Geometric Interpretation of Rotated basis of Hamiltonian and collective Dicke states

Question

Suppose I start with a basis of states for a two spin-1/2 particle system, namely, $\{\left|\uparrow\uparrow\right\rangle, \left|\downarrow\downarrow\right\rangle, \left|\uparrow\downarrow\right\rangle, \left|\downarrow\uparrow\right\rangle \} $, and then I apply a Hamiltonian that gives me the new basis, $\{\left|\uparrow\uparrow\right\rangle, \left|\downarrow\downarrow\right\rangle, \cos(\gamma)\left|\uparrow\downarrow\right\rangle +\sin(\gamma)\left|\downarrow\uparrow\right\rangle, \sin(\gamma)\left|\uparrow\downarrow\right\rangle -\cos(\gamma)\left|\downarrow\uparrow\right\rangle \} $

Which gives us a transformation matrix linking the original basis kets to the new basis kets which corresponds to an inversion and rotation of the last two kets in the original basis. A few questions:

Is there any physical significance to the inversion, or is it just an "arbitrary phase"?

If not, what is the most convenient way to see a geometric interpretation of the state? What would be an arbitrary collective Bloch vector for this space?

I have heard of so-called Dicke states, in which a system of $N$ spin-1/2 particles can be thought of as a single spin-$N/2$ particle, in state $\left|N/2,M\right\rangle $, where $M= -N/2,-N/2+1,...,N/2-1,N/2$.

Does this entail the use of a collective Bloch vector for all possible states solely excluding the singlet state $\left|0,0\right\rangle$?

Answer 1

Ok--I think I understand where my confusion was coming from.

The change in basis is simply convenient when attacking problems from a variational point of view. The factors of $\cos \varphi$ and $\sin \varphi$ are simply there to guarantee that the basis is orthonormalized.

With Dicke states as eigenstates of the total spin squared $S^2=\left(\sum_iS_i\right)^2$, we can use these states to describe a collective Bloch vector in the case that we are working with coherent states, which are superpositions of Dicke states: $$\left|\theta,\phi\right\rangle=\sum_Mc_M\left|J,M\right\rangle$$

So merely by a change of basis, we can get a relation between coherent states $\phi$ and the previously variational parameter $\varphi$, so long as our variational ansatz is an eigenstate of $S^2$.