How Do You Find the Eigenenergies of a Rotor in an Electric Field?

[valtorEN]

Homework Statement

A plane (x-y plane) rotor has moment of inertia I and electric dipole moment P.

A uniform electric field is applied along the x-direction (Ex, so the interaction

energy can be written as

H'=P*E cosΦ, where phi is the angle between the E-field and the dipole P

Find the eigenenergies of the rotor for arbitary eigenstate up to 1st order perturbation

Homework Equations

torque of dipole in unifrom electric field
τ=p*E

dipole
P=Q*D

S=I*omega (spin angular momentum)

The Attempt at a Solution

from E&M
A dipole placed in a uniform electric field has no force on it however a torque IS induced that tends to align the dipole up parallel to the E-field

how do i calculate the hamiltonian? i am given the perturbation but do not have the wavefunctions for this system.

any help would be impressive as i am not sure how to even approach the problem
cheers
nate

 

[nrqed]

Homework Statement

A plane (x-y plane) rotor has moment of inertia I and electric dipole moment P.

A uniform electric field is applied along the x-direction (Ex, so the interaction

energy can be written as

H'=P*E cosΦ, where phi is the angle between the E-field and the dipole P

Find the eigenenergies of the rotor for arbitary eigenstate up to 1st order perturbation

Homework Equations

torque of dipole in unifrom electric field
τ=p*E

dipole
P=Q*D

S=I*omega (spin angular momentum)

The Attempt at a Solution

from E&M
A dipole placed in a uniform electric field has no force on it however a torque IS induced that tends to align the dipole up parallel to the E-field

how do i calculate the hamiltonian? i am given the perturbation but do not have the wavefunctions for this system.

any help would be impressive as i am not sure how to even approach the problem
cheers
nate

The unperturbed hamiltonian is simply the energy of a spinning rotor with moment of inertia I. So it's basically just rotational kinetic energy. You may write this as something proportional to [itex]{\vec L}^2[/itex] (I will let you figure out the constant, which will of course contain I). This is your unpertubed hamiltonian. You then know the eigenstates and eigenvalues since you surely already know the eigenstates and eigenvalues of [itex]{\vec L}^2[/itex] .
Then you apply perturbation theory with your H' (notice that there is degeneracy in the unperturbed solution)


Patrick

 

[valtorEN]

ok patrick i thank u. i am still i tad confused...

i know the rotational K.E. is 1/2*I*ω^2 and the potential is just from the E-field = q*φso, the unperturbed Hamiltonian is T+V = 1/2*I*ω^2 +q*φ? i made a mistake on the perturbed hamiltonian, there should be a negative sign

H'=-P*E*cosφ

<ψn|H'|ψm>

what are the ψ's for this?

 

[nrqed]

ok patrick i thank u. i am still i tad confused...

i know the rotational K.E. is 1/2*I*ω^2 and the potential is just from the E-field = q*φ


so, the unperturbed Hamiltonian is T+V = 1/2*I*ω^2 +q*φ? i made a mistake on the perturbed hamiltonian, there should be a negative sign

H'=-P*E*cosφ

<ψn|H'|ψm>

what are the ψ's for this?

The E field here is treated as a perturbation so there is no electric field contribution to the unperturbed hamiltonian.

You have the right kinetic energy. Now, recalling that the magnitude of the angular momentum is [itex] L = I \omega [/itex], you can rewrite [itex] H_0 [/itex] in terms of the angular momentum (which will be squared, obviously). Since you know the eigenvalues and eigenstates of [itex] \vec{L}^2 [/itex], you basically have solved the unperturbed hamiltonian.


Patrick

 

[valtorEN]

i see u r saying that u JUST need the rotational kinetic energy and no potential energy V=0
H=T+V = 1/2*I*omega^2
I=1/3ML^2 (for rotation about end of rod of length L)

now i need the eigenvalues, but i again have a question!

this is a quantum test but do i don't need quantum mechanics to solve this?

this is just perturbation theory, the Hamiltonian still being the total energy E of the system in question.

for 1st order correction, i use power series to the 2nd term, Ao + εA1?

