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Physics 322: Modern Physics
Spring 2015
Problem Set #9
(Keeping it Real in 3-D, getting to Hydrogen)
Due WEDNESDAY, April 8 in Lecture
ASSUMED READING: Before starting this homework, you should have read
Chapter 7, Sections 1 to 6 of Harris’ Modern Physics. 1. [Harris 7.1 modified] What is a quantum number, and how does it
arise? In answering this question, clarify what it physically represents. 2. [Harris 7.3 tweaked] Consider a 2D infinite well whose sides are of
unequal length with the “horizontal” (x-axis) length being longer than the
“vertical” (y-axis) length.
a. Sketch the probability density – as density of shading – for the
ground state (in other words, darker shades in a sketch mean
higher probability density).
b. There are two likely choices for the next lowest energy. Sketch the
probability density and explain how you know this must be the next
lowest energy. (Focus on the qualitative idea, avoiding unnecessary
reference to calculations). 3. [Harris 7.21 tweaked] An electron is trapped in a (cubical) quantum
dot, in which it is confined to a very small region in all three dimensions.
If the lowest-energy transition is to produce a photon of 450 nm
wavelength, what should be the width of the well (assumed cubic)?
NOTE: You can treat the quantum dot as a 3D infinite well. Also, explain
how you know the transition you picked is the lowest-energy transition. 4. [Harris 7.22] (10 points) Consider a cubic 3D infinite well.
a. How many different wave functions have the same energy as the
one for which (nx, ny, nz) = (5, 1, 1)? HINT: Table 7.1 might help
speed things up.
b. Into how many different energy levels would this level split if the
length of one side (say the z-axis) were increased by 5%?
c. Make a scale diagram, similar to Figure 7.3, illustrating the energy
splitting of the previous degenerate wave functions.
d. Is there any degeneracy left? If so, how might it be “destroyed”? – Page 1! of 3! –
Physics 322: Modern Physics
Spring 2015
5. [Harris 7.37 extended] An electron is in the " ℓ = 3 state of the hydrogen
atom.
a. What possible angles might the angular momentum vector make
with the z- axis? b. What possible angles might the angular momentum vector make
with the x- axis? y-axis? NOTE: To be clear, I am not suggesting
you know the angle between the angular momentum vector and the
z-axis while also knowing the other ones. I am suggesting if you
were to measure along the x- or y- axis instead of the z- axis, what
would you measure? 6. [Harris 7.39 extended] In Section 7.5, " eimℓφ is presented as our
preferred solution to the azimuthal equation, but there is a more general
one that need not violate the smoothness condition, and that in fact covers
not only complex exponentials, but also, with suitable redefinitions of
multiplicative constants, sine and cosine " Φ mℓ (φ ) = Ae+imℓφ +B e −imℓφ
a. Show that the complex square of this function is not, in general,
independent of" φ .
b. What conditions must be met by A and/or B or the probability
density to be rotationally symmetric – that is, independent of " φ ?
c. The original version of this problem in Harris parenthetically notes
“This highlights another reason, besides their being of well-defined
Lz, why we like our preferred solutions.”
Why is rotational
symmetry a desired property for a solution to the wave equation for
a central force? 7. [Harris 7.45 tweaked] An electron is in an n=4 state of the hydrogen
atom.
a. What is its energy (Verify equation 7-14 comes from equation 7-12
and you can use 7-14)?
b. What properties besides energy are quantized, and what values
might be found if these properties were to be measured? 8. [Harris 7.48] Show that the normalization constant " 15 / 32π given in
Table 7.3 for the angular parts of the " ℓ = 2, mℓ = ±2 wave function
15
sin 2 θ e±2iφ is correct. Hint: You are working in
32π
spherical coordinates, so you need the normalization condition given in
equation (7-37). Justify your use of this equation, don’t just use it because
" Θ 2, ±2 (θ ) Φ ±2 (φ ) =
– Page 2! of 3! –
Physics 322: Modern Physics
Spring 2015
I told you to. To “justify” your use, you must answer “why would this
equation be at all appropriate” and “what does an equation need to do to
be a normalization condition?” Hint #2: sin5θ can be written as (1cos2θ)2sinθ, which if you expand, can make for an easier to tackle integral. – Page 3! of 3! –