Observable [tex]G_{z}= -(xL_x+yL_y)[/tex]

[castlemaster]

Homework Statement

Consider the space of the states of a particle with no spin under an isotropic harmonic oscillator potential and the observable [tex]G_{z}= -(xL_x+yL_y)[/tex]
Obtain the spectrum of Gz. Will be ever he particle in a normalisable state where Gz is defined?

Can you interpret phisically Gz?

Homework Equations

The Attempt at a Solution

I think [tex]G_{z}= zL_z[/tex]

but how do I calculate the possible values of the observable.I tried with anhilitation and creation operators, but what represents Gz?

Thanks

 

[mark726]

for your question! It seems like you are on the right track with your understanding of Gz. In this case, Gz represents the z-component of the angular momentum operator, which is defined as the cross product of the position vector and the linear momentum vector.

To calculate the possible values of Gz, you can use the eigenvalue equation for the angular momentum operator, which is given by L_z |l,m> = mħ|l,m>, where m is the eigenvalue and ħ is the reduced Planck's constant. For a particle with no spin, the possible values of m are integers ranging from -l to l, where l is the orbital angular momentum quantum number.

So in this case, the spectrum of Gz would be all possible values of m, which can be calculated by plugging in different values of l. As for the normalizability of the particle, it is possible for the particle to be in a normalizable state where Gz is defined, as long as the wavefunction is finite and square integrable.

Interpreting Gz physically, it represents the component of the angular momentum along the z-axis. This can have implications for the orientation and stability of the particle in the harmonic oscillator potential. I hope this helps clarify things for you!