Quantum energy of a particle in 2 dim space

[Apashanka]

Homework Statement

Homework Equations

Doing this problem like e.g setting the determinant of potential matrix and the ω²*kinetic matrix equal to 0 ,det(V-ω²T)=0,I got the frequency of the normal modes of vibration to be 2ω₀ and ω₀ where ω₀ is the natural frequency,
But sir how to treat this problem quantum mechanically?
The term z is a typographical error..

The Attempt at a Solution

 

[haruspex]

You might do better to post this in the Advanced Physics homework forum and mention quantum in the title

 

[Orodruin]

Your potential is harmonic. What is the ground state energy of a (quantum) harmonic oscillator of frequency $\omega$?

 

[Apashanka]

Your potential is harmonic. What is the ground state energy of a (quantum) harmonic oscillator of frequency $\omega$?

Sir that's .5ħω for 1d motion and for 2d it is only ħω( for two independent coordinates)
But here the two independent coordinates x and y are coupled to each other

 

[Orodruin]

and for 2d it is only ħω( for two independent coordinates)

That is only true if the oscillator has the same eigenfrequencies in both eigendirections. In your case you have a problem consisting of two independent one-dimensional harmonic oscillators with different frequencies.

 

[Apashanka]

That is only true if the oscillator has the same eigenfrequencies in both eigendirections. In your case you have a problem consisting of two independent one-dimensional harmonic oscillators with different frequencies.

Yes sir that's true but what about the cross term of xy coming in the potential term??

 

[Orodruin]

Yes sir that's true but what about the cross term of xy coming in the potential term??

That just tells you that the eigendirections are not the x and y directions. If you have diagonalised the potential properly (I did not check the numerics but it seems reasonable that the problem constructor would choose values such that your frequencies come out to be integers), then you have found the frequencies in the eigendirections, which are orthogonal because the potential matrix is symmetric.

 

[Apashanka]

That just tells you that the eigendirections are not the x and y directions. If you have diagonalised the potential properly (I did not check the numerics but it seems reasonable that the problem constructor would choose values such that your frequencies come out to be integers), then you have found the frequencies in the eigendirections, which are orthogonal because the potential matrix is symmetric.

Thanks sir

 

[Apashanka]

That just tells you that the eigendirections are not the x and y directions. If you have diagonalised the potential properly (I did not check the numerics but it seems reasonable that the problem constructor would choose values such that your frequencies come out to be integers), then you have found the frequencies in the eigendirections, which are orthogonal because the potential matrix is symmetric.

Sir will you please suggest something about the potential matrix diagonalization ,the matrix terms which will appear and what actually the eigenvalues mean in this particular problem??