficients is recovered from the quantum mechanical result in the limit E»T. It is also recovered for finite specific E>T. Determine the values of E where this occurs in terms of the width L and height l' of the potential. [4 marks)
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