Representing harmonic oscillator potential operator in. Cartesian basis

[Apashanka]

My question is given an orthonormal basis having the basis elements Ψ's ,matrix representation of an operator A will be [Ψ_iIAIΨ_j] where i denotes the corresponding row and j the corresponding coloumn.
Similarly if given two dimensional harmonic oscillator potential operator .5kx²+.5ky² where x and y are the operators ,what will be the matrix representation of this operator in the 2d Cartesian basis where i and j are the unit vectors and this is an orthonormal basis??
What is the way to find these types of problems??

 

[king vitamin]

You may be confusing the basis of real space (unit vectors along x and y) with the basis of the Hilbert space of your problem (square-integrable functions on the plane). To get the sort of matrix you are interested in, you need to find a complete set of square-integrable functions f_i(x,y) and evaluate $\int dx dy f_i(x,y)^{\ast} A f_j(x,y)$ for your desired operator A.

 

[Apashanka]

You may be confusing the basis of real space (unit vectors along x and y) with the basis of the Hilbert space of your problem (square-integrable functions on the plane). To get the sort of matrix you are interested in, you need to find a complete set of square-integrable functions f_i(x,y) and evaluate $\int dx dy f_i(x,y)^{\ast} A f_j(x,y)$ for your desired operator A.

Then what is actually the matrix we calculate taking the coefficients of x² and y² and also xy,then calculate it's eigenvalue giving the coefficients of square of new coordinates
An example may be given hamiltonian

Calculating it's eigenvalue value we get the new potential term as .5kX²+.5(4k)Y² where X (unit vector) and Y(unit vector) are elements of another orthonormal basis ,k and 2k are the eigenvalue of the potential matrix
Actually what is actually done in these process??

 

[king vitamin]

The question in the the picture appears to be very different than the question I thought you were originally asking, so now I'm a bit confused.

Are you asking about the transformation from the variables $x$ and $y$ given in the problem to new variables $X$ and $Y$ which put the potential into the form $V(x,y) \propto X^2 + Y^2$? Because this is also a linear algebra problem with some conceptual similarities to what I thought you were asking above, although it is much simpler (the vector space is two-dimensional instead of infinite-dimensional). And if you are asking about this, I am currently unclear about your specific question. (You're also referencing "k" and I do not know what that is.)

Can you think of a more concrete way to phrase your question? What is the current problem which you are trying to solve which you do not understand?

 

[Apashanka]

The question in the the picture appears to be very different than the question I thought you were originally asking, so now I'm a bit confused.

Are you asking about the transformation from the variables $x$ and $y$ given in the problem to new variables $X$ and $Y$ which put the potential into the form $V(x,y) \propto X^2 + Y^2$? Because this is also a linear algebra problem with some conceptual similarities to what I thought you were asking above, although it is much simpler (the vector space is two-dimensional instead of infinite-dimensional). And if you are asking about this, I am currently unclear about your specific question. (You're also referencing "k" and I do not know what that is.)

Can you think of a more concrete way to phrase your question? What is the current problem which you are trying to solve which you do not understand?

Yes that's the question I am asking about the transformation of the potential to V(x) ∝X²+Y² quantum mechanically( e.g representing the potential term in terms of X and Y instead of x and y which is given in the problem) ,the method to the process actually I want to know and the reason behind it .

would it be incorrect to say the representation of the potential term in the basis (x,y) to the basis of (X,Y) ,I want to clarify??
Thank you

 

[king vitamin]

I see. Let's say we have a Hamiltonian written in terms of position, $x_i$ and momentum, $p_i$. If we want to change to some new variables $X_i$ and $P_i$, which can be useful because (for this case) we might reproduce a known Hamiltonian which we have already solved, we will want to preserve the commutation relations:
$$[x_i,p_j] = [X_i,P_j] = i \delta_{ij} \hbar$$
Therefore, if we make some transformation $X_i = f_i(x,p)$ and $P_i = g_i(x,p)$, we better make sure that $[f_(x,p),g_j(x,p)] = i \delta_{ij} \hbar$. (For your future reference, this is called a canonical transformation.)

Now let's look at your specific problem. Your potential is such that there exists some transformation $X = ax + by$, $Y = cx + dy$ which puts the potential into the form you want. (From your posts I gather that you already know the explicit values of the coefficients.) Then to preserve the commutation relations, and therefore end up with a quantum Hamiltonian where the momentum and position operators have the correct commutation relations, how should the momenta be transformed?

 

[Apashanka]

I see. Let's say we have a Hamiltonian written in terms of position, $x_i$ and momentum, $p_i$. If we want to change to some new variables $X_i$ and $P_i$, which can be useful because (for this case) we might reproduce a known Hamiltonian which we have already solved, we will want to preserve the commutation relations:
$$[x_i,p_j] = [X_i,P_j] = i \delta_{ij} \hbar$$
Therefore, if we make some transformation $X_i = f_i(x,p)$ and $P_i = g_i(x,p)$, we better make sure that $[f_(x,p),g_j(x,p)] = i \delta_{ij} \hbar$. (For your future reference, this is called a canonical transformation.)

Now let's look at your specific problem. Your potential is such that there exists some transformation $X = ax + by$, $Y = cx + dy$ which puts the potential into the form you want. (From your posts I gather that you already know the explicit values of the coefficients.) Then to preserve the commutation relations, and therefore end up with a quantum Hamiltonian where the momentum and position operators have the correct commutation relations, how should the momenta be transformed?

Okay thanks but apart from this linear algebra if you could suggest something about the potential matrix diagonalization ,etc
Thanks

 

[king vitamin]

What matrix do you want to diagonalize? Are you trying to find the constants a, b, c, and d, specified in my post, using matrix methods?