 

[nrqed]

i see u r saying that u JUST need the rotational kinetic energy and no potential energy V=0
H=T+V = 1/2*I*omega^2
I=1/3ML^2 (for rotation about end of rod of length L)

There is no need to introduce an explicit form for I. Just leave it as "I" (we don't know the exact mass distribution so we can't use any specific formula)


What you need now is to rewrite the unperturbed Hamiltonian in terms of the angular momentum L. see my previous post.

now i need the eigenvalues, but i again have a question!

this is a quantum test but do i don't need quantum mechanics to solve this?

You do. You will use that to identify the unperturbed eigenstates and eigenvalues.

this is just perturbation theory, the Hamiltonian still being the total energy E of the system in question.

for 1st order correction, i use power series to the 2nd term, Ao + εA1?

Yes...

 

[valtorEN]

H=Ho+H'

Ho=1/2*I*ω^2, but L=I*ω, so i can rewrite into
Ho=1/2*(I*ω)*ω=1/2*L∙ω

now
Lz|lm>=mhbar|lm>
and
L^2|lm>=l(l+1)*hbar^2|lm>

so the eigenvalues and eigenstates of L^2 are l(l+1)hbar^2 with (2l+1) possible eigenvalues for Lz

m=-l,-l+1...,0,...l-1,l

so the total Hamiltonian is

H=1/2*L*ω-P*E*cosφ

 

[nrqed]

H=Ho+H'

Ho=1/2*I*ω^2, but L=I*ω, so i can rewrite into
Ho=1/2*(I*ω)*ω=1/2*L∙ω

Wait...


Omega is still a variable here. You want to rewrite the hamiltonian in terms of L and constants only.

 

[valtorEN]

L^I*omega;

L^2=I^2*omega^2
OR
L^2/I=I*omega^2=Ho
yes?

 

[nrqed]

L^I*omega;

L^2=I^2*omega^2
OR
L^2/I=I*omega^2

Yes

L^2/I=I*omega^2=Ho
yes?

Not quite (that's not H_0) but you are just missing a factor of 1/2

After that, what you said in your previous post was right...The eigenstates of your Hamiltonian will be the same as the eigenstates of [itex] L^2 [/itex] and the energies will be the eigenvalues of [itex] L^2 [/itex] multiplied by 1/(2 I). So your unperturbed system is now solved.

 

[valtorEN]

H0=l^2/2*i?

 

[valtorEN]

H=L^2/(2I)-PEcosφ is the correct unperturbed Hamiltonian.

now i have H=L^2/2I-PEcosφ

i know L^2 has l(l+1)hbar^2 for its eigevalues but what do i do with the H' term?

again thanks for the help this problem is due Tomorrow!

 

[valtorEN]

ok i got it! (i hope!)
E(l,m)=(l(l+1)hbar^2/(2*I))-m*hbar!
VIOLA!
now i have to plug into the degenerate perturbation theory
cheers
nate

 

[nrqed]

ok i got it! (i hope!)
E(l,m)=(l(l+1)hbar^2/(2*I))-m*hbar!
VIOLA!
now i have to plug into the degenerate perturbation theory
cheers
nate

I don't think that the electric field piece shoul dbe there at all in the unperturbed energy. You should only have the first part on the left (andI don't know how you could get [itex] - m \hbar [/itex] )

 

[valtorEN]

so its just (l(l+1)hbar^2/(2*I)?

 

[nrqed]

so its just (l(l+1)hbar^2/(2*I)?

Yes, this is the unperturbed energy

 

[valtorEN]

anyone still with me on this problem?

i need to find the solutions for 1st order perturbation for this (degenerate) system!

ouch!

