Write the matrix representation of the raising operators...

  • #1
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Homework Statement



Hi, guys. The question is: For a 3-state system, |0⟩, |1⟩ and |2⟩, write the matrix representation of the raising operators ## \hat A, \hat A^\dagger ##, ## \hat x ## and ##\hat p ##.

Homework Equations



I know how to use all the above operators projecting them on eigenstates.

The Attempt at a Solution



I don't knowhow to representate this 3-state system in the matrix form. I know that ## \langle m | A| m' \rangle ## is used to get the matrix, but I don't how to get the matrix form for this 3-state. Is it a colum vector?
 

Answers and Replies

  • #2
Ray Vickson
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Homework Statement



Hi, guys. The question is: For a 3-state system, |0⟩, |1⟩ and |2⟩, write the matrix representation of the raising operators ## \hat A, \hat A^\dagger ##, ## \hat x ## and ##\hat p ##.

Homework Equations



I know how to use all the above operators projecting them on eigenstates.

The Attempt at a Solution



I don't knowhow to representate this 3-state system in the matrix form. I know that ## \langle m | A| m' \rangle ## is used to get the matrix, but I don't how to get the matrix form for this 3-state. Is it a colum vector?
No: a matrix is not a vector; it is a two-dimensional array.

If you have never seen matrices before, you should do some reading before attempting to do questions involving them. Google "matrix".
 
  • #3
DrClaude
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Hi, guys. The question is: For a 3-state system, |0⟩, |1⟩ and |2⟩, write the matrix representation of the raising operators ## \hat A, \hat A^\dagger ##, ## \hat x ## and ##\hat p ##.
Is that the full question? Because without additional details, the operations of ##\hat x## and ##\hat p## are unknown.
I know that ## \langle m | A| m' \rangle ## is used to get the matrix, but I don't how to get the matrix form for this 3-state.
I don't understand what you don't understand! You just stated how to calculate the matrix elements, so all that is left is for you to calculate them.
 
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  • #4
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Is that the full question? Because without additional details, the operations of ##\hat x## and ##\hat p## are unknown.

I don't understand what you don't understand! You just stated how to calculate the matrix elements, so all that is left is for you to calculate them.
Yes, that is the full question. I know what a matrix is. This question is really confusing me. I don't understand how to representate this 3 state system as a matrix. It is a superposition of 3 states, so ## |x⟩=|0⟩+|1⟩+|2⟩ ## can be represented as a column matrix? $$ \begin{pmatrix}
\langle 0 | \hat A | 0 \rangle \\
\langle 1 | \hat A | 1 \rangle \\
\langle 2 | \hat A | 2 \rangle
\end{pmatrix} $$.

So, $$
\begin{pmatrix}
\langle 0 | 0 \rangle \\
\langle 1 | 0 \rangle \\
\langle 2 | 1 \rangle
\end{pmatrix} $$, $$
\begin{pmatrix}
1 \\
0 \\
0
\end{pmatrix} $$?
 
  • #5
Ray Vickson
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Yes, that is the full question. I know what a matrix is. This question is really confusing me. I don't understand how to representate this 3 state system as a matrix. It is a superposition of 3 states, so ## |x⟩=|0⟩+|1⟩+|2⟩ ## can be represented as a column matrix? $$ \begin{pmatrix}
\langle 0 | \hat A | 0 \rangle \\
\langle 1 | \hat A | 1 \rangle \\
\langle 2 | \hat A | 2 \rangle
\end{pmatrix} $$.

So, $$
\begin{pmatrix}
\langle 0 | 0 \rangle \\
\langle 1 | 0 \rangle \\
\langle 2 | 1 \rangle
\end{pmatrix} $$, $$
\begin{pmatrix}
1 \\
0 \\
0
\end{pmatrix} $$?
What you have written are vectors, not matrices as usually understood.

I don't see what your issue is. You say you can calculate the quantities ##A_{m,m'} = \langle m |A| m' \rangle## for your operators ##A## and your states ##|1\rangle, |2\rangle, |3\rangle##. The quantities ##A_{m,m'}## are just some (complex) numbers that could be computed from problem input information about the nature of the states and the operators.

There are a total of 9 quantities ##A_{m,m'}## because we can have ##m = 1,2,3## and ##m' = 1,2,3## selected independently of one another. Those 9 quantities ARE the matrix, when displayed appropriately. That's all there is to it: no more, no less.

There is nothing at all in the stated problem about a "superposition of states" or anything like that, unless it is in some parts of the question you did not tell us about.
 
Last edited:
  • #6
DrClaude
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To add to what @Ray Vickson said, note that contrary to what you seem to imply, ##\hat A | 0 \rangle \neq | 0 \rangle##.
 
  • #7
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Thank you guys.
 

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