Exponential operator multiplication

In summary, the conversation revolves around a question on how to combine operators inside exponentials in quantum mechanics. The solution is to use the Taylor Expansion of exp with the U-operator as argument, and also an exponential identity that involves the commutator of operators. The conversation also includes a specific example of operators and their corresponding creation and annihilation operators.
  • #1
Dracovich
87
0
1. I have a fairly straight forward problem (or basically, i need help to just get started on my problem), i have forgotten all my QM and am in a "bit" over my head here. Basically i am having problems remembering how one would go about treating operators that are inside exponentials.



2. So i have [tex]e^{\hat{U^{\dagger}}}\hat{A}e^{\hat{U}}[/tex] and basically was just wondering how one combines the A and U operator in some way



3. I honestly didn't get very far, looking through some of the basic coursebooks for QM i didn't find it at least in an obvious place (as in where they explain the basic properties of operators) and looking around/googling etc the closest i got was to represent it as a power series, i tried writing that out and see if it got me somewhere but i couldn't see how it was suppose to help me (although I'm sure it should).
 
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  • #2
Use the Taylor Expansion of exp with the U-operator as argument.

[tex] \mathrm{e}^{x} = \sum^{\infin}_{n=0} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots [/tex]
 
  • #3
Thanks, i had kind of gotten that far (as noted in 3, perhaps it's not called power series, sorry about that). Guess i got it right but I'm just not good enough to go from there lol, well at least i know that's definitely what i should be working with, cheers!
 
  • #4
Is there a particular A and U that you have in mind?
 
  • #5
Yes, didn't mention it since i figured i'd try to just get a boost here and finish on my own, but the actual thing looks like this:

[tex]e^{-i\theta \hat{J}^{\dagger}}\big(\frac{\hat{a_{0}}}{\hat{a_{1}]} \big) e^{i\theta \hat{J}}[/tex]

I can't find the command for vector notation so i had to use \frac, but that's not a fraction, it's a vector

[tex]\hat{J}=(\hat{a_{0}}^{\dagger}\hat{a_{1}]+\hat{a_{1}}^{\dagger}\hat{a_{0}})/2[/tex]

where [tex]a_0[/tex] and [tex]a_1[/tex] (and the daggers that is) are the creation and annihilation operators.

I tried writing it out and seeing what happens when the operators J and a get multiplied, but I'm honestly horrible at this, it was probably a mistake taking this course seeing as how it's been over 2 years since i last even looked in a quantum book and wasn't that great to begin with hah, but there you go, might as well stick it out as i can't opt out anymore.
 
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  • #6
\vec{your vector}

Why don't you show us a bit of your attempt to solution?

It is required to get help you know..
 
  • #7
I would if i had much to show :) I'm afraid I'm fairly clueless, like i mentioned i had only gotten so far as writing the exponential operator out in a power series, and also writing out explicitly the a operators multiplying with a single J operator in hopes that it would give me some insight, but no luck. But don't worry about it, the class is today so it's too late anyway :) Thanks though, i really do appreciate the help given so far.
 
  • #8
There's an exponential identity that you can use instead of expanding it all out.

[tex]
e^{\hat{-\hat{B}}}\hat{A}e^{\hat{B}} = \hat{A} + [A,B] + 1/2![A,[A,B]] + 1/3![A,[A,[A,B]]] + ...
[/tex]

Because often times, the dagger of a unitary operator is just the negative of it. And the brackets denote the commutator. In many problems, the commutator [A,B] = scalar so only the first two terms are non-zero since [A,(any constant)] = 0.
 

Related to Exponential operator multiplication

1. What is the exponential operator in multiplication?

The exponential operator is a mathematical symbol represented by the caret (^) that denotes repeated multiplication of a number by itself for a certain number of times. For example, 3^4 means 3 multiplied by itself 4 times, resulting in 81.

2. How is the exponential operator used in multiplication?

The exponential operator is used to make calculations easier when there are large numbers involved. It is also commonly used in scientific notation to represent very small or large numbers.

3. What is the difference between exponential operator multiplication and regular multiplication?

The exponential operator multiplication is a shortcut for writing out repeated multiplication of a number by itself. Regular multiplication involves multiplying two or more numbers together to get a product.

4. Can the exponential operator be used with other operations besides multiplication?

Yes, the exponential operator can also be used for repeated addition (ex. 2+2+2 can be written as 2^3) and repeated division (ex. 8/2/2 can be written as 8^(-2)).

5. How is the order of operations applied when using the exponential operator?

The exponential operator follows the same order of operations as regular multiplication and division, which is done before addition and subtraction. However, if there are parentheses involved, the operations inside the parentheses are done first.

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