Phys 2203 Homework 9, due Wednesday November 6.
The homework consists of a mix of problems from the textbook (at the back of each chapter), and
some extra problems typed out below.
(1) This problem involves looking at the spherical harmonic functions like we did in class Oct 23.
The applet we used can be found at: http://www.falstad.com/qmatom/ You should choose to look at
the Complex Orbitals (Physics) in the first selection box. If that doesn’t work you can see smaller
images of the spherical harmonics here (look at the complex functions on the right hand side, not the
real functions on the left hand side): http://oak.ucc.nau.edu/jws8/dpgraph/Yellm.html
(a) Start by sketching or describing what classical motion gives an angular momentum vector
that points in the positive z direction.
(b) Now think about the space quantization picture we have of how the orientation of the
quantum mechanical L-vector is quantized with the different values of the ml quantum
number. Draw such a picture for  = 1,  = 2, and  = 3.
(c) Which ml quantum number (for a given ) gives rise to the “most vertical” angular
momentum vector?
(d) Look at the spherical harmonics Y11(θ,φ),Y22(θ,φ), and Y33(θ,φ). Are your answers in (a), (b),
and (c) consistent with what you see?
(2) Consider a hydrogen atom in the 5f sub-shell. In this problem we will not consider electron spin.
(a) How many different states have the same energy in this sub-shell?
(b) What is the energy of all the 5f states
(c) Now we expose the atom to a 3T magnetic field. Write down the new energies of all the 5f
states in the magnetic field.
(3) Show that for transitions between any two states of hydrogen, no more than three different
spectral lines can be obtained for the (normal) Zeeman effect (ignore spin). (Pick some specific
example to work from).
(4) Are the following transitions allowed, between hydrogen states (n, l, ml)
(a) (5,2,1) -> (5,2,0)
(b) (4,3,0) -> (4,2,1)
(c) (5,2,-2) -> (1,0,0)
(d) (2,1,1) -> (4,2,1)
(5) Including spin, show that the degeneracy for energy level n in hydrogen is 2n2. (Hint: use the
series summation 1+3+5+…+(2n-1) = n2.)
(6) Problem 8.4
(7) Problem 8.6
(8) Problem 8.8
(9) Problem 8.15(b) and (c) only
(10) Problem 8.16