Understanding Energy and Momentum in Quantum Mechanics

[AKG]

I am taking an intro to QM course, but it seems way beyond intro. The book starts immediately by discussing concepts that I haven't been introduced to in my previous physics course. I have a number of questions to answer. Of course, I don't expect answers to the problems, as I wouldn't understand them anyways, I would just like to know what the heck these problems mean, what equations I should know and be using, and where I can find some quick online material that explains the basic underlying concepts of these problems in a simpler manner:

4. The maximum energy of photoelectrons from aluminum is 2.3 eV for radiation of 200 nm and 0.90 eV for radiation of 258 nm. Use these data to calculate Planck's constant and the work function of aluminum

(To give you an idea of where I'm at, I don't know what a photoelectron is, I don't know what it means for the energy of radiation of X wavelength to be Y eV, I have no idea how I would use this to calculate Planck's constant, and I don't know what a work function is ).

5. A 100 MeV photon collides with a proton that is at rest. What is the maximum possible energy loss for the photon?

(My best guess is that 100 MeV tells you the mass of the photon, and you know the speed is c, and you know the initial speed of the proton is 0 and it's mass is some constant - 1g/6.02 x 10²³, and perhaps I should assume conservation of momentum and energy. But I also am not sure which equation applies where. Is the kinetic energy of the proton ½mv² and the momentum mv? What's with the equation E = m²(c²)² + p²c²? Is that the kinetic energy of the photon? Not sure what I should be doing here exactly)

7. An electron of energy 100 MeV collides with a photon of wavelength 3 mm (corresponding to the universal background of blackbody radiation). What is the maximum energy loss suffered by the electron?

(I'm supposed to know what blackbodies are but I have no idea. I'm also supposed to know something about how they radiate, again, no idea. I don't get the stuff in brackets in the question above, and only have a vague idea of the rest of it, similar to my understanding of question 5)

9. A nitrogen nucleus (mass ~ 14 x proton mass) emits a photon of energy 6.2 MeV. If the nucleus is initially at rest, what is the recoil energy of the nucleus in eV?

(Again, this seems like a conservation of momentum/energy question. Let's see if I have any idea: initial energy and momentum are zero. When the photon is fired off, it has some known momentum and energy. We know the mass of the nucleus, and we know that the momentum of the nucleus will be equal in magnitude to that of the photon, and the same goes for it's energy. Do we have two equations with one unknown, the speed of the nucleus? And if so, we can find the speed and thus the energy. And am I right to assume that the classical equations for momentum and energy hold for the nucleus. Actually, I won't even say classical equations, rather, high school equations.)

11. The smallest separation resolvable by a microscope is of the order of magnitude of the wavelength used. What energy electrons would one need in an electron microscope to resolve separations of (a) 15 nm, (b) 0.5 nm?

(I assume that the energy of the electrons is a direct function of their wavelength? So if E = f(λ), I just need to figure out f(15 nm) and that's my answer? And if I'm right, what is that function f(λ)?)

14. Use the Bohr quantization rules to calculate the energy levels for a harmonic oscillator, for which the energy is p²/2m + mw²r²/2; that is, the force is mw²r. Restrict yourself to circular orbits. What is the analog of the Rydberg formula? Show that the correspondence princple is satisfied for all values of the quantum number n used in quantizing the angular momentum.

(?)

16. The classical energy of a plane rotator is given by E = L²/2I where L is the angular momentum and I is the moment of inertia. Apply the Bohr quantization rules to obtain the energy levels of the rotator. If the Bohr frequency condition is assumed for the radiation in transitions from sttes labled by n1 to states labled by n2, show that (a) the correspondence principle holds, and (b) that it implies that only transitions Δn = ±1 should occur.

 

[Tide]

That's a lot of questions!

I'll pick and choose the easiest! ;-)

A photoelectron is an electron that has been removed from a solid (like a metal) as a result of a photon striking it. The work function is the amount of energy required to removed an electron from the solid.

A photon does not have mass. However, it does have energy and momentum. The 100 MeV refers to the photons energy from which you can determine its frequency and momentum. In scattering off a proton both energy and momentum will be conserved. You can use that to determine the maximum energy loss of the photon. That should help with the next two problems as well.

