### PHYS307 (Intermediate Quantum Mechanics) **Assignment Sheet Fall 2014**
Text: Griffiths __Introduction to Quantum Mechanics__ [second edition

**Assignment 1** (Due Wed 3 Sep)
1. Work out the numberical values for the wavelengths (in nm), frequencies (in THz) and photon energies (in eV) for
alpha line, the beta line, and the series limit, for the Lyman series and the Brackett series of hydrogen.
2. The momentum of the electron in a Hydrogen atom orbit *n* is just *μv*_{n}. Show that the number
of de Broglie wavelengths appropriate to that speed and orbit that can be 'wrapped around' the circumference of that orbit (2*πr*_{n} is just
*n*.
3. We have made use of two handy series sum formulas when working out the black body results. Show how the second one of them
can be obtained from the first one by taking *x* times *d/dx* on the first one: *x d/dx*(Σ *x*^{n}) =
(Σ *n x*^{n}. So, the same thing can be done to what the series actually sum to, as you should verify too.
[Hint: the derivative can be pushed under the summation.. those operations 'commute']
4. From the Planck formula for the energy distribution function *ε*(*f*), derive Wien's law, which states the the ratio of the temperature
to the frequency at which the distribution reaches a maximum is a constant (or, the product of the temperature times the wavelength of the peak is a
constant). For either version, you do *not* have to evaluate the constant, which is nasty!
Solutions to assgt 1
**Assignment 2** (Due Mon 15 Sep)
1. Griffiths 1.4
2. Grittiths 1.7 -- and what is the 'classical' version of this statement?
3. For the 1*d* infinite square well, show that <*x*> = *a*/2, by integration by parts.
4. For the 1*d* infinite square well, show that <*p*> = 0, by using the momentum operator and doing the appropriate integral.
5. Griffiths Ch 2.5
6. Griffiths Ch 2.7 [for (b), find an expression for the expansion coefficients {*c*_{n}}, knowledge of which is equivalent to
knowing what &Psi(*x,t*) is; for (d), is it possible to actually "do" the required summation? Read example 2.3]
Solutions to assgt 2
**Assignment 3** (Due Wed 24 Sep)
1. Griffiths 2.10c [you can write them down from table 2.1 on page 56 and using eq 2.85; note which ones are even about the origin and which are odd
about the origin; the hint in the problem means that you only have to integrate one possibility; then pick up outside my office the guide sheet
for Gaussian integrals and apply the relevant formulas!]
2. 2. Suppose that, for the finite square well, the potential height *V*_{0}
is the minimum for there to be three bound states (in lecture, we did the case for just barely two bound states).
From this, find the values of *E*_{1} and *E*_{2} as a multiple of *V*_{0}.
Also, find the values of *ka* for the two states and compare to the values for the infinite square well of the same width.
Just for definiteness, take *a* = 1 nm and *m* = 10-30 kg, and take *V*_{0} = 1.0 eV.
3. Griffiths 2.11 [to do (c), use your results from (a)]
4. Griffiths 2.13abd [the HO wavefunctions are *orthonormal*, so (a) is easy!]
5. Griffiths 2.16
Solutions to assgt 3
**Assignment 4** (Due Wed 1 Oct)
1. Griffiths 2.18 [this will increase your comfort level with the two equivalent forms; all relies on the wonderful Euler formula (see http://en.wikipedia.org/wiki/Euler%27s_formula)]
2. read Griffiths 2.20 and 2.22 and compare to lecture notes
3. Griffiths 2.21 [you will need to express the wave function as *e*^{-ax} for *x* > 0, and as *e*^{ax} for *x* < 0]
4. Griffiths 2.44 [this one is very cool and not hard, really... note that the odd solutions are not affected by the delta function and so are exactly the
infinite square well solutions, while the even solutions require more work]
Solutions to assgt 4
**Assignment 5** (Due Mon 20 Oct)
1. Griffiths 4.2 [you *can* do it and you *should* do it. You might want to work in a group as you sort out the energies of the first 14 or so
energy eigenvalues! You will have noticed that if the box is not a cube, but a rectangular piped (shoebox) of three *differing*
dimensions, there are fewer to no opportunities for degeneracy to kick in]
2. Griffiths 4.3 [just to get you familiar with the mathematics of the spherical harmonics]
3. Griffiths 4.4
**Assignment 6** (Due Mon 27 Oct)
1. Using the expressions for the operators *L*_{z} and *L*^{2} in spherical coordinates, by acting on some function (to keep
track of the derivatives using the product rule) *f*(*θ*,*φ*), show that the commutator of the those two operators is zero.
2. Griffiths 4.18
3. Read the book to see equations 4.130. You will note that they are adjoints of one another, in that the sign of *i* is flipped, *and* the sign
of the derivatives are flipped too [the old partial integration trick]. Then do Griffiths 4.21
4. Griffiths 4.23
5. Griffiths 4.13. You may use the web to look up the required integrals, of course.
**Assignment 7** (Due Wed 12 Nov)
1. Griffiths 4.26
2. Griffiths 4.27
3. Griffiths 4.29 (the eigenvalues and eigenspinors are in the notes)
4. Griffiths 4.31