Test 3

 

This is a Take-Home Test to be taken under the conditions specified in Hewett's Take-home Exam Instructions sheet. Furthermore, this test is an Answer-Only test. Please submit your answers to me by e-mail by Wednesday 28, 2009.

 

1.         Which of the following waveforms are possible solutions to Schrödinger’s wave equation for the indicated potential functions U and the indicated energy levels. In each case, the respective energy level line is used as the x-axis for the respective waveform. En indicates the n'th allowed energy level while E without any subscript indicates an energy level with unspecified quantum number n. If you feel that a particular wave function is not possible, state why.

 

 

2.         Given the following functions:
(a) y(x)  =  A  cos (kx)

(b) y(x)  =  A sin (kx) - A cos (kx)

(c) y(x)  =  A cos (kx) + iA sin (kx)

(d) y(x)  =  A eik(x-a)

(e) y(x)  =  A  d(x-xo)

2.1       Which are eigen functions of the position operator [x]?

2.2       Which are eigen functions of the momentum operator [p]?

2.3       Which are eigen functions of the potential energy operator [U=0]?

2.4       Which are eigen functions of the kinetic energy operator [K]?

2.5       Which are eigen functions of the Hamiltonian operator [H]?

2.6       Which are eigen functions of the total energy operator [E]?

2.7       What are the eigenvalues of function (a), where appropriate?

2.8       What are the eigenvalues of function (b), where appropriate?

2.9       What are the eigenvalues of function (c), where appropriate?

2.10     What are the eigenvalues of function (d), where appropriate?

2.11     What are the eigenvalues of function (e), where appropriate?

 

 

3.         A free electron has a wavefunction y(x)  =  A sin (kx), where k = 2x1010 /m.

3.1       What is the electron's de Broglie wavelength in meters?

3.2       What is the electron's momentum in kg.m/s?

3.3       What is the electron's energy in electron volts?

 

4.         A quantum oscillator has a potential energy function U(x) = mw2x2, where m is the mass of the particle, w is the classical frequency of vibration, and x is the displacement from equilibrium.

4.1       For what value of a does the wavefunction  satisfy Schrödinger’s wave equation for this oscillator?

4.2       What is the energy of the state corresponding to this wave function?

4.3       Using the fact that    a > 0, determine the normalization constant C.

 

5.         A ruby laser emits light of wavelength 694.3 nm. If this light is due to transitions from the n = 2 state to the n = 1 state of an electron in a box, find the width of the box.

 

6.         A particle of mass m is located in a box of length L and found to be in its ground state.

6.1       What is the probability of finding the particle between x=0 and x=L/4?

6.2       What is the expectation value for the position of the particle?

6.3       What is the expectation value <x2>?

6.4       What is the uncertainty in the position of the particle?

6.5       What is the kinetic energy of the particle?

6.6       What are the two possible values of the momentum for the particle?

6.7       What is the expectation value for the momentum of the particle?

6.8       What is the expectation value <p2>?

6.9       What is the uncertainty in the momentum of the particle?

6.10     What is the uncertainty product Dx Dp?

6.11     How does this result compare with uncertainty principle?

 

7.         Consider a particle in a superposition of states given at time t = 0 by Y(x,0) = C(y1(x) + y2(x)), where y1 (x) and y2(x) are the stationary states with energies E1 and E2, respectively.

7.1       If y1(x) and y2(x) are orthonormalized, what value of C is required to normalize Y(x,0)?

7.2       What is the equation for Y(x,t) at a later time?

7.3       Is Y(x,t) an eigenfunction of the energy operator [E]?

7.4       Is Y(x,t)  a stationary state?

7.5       What is the expectation value for the energy <E>?

7.6       Does this expectation value change with time?

 

 

8.         A particle of mass m and energy E is incident from the left upon a potential step function of height U = 3/4 E as shown at right. Assume that the incident wave has an amplitude A, the reflected wave has an amplitude B, the transmitted wave has an amplitude C, the wave on the left side of the step has a wave number k1, the wave on the right side has a wave number k2, and the frequency of the wave is w.

 

(a) Write the equation giving Y(x,t) on the right side of the step in terms of amplitudes, wave numbers, and frequency.

 

(b) Write the equation giving Y(x,t) on the left side of the step in terms of amplitudes, wave numbers, and frequency.

 

(c) Find the values of k1 and k2 in terms of m and E.

 

(d) What are the equations relating A, B, and C that result from the boundary conditions at the position of the step (i.e. at x=0).

 

(e) Solve the above equations for the numerical values of the amplitude ratios B/A and C/A. (Hint: Eliminate all variables but these two ratios.)

 

(f) What is the reflection coefficient for this wave function?

 

(g) Because the kinetic energy of the particle when on the right is only 1/4 of its kinetic energy when on the left, the transmission coefficient is not simply T = (C/A)2 but rather T = K(C/A)2. What are the correct values for T and K for this wave function? (Hint: First use the reflection coefficient determined above to find the correct transmission coefficient.)