# More quantum homework

#### Shackleford

Last week, I printed out several notes I found online on Dirac notation and operator methods in quantum mechanics. I made it a point today to read all of them and glean what knowledge I could outside of our terrible Gasiorowicz book. After reviewing all the properties, useful derivations and so forth, I feel I should be able to knock out these two problems with ease. However, that's not happening. I feel a mental block and disconnect from the material I covered in using it to work out these two problems. I'm not sure which direction to go with the problem to show that it's zero unless n = m +/-1. Do I insert the explicit quantum operator? Conjugate the inner product and play with a few identities?

http://i111.photobucket.com/albums/n149/camarolt4z28/2010-10-20221959.jpg?t=1287631877 [Broken]

http://i111.photobucket.com/albums/n149/camarolt4z28/2010-10-20222336.jpg?t=1287631902 [Broken]

http://s111.photobucket.com/albums/n149/camarolt4z28/?action=view&current=2010-10-20222322.jpg

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#### truth is life

Ah ha ha ha ha, that's exactly the problem I'm working on right now. There's a very important fact you need to know to solve the problem (and the next one, and number 10, at least): n and m are eigenstates of the harmonic oscillator, something Gasiorwicz fails to state anywhere in the problem. Now, go back and read section 6-2 very carefully...there might be something there that will let you define the x and p operators more usefully...

#### Shackleford

Ah ha ha ha ha, that's exactly the problem I'm working on right now. There's a very important fact you need to know to solve the problem (and the next one, and number 10, at least): n and m are eigenstates of the harmonic oscillator, something Gasiorwicz fails to state anywhere in the problem. Now, go back and read section 6-2 very carefully...there might be something there that will let you define the x and p operators more usefully...
We're assigned 1,5,6,10, and 13.

Eigenstates of the harmonic oscillator, you say! That's great. That's the material I had not read yet because we haven't covered it in class. We might have today, but I couldn't make it because I thought I had a bad starter. It was simply stupid corrosion on the positive terminal.

At any rate, let me read 6-2 carefully. I may need to get outside notes from online to supplement the "material" in G.

#### fzero

Homework Helper
Gold Member
It's not clear from the images you linked, but I'm assuming this is about the harmonic oscillator and the ladder operators? If true, there is a linear relationship between the ladder operators and the position and momentum operators:

$$\hat{a} = \alpha \hat{x} + i \beta \hat{p}.$$

We can use the inverse relationships to compute the expectation values in question.

#### Shackleford

It's not clear from the images you linked, but I'm assuming this is about the harmonic oscillator and the ladder operators? If true, there is a linear relationship between the ladder operators and the position and momentum operators:

$$\hat{a} = \alpha \hat{x} + i \beta \hat{p}.$$

We can use the inverse relationships to compute the expectation values in question.
I apologize. I didn't realize these problems cover material I have not covered yet. I'll have to get back to it tomorrow.

#### truth is life

We're assigned 1,5,6,10, and 13.

Eigenstates of the harmonic oscillator, you say! That's great. That's the material I had not read yet because we haven't covered it in class. We might have today, but I couldn't make it because I thought I had a bad starter. It was simply stupid corrosion on the positive terminal.

At any rate, let me read 6-2 carefully. I may need to get outside notes from online to supplement the "material" in G.
So was my class...hmm, this is getting suspicious...who's your professor? (Last name's fine)

I agree that G. is pretty useless (and in fact have refused to even consider going to Minnesota for grad school); I've bought at this point 4 other books (Griffith's, Dirac's, Bohm's, and now Sakurai's) to supplement it so that I can actually do the problems. Dirac is really good, but doesn't have any problems, so it's really hard to apply that in class. The other two I actually have at the moment don't really have anything on the Heisenberg/Dirac approach, which makes them rather useless at this stage.

#### Shackleford

So was my class...hmm, this is getting suspicious...who's your professor? (Last name's fine)

I agree that G. is pretty useless (and in fact have refused to even consider going to Minnesota for grad school); I've bought at this point 4 other books (Griffith's, Dirac's, Bohm's, and now Sakurai's) to supplement it so that I can actually do the problems. Dirac is really good, but doesn't have any problems, so it's really hard to apply that in class. The other two I actually have at the moment don't really have anything on the Heisenberg/Dirac approach, which makes them rather useless at this stage.
HU.

One of my friend's has an account on here but doesn't really use it. However, we're far away from MINI-SODA!

It sucks for me because I have work in the morning! Otherwise, I'd be up working on this and classical mechanics.

#### truth is life

HU.

