# Geometric Interpretation of Rotated basis of Hamiltonian and collective Dicke states

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 $$\left|0,0\right\rangle$$?

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 $S^2$.