I have as the eigenenergies E(l,m)=l(l+1)*hbar^2/(2*I)

now how do i use denegerate perturbation theory to solve for the 1st order

correction to an eigenstate??

i assume |lm> as the notation for eigenstate (eigenvector) for angular

momentum operator L^2|lm>=l(l+1)*hbar^2|lm> and Lz|lm>=m*hbar|lm>

HELP if u can, as i am spinning my wheels (rotors?!) at this point (haha, pun VERY intended lol)

 

[nrqed]

anyone still with me on this problem?

i need to find the solutions for 1st order perturbation for this (degenerate) system!

ouch!

I have as the eigenenergies E(l,m)=l(l+1)*hbar^2/(2*I)

now how do i use denegerate perturbation theory to solve for the 1st order

correction to an eigenstate??

i assume |lm> as the notation for eigenstate (eigenvector) for angular

momentum operator L^2|lm>=l(l+1)*hbar^2|lm> and Lz|lm>=m*hbar|lm>

HELP if u can, as i am spinning my wheels (rotors?!) at this point (haha, pun VERY intended lol)

You have to write the perturbation hamiltonian in a usable form so that the matrix element [itex] <l'm'| H_{pert} |l,m>[/itex] may be easily computed.

Your perturbation hamiltonian is proportional to [itex] cos (\phi) [/itex] . The best thing to do probably is to rewrite this in terms of the ladder operators [itex] L_+ [/itex] and [itex] L_- [/itex]. Then the expectation values will be easy to calculate.

 

[valtorEN]

got some of it! (i hope!)

ok, i assume my question is a tough one, or that no one really understands

what i am asking!



here is what i have so far

1st i need B in spherical coordinates

aside from that i am using B=B0x+B0y+B0z in cartesian coordinates

say the B-field is in the +x direction, i.e., B=B0x

the interaction of the spin 1/2 particle( i assume electron) with the magnetic

field is due to the magnetic

dipole moment μ=(2*e*B0/m*c)*S ,where S is the spin vector (Sx,Sy,Sz)

therefore μ=(2*e*B0/m*c)*Sx

the Hamiltonian is 2*e*B0/(m*c)*Sx

where Sx is the x-component of the spin vector

if in the z-direction, its the same, just replace Sx with Sz etc...

since i have to find the polarization direction <σ> (just the expectation value

of σ) of the electron at time t>0, i assume i have to use the

TIME_DEPENDENT S.E. -i*hbar(∂|ψ>/∂t)=H|ψ>

hoes it looking so far? lol!

 

[valtorEN]

oops!
I need to move my last post to a different problem, but thank you nrqed (is

the qed stand for what i think it does ) i thought that the ladder ops would be

important for the states |l'm'>

let me see what i can work out

cheers
nate

 

[valtorEN]

should i rewrite the eigenenergy as E=L^2/(2*I)? (to show its dependence of total angular momentum)

also, how should i go about writing the cos(φ) into L+and or L- operators?

L+=Lx+i*Ly

L-=Lx-i*Ly

 

[nrqed]

oops!
I need to move my last post to a different problem, but thank you nrqed (is

the qed stand for what i think it does ) i thought that the ladder ops would be

important for the states |l'm'>

let me see what i can work out

cheers
nate

I was confused by you talking about a magnetic field!

You are welcome (yes, nrqed = nonrelativistic quantum electrodynamics, the subject of my phd thesis)

 

[nrqed]

should i rewrite the eigenenergy as E=L^2/(2*I)? (to show its dependence of total angular momentum)

also, how should i go about writing the cos(φ) into L+and or L- operators?

L+=Lx+i*Ly

L-=Lx-i*Ly

You will need the expressions of L+ and L- (or Lx and L_y) as differential operators in spherical coordinates.

Pat

 

[valtorEN]

L_+=hbar*e^(i*φ)*(∂/∂θ+i*cot(θ)*∂/∂φ)

L_-=hbar*e^-(i*φ)*(-∂/∂θ+i*cot(θ)*∂/∂φ)

ok, i got them squared away

how do i write cos(φ) in terms of L_+ and L_-?

also,
i would be interested in reading your thesis
cheers
nate