Deuteron Ground State / Perturbation Theory

[EEWannabe]

Homework Statement

The deuteron ground state is made up of l = 0 and l = 2 states;
a)Show this mixture cannot be an eigenstate of a central potential Hamiltonian
b)Using first-order time independent perturbation theory, argue the potential must contain a term proportional to some combination of Y2m for this mixing to be possible.
c) However Y2m are not scalars, they're components of rank 2 tensors. The Hamiltonian must be a scalar so how can one construct a perturbation that is a scalar yet contain this rank 2 tensor?

Homework Equations

$H = (\frac{p^{2}}{2m} + V(r))$
$\psi_{total} = c\psi_{l=0} + d\psi_{l=2}$
$\psi_{l=0} = \frac{u_{0}(r)}{r} Y_{0m}$
$\psi_{l=2} = \frac{u_{2}(r)}{r} Y_{2m}$

The Attempt at a Solution

a) Since r is the same for both of them the fact that $H \psi = E \psi$, this implies that

$\frac{p^{2}}{2m}c \psi_{l=0} + \frac{p^{2}}{2m}c \psi_{l=2} = D (c \psi_{l=0} + d \psi_{l=2}) = (E - V(r))(c \psi_{l=0} + d \psi_{l=2})$

where D is a constant, however this is not true as as the two psi functions are not of the same form, hence they are not an eigenstate if the potential is constant.

b) I'm really stuck on this bit, I'm not really very familiar with time independant pertubation theory but I've been trying to teach myself, so I let the hamiltonian equal sometime of the form;

$H_{new} = H_{0} + V$

I've tried playing around with the algebra trying to figure out why V(r) would be proportional to Y2m but I'm really stuck, and I can't envisage why it'd be proportional to Y2m but not Y0m anyway.

I'm sorry but I really don't know where to go, if anyone could point me in the right direction that'd be great thanks.

Thank you for your time.

 

[mark726]

I would like to offer some insights on your questions.

a) You are correct in saying that the mixture cannot be an eigenstate of a central potential Hamiltonian. This is because the two states, l=0 and l=2, have different angular momentum values and cannot be combined to form an eigenstate.

b) In order for the mixing to be possible, the potential must contain a term that is proportional to a combination of Y2m. This is because Y2m represents the spherical harmonics, which are functions of the polar angles (θ and φ). These functions are related to the angular momentum of the system, and in this case, the l=2 state. So, in order to mix the two states, the potential must contain a term that is proportional to Y2m.

c) You are correct in saying that Y2m are components of rank 2 tensors and the Hamiltonian must be a scalar. In order to construct a perturbation that is a scalar yet contains this rank 2 tensor, we can use the tensor product of two spherical harmonics. This will result in a scalar quantity that contains the information of both Y0m and Y2m. This can then be used as a perturbation in the Hamiltonian.

I hope this helps clarify your questions. If you have any further questions, please let me know.