Homework due PH 401

Due Friday, December 8
Chapter 1: 2, 9 (remember: "What's n? c/l" only counts for light!)
Chapter 2: 2, 4, 15, 20 (If you need Gaussian integral click here)
Additional:  At t=0, a pulsed beam of free electrons (which can be approximated by a minimum uncertainty Gaussian wavefunction) has a speed of 10 km/s and a pulse width of 5 cmd of 10 km/s and a pulse width of 5 cm.
a)  Find expressions for the uncertainty in momentum and the the uncertainty in position at time t.
b)  At what time will the width of the pulse have spread by 10%?

Due Wednesday, December 13
Chapter 3:  5, 7, 8, 14, 25.
Some hints:
<E=<Eop=<Hop=S|An|2En
Integral from 0 to L of:
sin2(npx/L)=L/2
xsin2(npx/L)=L2/4
sin(npx/L)sin(mpx/L)=0, if m is not equal to n
xsin(npx/L)sin(mpx/L)=(L2/p2)[1/(n+m)2 - 1/(n-m)2], if m is not equal to n, and m+n is odd (equals zero if m+n is even)

Due Wednesday, January 10

1.  Given a=2i+3j+2k, and b=3j+4k
a) Find the dot product of the two vectors (any way you like).
b) Write each vector in matrix notation.
c) Write out the explicit matrix multiplication to find the dot product.

2.  A particle in an infinite square well (width=L) is in state |a.  The normalized wavefunction describing that state is <x|a=(2/L)1/2 [0.5sin(px/L)+0.5isin(2px/L)+0.5sin(4px/L)-0.5isin(5px/L)].
a)  What is <x1|a, where x1=0?  (number)
b)  What is <a|x?  (expression)
c)  What is <a|x<x|a? (expression)  What is it's physical meaning?
d)  What is integral over all x of <a|x<x|a?  (number)

3.  Referring to the state described in #2.
a)  What is <E1|a (|E1 is the ground state)?  (number)
b)  What is <a|E1<E1|a?  (number) What is it's physical meaning?
c)  What is the sum over all n of <a|En<En|a?

4.  Referring to the state described in #2.
a)  Express the state as a column matrix in the energy representation.
(at the end you may write 0..., since I don't want you wasting trees writing infinitely many zeroes.)
b) Find the norm by explicit matrix multiplication.

Due Monday, January 15
Chapter 4:  8,
Chapter 5:  12,
Chapter 6:  Exercises 3, 4, problems 3 -- also calculate the expectation values of the kinetic energy and the potential energy for the n=4 state for this molecule, 5.
Some hints:
Ch6E3 use eqns 9 & 10
Ch 6 don't do integrals for expectation values, use a and a-dagger operators

Due Friday, January 26
Chapter 7:  Exercises 4* & 26,
Chapter 7:  Problem 10,
1.  A beam of atoms with l=1 is traveling along the y-axis.  The beam passes through a Stern-Gerlach magnet with its mean magnetic field along the x-axis.

The magnet splits the original beam into three sub-beams: one with Lx-component = -hbar, one with zero, one with +hbar.
(a)  The sub-beam with Lx-component = +hbar is passed through another Stern-Gerlach magnet, this time with the magnetic field in the z-direction.  Into how many beams is the sub-beam further split, and what are the relative numbers of atoms in them?
(b)  Repeat (a) for the sub-beam with zero Lx-component.

2.  Consider the following description of the measurements in #1:  The first Stern-Gerlach magnet measures the x-component of angular momentum, and the second measures the z-component.  Therefore, using  the two magnets we now know both the x- and z- components.  Is this correct?  If not, give a correct description.  If the beam from #1 (a) is passed through a third magnet, again in the x-direction, will we find that the x-component is 100% +hbar?  If not, why not?

3.  Consider a particle in the 3rd excited state of a 3-D simple harmonic oscillator.
(a)  List all the different possible states |nx ny nz >
(b)  List all the different possible states |n l m >

*If your first name begins with A-C, do the x-component, if your first name begins with D-J, do the y-component, otherwise, do the z-component.

Due Wednesay, January 31
Chapter 8:  Problem 9,
Chapter 9:  Exercise 5, Problems 1, 16,
1) Prove that L+|q l m> is an eigenstate of Lz
(hint: similar to proving that a+|n> is an eigenstate of H)

2) a) List all the states |j1 j2 m1 m2> that have j1=1 j2=2
b) List all the states |j1 j2 j m> that have j1=1 j2=2
c) Verify that the number of states is (2j1+1)(2j2+1) either way.

Due Friday, Feb 16 homework will include, but is not limited to:
Chapter 9:  Problem 14,

1) Two electrons are in l=1 states of an atom.  Each has three possible states (ml's) it could be in.
a) List all possible symmetric 2-electron spatial states (ignore spin for now).
b) List all possible antisymmetric 2-electron spatial states.
Note: there should be 3X3=9 different 2-electrons states, adding # in (a) and (b).
2) These two electrons each have two possible spin states (up or down).
a) List all possible symmetric 2-electron spin states (ignore space for the moment).
b) List all possible antisymmetric 2-electron spin states.
Note: there should be 2X2=4 different 2-electrons states, adding # in (a) and (b).
3) Since the full (spin & spatial) state of these two electrons must be antisymmetric, how many 2-electron states are available?  Either list them or explain clearly how you determined the number.

4) Consider the Rutherford scattering experiments, with alpha particles incident on gold foil.  The alpha particles have a charge of +2e.  The gold atoms were original thought to conform to the plum-pudding model -- the electrons were interspersed among the positive charges.  If this were the case, the incoming alpha particle would rarely experience a net nearby charge of more than about +e, since the negative and positive charges would tend to cancel.  We now know that the gold atoms consist of a charged nucleus surrounded by electron clouds, so if the alpha particle came close to the nucleus, it could experience essentially its full charge.  How much more likely is Coulomb scattering through 180o in the latter (realistic) case than in the former?

5) Refer to the Nd-YAG laser transition on the handout.  The initial state is 4F3/2.
a)  What are the S and L values for this initial state?  Verify that they can be produced by three d-subshell electrons.
b)  What are all the possible J values, given these S & L values?  Which is lowest in energy?  Is this consistent with the handout?

6)  Read the Scientific American article with an open mind.  Answer each of the following briefly, then choose one to answer in more detail.
a)  What are the differences between a quantum superposition, and classically just not knowing.
b)  What are the problems arising with the Copenhagen "collapse" idea?  Think of a He atom in ita ground state.  Two electrons, each 50-50 spin up and down.  If you measure one to be spin up, you collapits wavefunction.  What happens to the other electron?  What collapses its wavefunction?
c)  Does the many-worlds interpretation, which claims that all superpositions are real, explain why we only experience one alternative?
d)  Does decoherence explain why you get one result rather than the other (such as a card face up rather than down)?
e)  If physics did explain the reason for everything happening, would there be any free will?  Does the lack of a physical explanation for which possibility is actually realized leave an opening for free will?