Polarization of the wave function

[Javier141241]

Homework Statement

Given the wave function Ψ(θ,φ,r)= f(r,θ)·[cosφ+cos2φ-i(senφ+sen2φ)] for an electron.
(φ is the azimut)

-Does it spin arround the z axis?
-What kind of polarization has? It is dextrogyre or levoryre?
-What are the posible values of L_z and what are they respective probabilites?
-Can be determinated the kinetic energy of rotation?

Homework Equations

-Spherical harmonics (Y_l^m )

-L_z=mh with m=0,±1,±2,...

The Attempt at a Solution

The expression can be simplified as

Ψ(θ,φ,r)=f(r,θ)·[exp(-iφ)+exp(-2iφ)]

For the first question,since it has dependence on φ i would say It does spin arround the z axis.

For the third,If I would like to know the posible values of L_z and the probabilities , I would have to look at a table of spherical harmonics, express Ψ(θ,φ,r) as

Ψ(θ,φ,r)=A·R_nl·Y_l^-1+B·R_nl·Y_l^-2
And then apply that <L_z>=<ψ|L_z|ψ> , and using the orthonormality of the wave function i would get and expression for L_z

<L_z>=A²·(-ħ)+B²·(-2ħ)
Now my problem is I can't get from the statement expression to one of the kind
Ψ(θ,φ,r)=A·R_nl·Y_l^-1+B·R_nl·Y_l^-2
without having one of the terms A or B dependency on θ.

For the polarization question
Ψ(θ,φ,r)=f(r,θ)·[exp(-iφ)+exp(-2iφ)]=Ψ(θ,φ,r)=f(r,θ)·[1+exp(-iφ)]·exp(-iφ)=f(r,θ)·[exp(iφ)+1]exp(-2iφ)
I can see its not linear but I am not sure wheter or not is circular,eliptical, dextrogyre or levogyre. (I would go with dextrogyre and eliptical,but can't really justify it)

For the last question, since K_rot=L²/2mr² , and L² could be any value ( since it depends on l,and l can't be deduced from the statement),I can't determinate it.

 

[Simon Bridge]

You should be verifying your answers from intuition/by inspection by applying the appropriate operator.

Of course it is easiest to use the operator on the eignenvector decomposition of the wavefunction: write out the wavefunction as a linear sum of appropriate eigenfunctions ... you do this by exploiting the fact that the eigenfunctions form a basis, and you already know what they are. You are basically just changing basis.

You could also, in a pinch, apply the differential form of the operators and just do that math.