QM 1st order perturbation theory

[quantumdude10]

Homework Statement

Material contains $$10^{19}$$/$$cm^{3}$$ $$Cr^{3\frac{1}{2}}$$.
in the state $$\Psi$$(l=0, s = 3/2) with fourfold degenerate ground states. When a DC magnetic Field in x-direction is applied to the material, the spin degeneracy is lifted. At near Zero absolute temperature, only ground states are occupied.

The Question:

Determine the Value of the DC magnetic field if the resultant Zeeman level seperations will result in the absorption of a 10GHz signal propagating in the material

Homework Equations

Probably Angular momentum wavefunctions and equations, Spin Matrices

The Attempt at a Solution

Given that the Material is at Near Zero absolute temperature, I think the Wavefunctions that we have to worry about are for the following spins:

$\left|\frac{3}{2},\frac{3}{2}\right\rangle, \left|\frac{3}{2},\frac{1}{2}\right\rangle \left|\frac{3}{2},-\frac{1}{2}\right\rangle \left|\frac{3}{2},-\frac{3}{2}\right\rangle$
For this DC field, I think we have to apply first order perturbation theory in the following:

Perturbation Hamiltonian : hbar*g*($$\frac{e}{2m}$$)L.B

Any ideas if this is right or wrong?

Thanks

 

[Jhero]

in advance for any help!

Thank you for your post. Your approach to the problem seems reasonable. To determine the value of the DC magnetic field, we can use the following equation:

ΔE = gμBΔB

Where:
ΔE = the energy difference between two spin states
g = the Landé g-factor
μB = the Bohr magneton
ΔB = the change in magnetic field

In this case, we are given that the material contains 10^19/cm^3 Cr^(3/2) in the state Ψ(l=0, s=3/2), which has a fourfold degenerate ground state. At near zero absolute temperature, only the ground states are occupied. This means that we only need to consider the four spin states you mentioned in your attempt at a solution.

To determine the energy difference between these states, we can use the following equation:

ΔE = E_β - E_α = gβμBΔB - gαμBΔB = (gβ - gα)μBΔB

Where:
E_β = the energy of state β
E_α = the energy of state α
gβ = the Landé g-factor of state β
gα = the Landé g-factor of state α
μB = the Bohr magneton
ΔB = the change in magnetic field

Since we are interested in the absorption of a 10GHz signal, we can set the energy difference equal to hν, where h is Planck's constant and ν is the frequency of the signal. This gives us the following equation:

hν = (gβ - gα)μBΔB

Solving for ΔB, we get:

ΔB = hν/((gβ - gα)μB)

Now, we need to determine the Landé g-factors for the four spin states. This can be done using spin matrices. The g-factor for a spin state can be calculated using the following formula:

g = 1 + \frac{J(J+1) - S(S+1) + L(L+1)}{2J(J+1)}

Where:
J = total angular momentum quantum number
S = spin quantum number
L = orbital angular momentum quantum number

For the spin state Ψ(l=0, s=3/