Can Amplitudes Be Assigned Between Subspaces in Quantum Mechanics?

[dEdt]

The amplitude for a state $|\psi\rangle$ to be in the state $|\chi\rangle$, 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 $|\psi\rangle$ is found to be in the state $|\chi\rangle$ after performing a suitable measurement. My question is, is it possible to assign an amplitude between two 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 $|+z\rangle$ and $|+x\rangle$ 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.

 

[fzero]

The amplitude you are concerned with is equal to the inner product $\langle \chi | \psi \rangle$, so it is better to speak of inner products. We can compute, as here, the inner product between two states, and as you say, the modulus squared $|\langle \chi | \psi \rangle|^2$ is the probability to find the system in the state $|\chi\rangle$ given that it was in the state $\psi$ before the measurement.

Given an appropriate orthonormal basis, it is even possible to express the probability that we find the system in one of the states of a subspace of the total space. Recall that, given a complete set of states $n\rangle$ forming an orthonormal basis for a Hilbert space $\mathcal{H}$, we can write the identity operator as

$$ \hat{1} = \sum_{n=1}^N | n \rangle \langle n|.$$

If we act on a normalized state $|\psi\rangle$ with this and then take the modulus squared, we can write

$$ 1 = \sum_{n=1}^N | \langle n| \psi \rangle |^2,$$

This is a fancy way of saying that, if we make a measurement on $|\psi\rangle$, we are bound to find the system in one of the state in the total Hilbert space $\mathcal{H}$.

Now if we have a subspace $A \subset \mathcal{H}$, with orthonormal basis $|a\rangle$, then we can always write

$$ |n \rangle = \sum_{a=1}^{N_A} c_{na} | a\rangle + \sum_{\alpha=1}^{N-N_A} c_{n\alpha} | \alpha \rangle,$$

where $\langle a | \alpha \rangle = 0$ for all $a,\alpha$. You should be able to convince yourself that

$$P_A = \sum_a | a \rangle \langle a| $$

is a projection operator from $\mathcal{H}$ to $A$. Therefore if we want to compute the probability that we find the system in the subspace $A$ given a measurement on some state $|\psi\rangle$, we can compute

$$ \mathrm{Prob}(A|\psi) = \left| P_A | \psi \rangle \right|^2 = \sum_a | \langle a| \psi \rangle |^2.$$

If we're told that $|\psi \rangle$ belongs to some other subspace $X$ that intersects $A$, then we can write

$$ | \psi \rangle = \sum_{x=1}^{N_X} \psi_x | x \rangle, $$

as well as

$$\mathrm{Prob}(A|\psi) = \sum_{a,x} |\psi_x|^2 | \langle a | x \rangle |^2.$$

The quantities $|\langle a | x \rangle |^2$ are the probabilities that we find the system in a particular basis state of $A$ given that the system started in a basis state of $X$. This collection of numbers (which can be written as a matrix) is probably as close as we can get to your "amplitude between two subspaces." These quantities are closely related to the same expressions that appear in any change of basis.

Also note that if we want to recover a c-number, rather than a matrix, we really need to supply the coefficients $|\psi_x|^2$ that specify the original state up to phase factors.