(no subject)
My Modern Physics take-home assignment:
Section I:
1. (8 pts) Determine the expectation for r for the (3,2,1) state [(n, l, ml) = short-hand notation]. Explain the difference between the most probable value for r and the expected value of r.
2. (8 pts) The most probable value for r can be found by taking dP/dr and looking for the maximum. For the (3,2,1) state, determine the most probable value for r. Explain the difference between the most probable value for r and the expected value of r.
3. (8 pts) Verify that the normalization constant for the 2p radial wavefunction is the same as the one listed in Table 6.1 of Beiser.
4. (8 pts) Determine the probability that an electron in the 2p state is found between 8ao and 10ao.
Section II:
5. (10 pts) Verify that the radial wavefunction for the hydrogen atom for the 3p state is a solution to TISE. Refer to Table 6.1 from Beiser for the appropriate radial wavefuction.
6. (10 pts) a) Starting with the transmission probability given in problem, determine T if aL >> 1, where a = sqrt(2m(Uo - E))L/h_bar. This would be what we called “Tapprox” discussed during class.
b) What are the conditions on E and Uo to obtain Tcrude ~ e-2aL from Tapprox?
7. (10 pts) Using another reference besides the course text (ie, Beiser), solve the Time-Independent Schrodinger Equation (TISE) for the Harmonic Oscillator in the ground state. Your solution should be designed to serve as a study guide for someone else in the course AND should help explain why other solutions (ie, the Hermite polynomials) are needed to explain higher energy states.
8. (10 pts) Equation (5.72) from Beiser provides the format for generating the wavefunctions for different states of the harmonic oscillator.
a) For the n = 3 state, verify that that the wavefunction for n = 3 satisfies TISE.
b) Discuss the similarities and differences of the energy levels for the harmonic oscillator with those from the hydrogen atom and from the infinite square well.
9. (10 pts) In class, we examined the locations in lab where one could possibly measure the split in 450 nm emission of some atom placed in a 0.300 T magnetic field. We used a 6000 lines/cm diffraction grating where the interference pattern is located 2.0 m from the grating. We found that the shift in the location of the emissions were on the order of 10 microns, not detectable in our traditional lab setting. However, for the same money, if a 1800 lines/mm grating and 1.30 T magnetic field were obtained, determine the locations of the spectral lines. Could this be observed in one of our labs? Explain. [Note: Even though L remains at 2.0 m, is there a way this could be altered without knocking down laboratory walls?]
Section III:
10. (12 pts) An electron is confined in an infinite well with a ground state energy of 0.10 eV.
What is the width of the well?
What is the probability that the electron would be found in the left-hand third of the well (0 < x < L/3)?
Determine the energy of the first excited state.
If the width of the well were changed to L = 1.0 mm, while keeping the energy at 0.10 eV, determine the principle quantum number for this energy state.
11. (12 pts) For the ground state of the infinite well, the uncertainty in position and the uncertainty in momentum can be found using statistics. For instance, Dx = sqrt( - 2). Using the ground state wavefunction for the infinite well, determine the expectation values , 2,
, and
2. Is the uncertainty principle satisfied?
12. (12 pts) The wavefunction for a particular situation is 2 a3/2 x e-ax for x > 0 and zero elsewhere.
a) Verify that the normalization constant is correct.
b) Sketch the wavefunction.
c) Determine the particle’s most probable position. (Look at where the probability function is maximized, ie, dP/dx = 0, where P = |y(x)2|.
d) What is the probability that the particle would be found between x = 0 and x = 1/a?
e) Calculate the expectation value of the particle’s position.
13. (12 pts) Starting with the boundary conditions for the barrier (tunneling) situation where U = Uo in the range 0 < x < L and zero elsewhere, and the particle energy, E < Uo, determine the transmission coefficient = T = F*F/A*A, where F = amplitude of the wavefunction when x > L and A = initial amplitude. [Note: You should get T = 4 (E/Uo) (1 - E/Uo) {sinh2[sqrt(2m(Uo - E))L/h_bar] + 4 (E/Uo) (1 - E/Uo)}]
Challenge I: Extra Credit (5 pts) Verify that the n = 3, l = 2, and ml = 1 full wavefunction is a solution to Schrodinger’s equation. Refer to Table 6.1 from Beiser for the appropriate full wavefunction.
Challenge II: Extra Credit (6 pts) Research the 2-D infinite square box. Prepare a presentation that could be used by others in the class to find the wavefunctions and energy levels for this situation/system.
This isn't going to be very easy.