Fourth Readings and Problems.
Reading:
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For Wednesday, February 23, read Chapter 6 in Shankar. |
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For Thursday, February 24, read Sections 7.1-7.3 in Shankar. |
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For Monday, February 28, finish reading Chapter 7 in Shankar. |
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For Thursday, March 3, read Chapter 8 in Shankar. |
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For Monday, March 14, read Chapter 9 and Sections 10.1-10.2 in Shankar. |
Homework Problems:
[61] Is the operator
Hermitian?[62] Assuming that
is a real constant, it the operator 
Hermitian?[63] Calculate the following commutators: (i)
, (ii) 
, (iii) 
[64] An infinite square potential well has sides at x=0 and x=2a. The potential within the well is V=0. At a certain moment in time, the spatial wavefunction of a particle is given by:
, for x values between 0 and a, and by 
for x values between a and 2a. Obviously, a is a real positive distance here. The spatial wavefunction vanishes everywhere outside of the well.Determine the probability of finding this particle to have the ground state (lowest) energy at this time.
[65] Exercise 7.3.7, p.202, Shankar.
[66] Exercise 7.4.1, p. 212, Shankar.
[67] Exercise 7.4.2, p. 212, Shankar.
[68] Exercise 7.4.3, p. 212, Shankar.
[69] Exercise 7.4.4, p. 212, Shankar.
[70] A particle is in a state described by the momentum space wavefunction
, where A and â are constants. Determine whether or not this state is an eigenstate of the position operator.[71] A particle of energy E=(4V0/3) and mass m is incident upon a potential step from the left. The potential step occurs at x=0, and at this point, the potential steps down from its value of V=V0 on the left-hand side of the step to its value of V=0 on the right-hand side of the step.
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If the spatial wavefunction on the left-hand side of the step is given by the expression  , write an expression for k in terms of the parameters m, V0, and hbar. |
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Write the form of the wavefunction on the right side of the step,  , making sure to identify any constants you might use to write it. |
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Determine the reflection coefficient, R, and the transmission coefficient, T, for the particle when incident upon the step from the left. |
[72] Do the multiple choice question attached.
[73] Exercise 7.4.5, p. 212, Shankar.
[74] Exercise 7.4.7, p. 212, Shankar.
[75] Exercise 7.4.8, p. 213, Shankar.
[76] Exercise 7.5.2, p. 218, Shankar.
[77] Exercise 9.4.1, p. 244, Shankar.
[78] Exercise 9.4.2, p. 244, Shankar.
[79] Exercise 9.4.3, p. 244, Shankar.
[80] Exercise 9.4.4, p. 244, Shankar.