- #1
VantagePoint72
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Given two spin-1/2 particles, the overall spin of the pair decomposes into a spin singlet and a spin triplet. Using the Clebsch-Gordon series and referring to the z-axis, we find the spin singlet is:
##|\Psi^- \rangle = \frac{1}{\sqrt{2}}(|\uparrow_z \downarrow_z \rangle - |\downarrow_z \uparrow_z \rangle)##
The overall state is one of the Bell states and is given the standard label. Being a spin singlet is the same thing as being rotationally invariant. However, consider another of the Bell states:
##|\Phi^+ \rangle = \frac{1}{\sqrt{2}}(|\uparrow_z \uparrow_z \rangle + |\downarrow_z \downarrow_z \rangle)##
This state has an interesting property; it may also be expressed as:
##|\Phi^+ \rangle = \frac{1}{\sqrt{2}}(|\uparrow_x \uparrow_x \rangle + |\downarrow_x \downarrow_x \rangle)##
or, indeed,
##|\Phi^+ \rangle = \frac{1}{\sqrt{2}}(|\uparrow_{\hat{n}} \uparrow_{\hat{n}} \rangle + |\downarrow_{\hat{n}} \downarrow_{\hat{n}} \rangle)##
where ##{\hat{n}}## is an arbitrary unit vector. But isn't this property just a mathematical expression of rotational invariance? So how does that square with the fact that it is ##|\Psi^- \rangle## (not ##|\Phi^+ \rangle##) that is the spin singlet for a bipartite system of two spin-1/2 particles?
##|\Psi^- \rangle = \frac{1}{\sqrt{2}}(|\uparrow_z \downarrow_z \rangle - |\downarrow_z \uparrow_z \rangle)##
The overall state is one of the Bell states and is given the standard label. Being a spin singlet is the same thing as being rotationally invariant. However, consider another of the Bell states:
##|\Phi^+ \rangle = \frac{1}{\sqrt{2}}(|\uparrow_z \uparrow_z \rangle + |\downarrow_z \downarrow_z \rangle)##
This state has an interesting property; it may also be expressed as:
##|\Phi^+ \rangle = \frac{1}{\sqrt{2}}(|\uparrow_x \uparrow_x \rangle + |\downarrow_x \downarrow_x \rangle)##
or, indeed,
##|\Phi^+ \rangle = \frac{1}{\sqrt{2}}(|\uparrow_{\hat{n}} \uparrow_{\hat{n}} \rangle + |\downarrow_{\hat{n}} \downarrow_{\hat{n}} \rangle)##
where ##{\hat{n}}## is an arbitrary unit vector. But isn't this property just a mathematical expression of rotational invariance? So how does that square with the fact that it is ##|\Psi^- \rangle## (not ##|\Phi^+ \rangle##) that is the spin singlet for a bipartite system of two spin-1/2 particles?