[물리] Deuteron(2) [MIT]

(c) Assume that V₃ can be treated as a small perturbation. Show that in zeroth order the wave function of the state with J_z = +1 is of the form

where the ket vector is the spin state with s_pz = s_nz = +1/2.

What is the differential equation satisfied by ψ_0(r)?

sol)

In zeroth order, the perturbation = 0, L, S are good quantum numbers.

For the ground state. L = 0 and so L_z = 0 Then for the state J_z = L_z + S_z = 1 so S_z = 1 and S = 1

And for ground state, the wave function is spherically symmetric, i.e., ψ_0 = ψ_0(r).

Thus the wave function of state J_z = 1 is ψ_0(r), and

And

So we have

(Hamiltonian is given in the previous post.)

Thus the differential equation satisfied by ψ_0 is

For L ≠ 0, the wave functions of states with J_z = 1 do not have the above form.(Spherical derivate terms are alive!)

(d) What is the first order shift in energy due to the term in V₃? Suppose that to first order wave function is

where β ket is a state with S_z = -(1/2) and ψ_0 is as defined in part (c).

By selecting out the part of the Schroedinger equation that is first order in V3 and proportional to (α, α) find the differential equation satisfied by ψ₁(x).

Separate out the angular dependence of ψ₁(x) and write down a differential equation for its radial dependence.

sol)

In first order approximation, write the Hamiltonian of the system as

and the wave funcdtion is

The first order correction is given by

As

So we have

and therefore

Note that as S is conserved ans L is not, the wave function is a superposition of the spin triplet states

and

Therefore, in the first order approximation, 

we obtain

To calculate the perturbation term:

By considering the terms in the above equation that are proportional to (α, α), we can obtain the equation for the wave function ψ₁(x):

Writing ψ₁(x) = R₁(r)Φ₁(θ, φ), we can obtain from the above

Thus L²Φ₁ = 2(2+1)h²Φ₁. The equation for R₁(r) is

.