Write the matrix representation of the raising operators....

• Mutatis
In summary, the question asks for the matrix representation of the raising operators and the matrix elements for a 3-state system. The operator and state vectors are already given, so the only thing left to do is to calculate the matrix elements.
Mutatis

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?

Mutatis said:

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".

Mutatis said:
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.
Mutatis said:
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.

Mutatis
DrClaude said:
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, $$\begin{pmatrix} \langle 0 | 0 \rangle \\ \langle 1 | 0 \rangle \\ \langle 2 | 1 \rangle \end{pmatrix}$$, $$\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$$?

Mutatis said:
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:
To add to what @Ray Vickson said, note that contrary to what you seem to imply, ##\hat A | 0 \rangle \neq | 0 \rangle##.

Thank you guys.

What is the purpose of writing the matrix representation of the raising operators?

The matrix representation of the raising operators is used to describe the behavior of quantum systems and their excitations. It allows scientists to mathematically model and manipulate these systems in order to make predictions and perform calculations.

What do the raising operators represent in matrix form?

The raising operators represent the creation of excitations or particles in a quantum system. In matrix form, they represent the action of increasing the energy or momentum of a system by a certain amount.

How are the raising operators related to the lowering operators?

The raising operators and lowering operators are related through their commutation relations. This means that they can be used together to generate a complete set of operators that can fully describe a quantum system.

What is the notation used for writing the matrix representation of the raising operators?

The matrix representation of the raising operators is usually written using the ladder operator notation, where the raising operator is denoted by "a" and the lowering operator is denoted by "a†". The matrix elements are then represented as a and a† in the matrix.

How are the raising operators used in quantum mechanics?

The raising operators are used in quantum mechanics to describe the behavior of quantum systems and their excitations. They are an essential tool for making predictions and calculating properties of quantum systems, such as energy levels and transition probabilities.

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