Questions and answers for homeworks in 3220, Fa 97

Homework #1

>Subject: Homework Gas 1-13

>This has proven to me to be a challenging problem.  That is, if I am in
>fact interpreting the author correctly.  Let's say the electron has an
>integral number of wavelenghts.  First of all I am having trouble
>expressing this model in terms of energy (depending on wavelenght) and
>equating it to the Rydberg's equation or E=nhf.  The only way I know
>how to express the electrons as a waveform is by use of the
>Schrodinger's equation.  If I treat the electron as a particle in an
>infinate square well potential then I think things might work out, but
>I feel that the solution might be simpler and that I am barking up the
>wrong tree.

>Thank You,


I think the solution is simpler - sounds like you "know too much",
remember this is still a Chapter 1 problem!  We don't know
Schrodinger's equation yet, and are still working with the crude idea
of de Broglie.

Draw yourself a picture of a circle (the classical orbit of an
electron), and then try to redraw that circle so that it has some tiny
amplitude (but not-necessarily tiny wavelength) "wiggles" to it,
representing the electron trying to travel around that circle, but at
the same time having some sort of wavelength. Problem 13 says that an
integer number of wavelengths have to "fit" in the orbit (whose
circumference you know), which the picture will also verify. So this
relates wavelength to circumference => radius. De Broglie relates
wavelength to momentum, so together you have a relation between
momentum and radius.  Stare at it - it should be equivalent to Bohr's
first postulate. That's it...

Does this make sense? It's a little hard to explain in words - that
little picture is worth at least a thousand here! If it doesn't, feel
free to stop by my office sometime this afternoon...

Cheers,
Steve

Homework #2

>Good morning, its 1:30 am and I doubt you are up....
>
>However, I hope you will be able to respond this weekend.
>
>My problem is that I seem to be unable to do the integral ( that is
neither
>can Mathematica) that will allow me to calculate N.  After finding A(k)
and
>substituting it into the integral to find f(x) and multiplying it by e^ikx
>I get the function
>(1/k)*[(e^(ik(x-a/2)))-(e^(ik(x+a/2)))] after taking out all the
constants.
 >I am integrating from -infinity to infinity in k space.  hopefully you
can
>find my error.  
>
>Hope to hear from you soon.
>
>Have a good weekend 



I'm not sure how to answer this one, probably best if you see me in
class, or after it, today. For #1b, you should be doing just the
integral of |f(x)|^2 dx, which has no exponentials or anything in it
all.  

The integrand you wrote in your email
> (1/k)*[(e^(ik(x-a/2)))-(e^(ik(x+a/2)))]

doesn't look quite right, so perhaps you've already made a mistake
(which will be relevant in part c?) That integrand is *close* to things
I get, I'm just not sure what it's supposed to be: f(x)?  A(k)? (It has
both k's and x's in it...) What it really looks like is the inverse Fourier
transform of A(k), which you don't have to evaluate, since I *gave* you
f(x).

(Note that your e^ik factors out, and what's *left* looks like (1/k)
sin(ka/2), which certainly does look familiar...)

Talk to you soon, sorry I couldn't help more by email.

Steve

Homework #3

FYI:
I've gotten a couple messages like the following;


>Dear Professor,

>        I wanted to ask if you would be available past three o'clock on
>Tuesday.  I can get there at 2, however, I feel that I dont understand
>much of what is going on as far as the homework is going, and I may need
>more of your time.  Maybe it is just that quantum is hard for me. E&M is
>easy compared to this, and that is a scary statement in itself....
>Please let me know if that works for you.


 The answer is: No problem at all, I can certainly be available past
3 on Tuesday, I'll be happy to stay as long as you like. (Last week I
was around till after 5) If you plan on *arriving* after 3, though,
please let me know ahead of time, just in case.  The same goes for Mondays -
office hours are officially till 4, but I've been staying quite a bit
longer, as long as people are interested! (There may, on occasion, be a
nuclear seminar Mondays at 4 which I have to attend, but these are infrequent)

By the way, quantum *is* hard, for many people it can be one of the
hardest undergrad physics courses, in part because the material is so
abstract.  But, don't despair, come talk to me and I'll bet you'll feel
that the homework isn't quite as incomprehensible as it looks when you
first sit down to it...

Homework #5

>I am having trouble with the frequency part of problem 1.  Is it okay
>to use classical energy E= m w^2 r^2 where w (omega) is the frequency
>we are looking for?  Or is it v = w/ 2 Pi?  If it is simply w, I am off
>by a factor of 2 (unless my comparative frequency is wrong).  Any
>suggestions?

 

I assume you are working on part f) of number 1? (Not part c, where the
calculation of frequency is basically direct, it's just the coefficient
of "time" in the cos)

In part f, I want you to think as classically as possible. If you have
a particle in a box, with some given energy (you're going to use the
result from part e as the given energy) it will simply bounce back and
forth.  The "frequency" is just the number of times it goes from one
side of the box to the other (and back again) per second. (Number of
"rattles"/sec, or slightly more technically, the number of
oscillations/sec) Given that, the "angular frequency", omega,  would
just be as usual omega = 2 pi nu. 

