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## Main Question or Discussion Point

The amplitude for a state [itex]|\psi\rangle[/itex] to be in the state [itex]|\chi\rangle[/itex], with both states represented as vectors in a complex Hilbert space, is a complex number whose modulus squared gives the probability that a system in the state [itex]|\psi\rangle[/itex] is found to be in the state [itex]|\chi\rangle[/itex] after performing a suitable measurement. My question is, is it possible to assign an amplitude between two

The reason I'm asking is that in practice physicists often do exactly that. For example, in discussions about spin physicists will talk about, say, the amplitude for an electron that's spin up in the z direction to be spin up in the x direction. But [itex]|+z\rangle[/itex] and [itex]|+x\rangle[/itex] aren't vectors in the electron's Hilbert space (which includes position eigenstates tensored with spin eigenstates), they're subspaces of the Hilbert space. Yet there doesn't appear to be any problem with assigning an amplitude between them.

I can give numerous other examples of this. It really is commonplace. Yet I haven't seen any procedure for constructing an inner product between subspaces in any textbook.

*subspaces*of a Hilbert space, rather than just two vectors in a Hilbert space?The reason I'm asking is that in practice physicists often do exactly that. For example, in discussions about spin physicists will talk about, say, the amplitude for an electron that's spin up in the z direction to be spin up in the x direction. But [itex]|+z\rangle[/itex] and [itex]|+x\rangle[/itex] aren't vectors in the electron's Hilbert space (which includes position eigenstates tensored with spin eigenstates), they're subspaces of the Hilbert space. Yet there doesn't appear to be any problem with assigning an amplitude between them.

I can give numerous other examples of this. It really is commonplace. Yet I haven't seen any procedure for constructing an inner product between subspaces in any textbook.