Friday, March 20, 2015

quantum mechanics - Entangled or unentangled?



I got a little puzzled when thinking about two entangled fermions.


Say that we have a Hilbert space in which we have two fermionic orbitals a and b. Then the Hilbert space H's dimension is just 4, since it is spanned by {|0,ca|0,cb|0,cacb|0}, where ci are the fermionic operators that create a fermion in orbital i.


Say we have a state cacb|0. Then if I partition my Hilbert space into two by looking at the tensor product of the Hilbert spaces of each orbital, i.e. H=HaHb, then my state can be written as ca|0acb|0b, from which it is obvious that this state is unentangled (|0=|0a|0b).


Now I was thinking about writing the state in first quantized i.e. a wavefunction. Let ϕa(r),ϕb(r) be the wavefunctions of the orbitals a and b. Then ψ(x1,x2)=x1x2|cacb|0=ϕa(x1)ϕb(x2)ϕa(x1)ϕb(x2). This is where I got confused. What object is ψ(x1,x2), i.e. what Hilbert space does it belong to? What exactly are we doing when we do x1x2|cacb|0? We seem to be changing/expanding our Hilbert space by taking the position representation?


Written in this way, and assuming the same partition HaHb, the unentangled nature of the original state is no longer manifest. I'm not sure what the partition HaHb even means in this context. Would that be saying ψ(x1,x2)=ψa(x1,x2)×ψb(x1,x2) where ψi(x1,x2) is a linear combination of ϕi(x1),ϕi(x2)? This does not seem right to me.


Regardless, now I have a state written in two different but supposedly equivalent ways, with the same partition of the Hilbert space, yet it is unentangled in one way and entangled in the other.


Help?




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