Don't use the E= m w^2 r^2 formula here - that's only for a harmonic
oscillator (which this isn't- there's no "spring" force in this
problem).

Hopefully, the "classical rattling frequency" will come out the same
order of magnitude as the "quantum rattling frequency" you found in
part c.

Does this help?

Steve

Homework #6

>Good day...
>
>I have two problems, the first one is that I am not quite sure how to think
>about #1 so I was hoping that you will hold office hours this afternoon. 
>Secondly, in problem #2 part b, are you giving us the R values?
>
>Take care...
>

 

I'll be in my office from about 3PM on today.

If you are confused about #1, you might also want to look at Griffiths,
who has a page devoted to this idea of the "S matrix" for scattering.

In problem #2b, I am giving you values of V_0 a^2, the product of the
potential and "dip width" squared, and of course E is V0/3. (So, this
is basically some information about q and k) I am not giving you R,
that's what the question is asking for.

Hope this helps,


>On Gas 5-14 I get Vo=247 MeV, however I think I might have gotten some
>signs mixed around.  Does this look about right?
>
>Live Long and Prosper
> 
 

V0 = 247 MeV looks quite high, compared to my numbers. Better check
those signs! 

(Another thing you might want to check is that you got the binding to be
the lowest possible state, and not the next higher one..?)



Dr. Pollock,

> I am a little stumped on which formula I use for R.  In class, we
> derived it for a well which ran from -a to positive a.  In our homework,
> the well runs from 0 to 2a.  I am not sure if this changes the R you
> derived in the notes.  I think it is going to change some.  I have tried
> to derive it myself based on the new boundary conditions, x<0, 0 and x>2a.  
> This system of 4 eqs and 4 unknowns is giving me fits.  If it
> is different, is there a simpler way to just modify the R in the notes,
> i.e. replace -a with 0 and a with 2a ???


Stare at the form of the answers in our notes (or Gas, Eq. 5-26). I
would claim that nowhere in those answers is there any indication of
what we *called* x=0, only the energy, and the depth and width of the
well. As it should be - the *physics* here is of a beam coming in from
negative infinity, hitting a well of a certain shape, and either
bouncing or not. How could it matter if I slide that well a little bit
to one side or another? (Infinity + a, how could you notice?)  So, I
don't think you should need to do any extra work, the formulas are
still perfectly fine. 

Talk to you soon, 
Steve

Homework #8

>Hi!
>
>I looked for Griffith's book in the library, but it had already been
>checked out.  Can you give me an extra hint so that I might be able to
>figure 5-6 out myself?
>
 
Griffith's book should be on "short term" reserve, so probably whoever
has it will be in the library (or will have to bring it back soon?)

Anyway, here's something to get you going by yourself:

Think about how the E>0 problem (which is what gave the formulas for R
and T) *changes* when E<0. For one thing, k goes imaginary (does that
tell you why we we let k = i kappa?) But more important, look at the
*form* of the wave function u(x) outside of the well. When E>0, there
are 4 terms (in the notation of Gas problem 1, these are the "A", "B",
"C", and "D" terms).  But, when E<0, there are only 2 terms (dying
exponentials - the other two would be bad at infinity) That's already
curious - how can the general solution in one case (E>0) have 4 terms,
and in the other case (E<0) only have 2 terms?  

Now stare at the equation in Gas.  Problem 1 (relating B and C to A and
D, via the "S matrix") and you will see that the above curiousity has
led to a very serious problem! In fact, I claim the "S matrix" equation
(in the case E<0) looks something like

(something not zero) = (S matrix)*(zero)

Think about that. How can this possibly be? What does it tell you abou
the S matrix? (In the case of this problem, what does it tell you about
R and T, which are after all just elements of the S matrix. Which
ones?)

Hope this helps. I'll be in my office around 12:30 or so, so you can
look at Griffiths there if it's not already back in the library.



>Hi there,
>
>I have a couple of questions.
>
>Regarding #4, is the N you are looking for the same N in E=hbarw(N+.5)?


Yes


>
>        I'm greatly missing the "straightforward"-ness of this problem.
>        I know that at x=0, all of the energy is KE


Yes


>       which is .5kw^2, no?


Well, No. .5 k w^2 isn't really anything.  (.5 kx^2 is potential
energy) What you want is .5 m v^2 = kinetic energy, and you know m and
v, I gave those...


>        and w=(k/m)^.5 which also = 2pi (because f = 1Hz). I tried to set
>        the above E equal to the KE


Good


>       and solved for N and got 3x10^21.
>        Is N supposed to be an integer?
>


Yes, but there's no reason it can't be a very *big* integer! (Although,
yours looks a *lot* bigger than than the answer I got!..)