I'll leave the others to anyone else who wants to jump in! :-)

 

[AKG]

Yes, I know I'm asking a lot of questions in one place, but it's really all the same question: I don't understand a thing about introductory QM, please help me! I know, that's not really a question. Anyways, thank you for that definition of photoelectron, it makes some more sense now. What is the work function a function of? How do I determine frequency and momentum from energy? Wait: E = h/λ and E = p²c² (since m=0)?

 

[Tide]

Yes, I know I'm asking a lot of questions in one place, but it's really all the same question: I don't understand a thing about introductory QM, please help me! I know, that's not really a question. Anyways, thank you for that definition of photoelectron, it makes some more sense now. What is the work function a function of? How do I determine frequency and momentum from energy? Wait: E = h/λ and E = p²c² (since m=0)?

The work function depends on the specific material and is, in fact, measured by the method suggested in your first problem.

A photon's energy is
$$E = h \nu = \frac {h c}{\lambda}$$
and just divide by c to get its momentum.

 

[AKG]

Right, right, it should have been E² = p²c², which would lead to what you're saying that E/c = p. Now, are you saying that for aluminum, the work function is a constant? And I don't understand the first problem. How would I use that (and my new knowledge of what a work function is) to find h and the work function?

 

[Tide]

Compare the maximum energy for a photoelectron with the photon's energy for both cases and see what you get and what you can infer.

 

[AKG]

Hmm... Like I said, I don't really get any of this, so it's just a guess:

When the photon collides with the aluminum and has 200 nm wavelength, it has a specific energy corresponding to that wavelength. When an electron is released, it has some maximum energy, plus some energy is lost in releasing the electron (the work function). So the work function plus 2.3 eV should equal the energy of the photon with wavelength 200 nm. Oh, and the energy of the photon is a function of wavelength and Planck's constant. So we have 2 equations (since we're given two different wavelengths) and two unknowns (the work function and h) so we can solve for the two? By the way, what's the standard notation for the work function? I'll just call it W. So my equations are:

W + 2.3 eV = hc/200 nm
W + 0.90 eV = hc/258 nm

If these equations are right, then I just have to solve for W and h, right?

 

[AKG]

I think I've got question 4 right, although I'd like to know the conventional symbol used to denote the work function. Doing question 4, I get values for h that agree with the values I've read, and the W value is pretty close.

I think I can get a couple of the other quesitons too, but I have another question:

$$E^2 = m^2c^4 + p^2c^2$$
$$E = \frac{1}{2}mv^2$$
$$E = h\nu$$

What exactly are the above equations? The second one is the kinetic energy of everything except a photon? The bottom one is the _______ (kinetic, total, what?) energy of a photon? What's the top one?

My attempt at 5:

Initial energy of the system = 100 MeV
Initial momentum of the system = 100 MeV/c

Final energy of the system = 100 MeV = $E_{\gamma} + \frac{1}{2}m_{proton}v_{proton}^2$, where $m_{proton}$ is some constant I should be able to look up, and $v_{proton}$ needs to be found.
Final momentum of the system = 100 MeV/c = $E_{\gamma}/c + m_{proton}v_{proton}$.

$E_{\gamma}$ and $v_{proton}$ are unknowns in 2 equations, so I can solve for them, and the answer to the question is 100 MeV - $E_{gamma}$.

My attempt at 7:

I'd still like to know what "corresponding to the universal background of blackbody radiation" means, but I have a feeling it's not relevant to getting the answer. Given the wavelength of the photon, I can get it's energy since $E = hc/\lambda$ and it's momentum since $E^2 = m^2c^4 + p^2c^2 \rightarrow p = E/c$. I'm given the energy of the electron, and I should be able to get it's momentum from this, but I'm not exactly sure how I'd do that. Also, I have a feeling that this isn't enough information. Do I have to know something more about the collision, or can I just solve this by setting up 2 equations like I did for the question above and solve for the two unknowns? I suspect that I will get a trivial answer where the collision has no effect, and one where there actually is a collision, and those numbers will be my answer. Like I said, though, I have a feeling that won't work.

Question 9:

Why wouldn't the recoil energy simply be 6.2 MeV, thereby conserving energy?

Question 11: was my approach stated in the original post right? If so, what is that $f(\lambda)$ I asked about?

Not even going to touch the last two for now.

Thanks.