One of my friend's has an account on here but doesn't really use it. However, we're far away from MINI-SODA!

It sucks for me because I have work in the morning! Otherwise, I'd be up working on this and classical mechanics.
Ah! You're in MY class! Houston, right?

I think I can guess (to 2 people) who you are, too...

#### Shackleford

Ah! You're in MY class! Houston, right?

I think I can guess (to 2 people) who you are, too...
Yeah, I think I sent you a text.

...maybe not.

G-MAN?

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#### Shackleford

It's not clear from the images you linked, but I'm assuming this is about the harmonic oscillator and the ladder operators? If true, there is a linear relationship between the ladder operators and the position and momentum operators:

$$\hat{a} = \alpha \hat{x} + i \beta \hat{p}.$$

We can use the inverse relationships to compute the expectation values in question.
Yes. I'm embarrassed to admit I haven't made any progress. I just got off work.

#### Mindscrape

At this point it seems like you have a ton of QM books, so I don't know if you need to buy anymore, as both Sakurai and Griffiths are pretty good. However, there is a book that if you can check it out then do so; the book is called Quantum Mechanics: Concepts and Applications by Zettili. Loads of examples, and doesn't skimp on the math formalisms like Griffiths does.

Anyway, if you rewrite the ladder operators in terms of x and p, then you can make an expectation value out of x and p that are in terms of the ladder operators themselves. Then you can use what the ladder operators do to your advantage.

#### Shackleford

At this point it seems like you have a ton of QM books, so I don't know if you need to buy anymore, as both Sakurai and Griffiths are pretty good. However, there is a book that if you can check it out then do so; the book is called Quantum Mechanics: Concepts and Applications by Zettili. Loads of examples, and doesn't skimp on the math formalisms like Griffiths does.

Anyway, if you rewrite the ladder operators in terms of x and p, then you can make an expectation value out of x and p that are in terms of the ladder operators themselves. Then you can use what the ladder operators do to your advantage.
No, no. I have only the Gasiorowicz book. The other guy has all the QM books.

#### Mindscrape

Ah, my bad. Well Shenkar is really good too, but it can be pretty advanced. Zettili is good if you aren't afraid of math (but if you are, I suggest griffiths).

Yeah, well, hopefully Gasiorowicz covers ladder operators. Sounds like he does a bit?

#### Shackleford

Ah, my bad. Well Shenkar is really good too, but it can be pretty advanced. Zettili is good if you aren't afraid of math (but if you are, I suggest griffiths).

Yeah, well, hopefully Gasiorowicz covers ladder operators. Sounds like he does a bit?

http://i111.photobucket.com/albums/n149/camarolt4z28/2010-10-22234312.jpg?t=1287809104 [Broken]

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#### Mindscrape

Hah, yeah, those x's and p's should have hats on them... Read about those for a bit, try to do what we've suggested, and if you have more trouble you're welcome to write back.

#### Shackleford

Hah, yeah, those x's and p's should have hats on them... Read about those for a bit, try to do what we've suggested, and if you have more trouble you're welcome to write back.
Here's what I've done so far. I assume I do the same thing for <m/p/n>.

http://i111.photobucket.com/albums/n149/camarolt4z28/2010-10-23101213.jpg?t=1287846830 [Broken]

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#### Mindscrape

Yeah, that's a really good start. You still have to do two things though. 1) show that it will vanish unless n=m±1 (one more line will quickly show that), and 2) calculate what the answer is (one additional line of work ought to then do this)

#### Shackleford

Yeah, that's a really good start. You still have to do two things though. 1) show that it will vanish unless n=m±1 (one more line will quickly show that), and 2) calculate what the answer is (one additional line of work ought to then do this)
Here's the same work for #6. I guess I'm not exactly sure how to do your 1) and 2).

http://i111.photobucket.com/albums/n149/camarolt4z28/2010-10-24134242.jpg?t=1287946078 [Broken]

Unless I'm just oblivious, I don't see how to work that out in Gasiorowicz. However, I think the following set of notes addresses it on page 54.

http://mysbfiles.stonybrook.edu/~klikharev/511-512/F08-S09/Ch4.pdf

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#### Mindscrape

Eqn 4.324 of those notes is the exact equation you're looking for.

#### Shackleford

Eqn 4.324 of those notes is the exact equation you're looking for.
Great. Thanks. I figured it was around there somewhere. This looks like a decent set of notes. However, I don't want to simply be "plugging and chugging." But it seems like the main issue is that I'm just missing a few key equations to do some of the homework problems.

Let me see what I can come up with now.

"More quantum homework"

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