Solving Quantum Mechanics Problems: Dirac Notation and Operator Methods

In summary: I'm in the same boat.I apologize. I didn't realize these problems cover material I have not covered yet. I'll have to get back to it tomorrow.
  • #1
Shackleford
1,656
2
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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  • #2
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...
 
  • #3
truth is life said:
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.
 
  • #4
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:

[tex] \hat{a} = \alpha \hat{x} + i \beta \hat{p}. [/tex]

We can use the inverse relationships to compute the expectation values in question.
 
  • #5
fzero said:
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:

[tex] \hat{a} = \alpha \hat{x} + i \beta \hat{p}. [/tex]

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.
 
  • #6
Shackleford said:
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.
 
  • #7
truth is life said:
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.
 
  • #8
Shackleford said:
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...
 
  • #9
truth is life said:
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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  • #10
fzero said:
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:

[tex] \hat{a} = \alpha \hat{x} + i \beta \hat{p}. [/tex]

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.
 
  • #11
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.
 
  • #12
Mindscrape said:
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.
 
  • #13
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?
 
  • #14
Mindscrape said:
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?

Are these the ladder operators?

http://i111.photobucket.com/albums/n149/camarolt4z28/2010-10-22234312.jpg?t=1287809104 [Broken]
 
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  • #15
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.
 
  • #16
Mindscrape said:
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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  • #17
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)
 
  • #18
Mindscrape said:
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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  • #19
Eqn 4.324 of those notes is the exact equation you're looking for.
 
  • #20
Mindscrape said:
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.
 

1. What is Dirac notation and how is it used in solving quantum mechanics problems?

Dirac notation, also known as bra-ket notation, is a mathematical notation used to represent quantum states and operators in quantum mechanics. It uses the symbols | and ⟩ to represent states, and ⟨ and | to represent dual states or bra vectors. This notation is useful in solving quantum mechanics problems because it allows for a more concise and elegant representation of complex mathematical concepts.

2. What are operators in quantum mechanics and how are they used to solve problems?

Operators in quantum mechanics are mathematical objects that represent physical observables, such as position, momentum, and energy. They are used to solve problems by acting on quantum states and producing new states, which can then be measured to determine the corresponding observable. In Dirac notation, operators are represented by matrices and are often used in conjunction with bra-ket notation to simplify calculations.

3. Can you explain the relationship between Dirac notation and the Schrödinger equation?

The Schrödinger equation is a fundamental equation in quantum mechanics that describes how a quantum state evolves over time. Dirac notation provides a convenient way to write and manipulate the terms in the Schrödinger equation, making it easier to solve for the time evolution of a quantum system. The bra-ket notation allows for the representation of both the state and the Hamiltonian (energy) operators in a compact form, making it easier to understand and apply the Schrödinger equation.

4. How do operator methods help in solving quantum mechanics problems?

Operator methods are a powerful tool in solving quantum mechanics problems because they allow for the manipulation of complex mathematical expressions using a set of rules and properties specific to operators. These methods can simplify calculations and provide a more intuitive understanding of quantum systems, making it easier to solve problems and obtain accurate results.

5. Are there any limitations to using Dirac notation and operator methods in solving quantum mechanics problems?

While Dirac notation and operator methods are widely used and have proven to be effective in solving a variety of quantum mechanics problems, they do have some limitations. For instance, they may not be suitable for describing certain physical phenomena, such as systems with infinite degrees of freedom. Additionally, they may not be as useful in certain mathematical formalisms, such as those used in quantum field theory. It is important to understand these limitations and use alternative methods when necessary.

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