>Regarding #5 (6-1), what does he mean by the operator of AB? There is an
>        operator acting on 2 other operators that have been multiplied
>        together? (Since he said that A and B themselves are operators) So
>        if I say O is the operator of AB, then I write ?
>


I'm not sure I understand your question. If A and B are each operators,
then the product AB *is* also an operator, and we're interested in its
properties. It's like when we worked out commutators, remember we had
e.g. [x,p] = xp - px.  In that expression we have "xp" which is the
product of two operators, and is *itself* an operator. When you say "O
is the operator of AB", what we mean is we simply view the product AB
as itself just being some (complicated) operator, and we give it the
name O. So, O = AB.  And the question Gas asks is, when is O
Hermitian?  (The *answer*, which you have to prove, is that O is
Hermitian if the individual operators A and B commute. And *also* vice
versa, if O is hermitian, then the individual operators A and B must
commute.

You don't need to write down any bra's or ket's for this problem. You
can just work with the operators A and B all by themselves. Remember
that the Hermitian conjugate (the "dagger") of a product is the product
of the daggers of the individuals, written in reverse order...

i.e (ST)+ = (T)+(S)+

The problem is, in the end, pretty much a "quicky"! 

>Thanks
>

Good luck, I'll be up on campus this afternoon.
Cheers,
Steve 

Homework #10

>Dr. Pollock,
>
>I was wanting to know if my differential eqn for (t) is correct. I
>have:
>
>d^2  / dt^2   +   w1^2    = (-1/2) w2.  I just wanted to know if
>this is correct before I try to solve it with Boas.
>
>Thanks.


I don't have the solutions in front of me, so I can't tell for sure if
I agree with the exact constant you have on the RHS, but the *FORM* of
your equation is indeed completely correct, so do go ahead and try to
solve it with Boas.

Homework #11

> Re: Gas 7-3
>I am not sure how to start the problem, my biggest dilemma is finding a
>form for um and un that I am happy with.  Am I on the right track ?


I'm not sure. You do not have to write down *any* explicit form for un
or um at any point in this problem! It's an operator problem. 

You want the matrix element of x, but we know x can be written as some
constant times (A+Adagger) So you only need to know matrix elements of
A (or Adagger) between states un and um. The main point of Ch. 7 was
that A is a lowering operator: when it hits a state um, it gives back
the state u(m-1).  (With a constant of proportionality that we worked
out in class for Adagger, and Gas' hint will tell you how to figure out
the constant of proportionality for A) 

In formulas:
A u_n = C_n u_(n-1)
Adagger u_n = C'_n u_(n+1)

We worked out C' in class. You can work out C, given C'.  The u's are
orthonormal. So, you never have to write them down, and you never need
to do any integrals.  You only need to know that constant of
proportionality, and there's not really any more work to be done.

For example, according to the above, 
braket(n|A|m) = braket(n |C_m| m-1),
the C_m comes out (it's just a constant), and the braket you're left
with is either 0 or 1!

Cheers,
Steve


>Professor Pollock,
>
>        I just had one real quick HW question.  On problem 1, do I need to
>rederive Eqns. 10-21 through 10-23 with the new potential and prove the
>general case, or do I just need to do the general case?


You do *not* have to rederive any answer (or part of an answer) that
has already been shown in Gasiorowicz (or class notes). E.g. we've
already shown that d/dt=0 IF V depends only on r (not the angles),
and you may make use of that fact to simplify your work.  But, you still
need to show that if V does depend on angles, then d/dt is given by
the formula in the homework. 

I don't know if this helps. I guess I'm saying that we've *already*
shown that the kinetic energy part of H commutes with L, and you don't
have to rederive that. So it's only the potential part of H that
requires new work in this problem.

Homework #14

>Professor Pollock,
>
>Just two real quick questions about problem 2 (Gas. 12-5).  For
>expanded wave functions, do I still find the expectation values the
>same way as we did before (i.e. integrate the square of the entire wave
>funtion with whatever I am trying to find stuck in the middle)?

You can do that, it will work out o.k. (Using orthonormality of the
hydrogen wave functions will help a lot!) There is an equivalent
formula that might save you a little labour, which in the 1-D case I
might have written as
expectation value of O  
	= Sum on n of:  (Value of O in state n)*
				(Probability of getting state n)
    	= Sum_n    (Expectation of O in state n) * |C_n|^2

(This formula only works if the "n"'s represent eigenfunctions of O, by
the way) Do you see how that is equivalent to what you're doing, only
maybe a bit quicker/simpler?


>Secondly, the way the wave function in the problem is shown implies
>that all the psi's are solely dependent on r (which means to my
>figuring that L^2 and Lz are both zero!). However the l's and the m's
>some of the psi's contain suggest that there is some theta and phi
>dependence.  Should I just assume the wavefunction is dependent on r
>only, or all three variables?

The wave functions in Gas 12-5 all depend on a BOLD r (look carefully!)
The bold r means the full vector r, i.e. r, theta, and phi (or x,y, and
z) They most certainly do (must!) depend on theta and phi, because, as
you said, the l's and m's are explicitly written, and not all zero...
(you would be right *if* the wave function depended only on little r,
then L^2 and Lz would indeed both be zero)