Where do our representations of quantum states come from? And how do we formally tie the wavefunction back to the more vector-y notions of quantum states?
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My first course on quantum mechanics began with Heisenberg's matrix mechanics. I took the class concurrent with linear algebra, and so I struggled through the mathematical formalisms: vector spaces, dual spaces, inner products, tensor products. Maybe because it was all so new to me I didn't really question why we chose the mathematical description we did.
That changed with Schrodinger. All of a sudden I had to throw out the inner products and the complex vectors in favor of integrals and functions. Then I began to wonder why. It seemed all so arbitrary. It took four more years of math and three more courses on quantum mechanics before I began to put together an answer to this question. At least one that satisfied me.
Now parts of the story came fairly early. From the get-go, I was taught that quantum systems are linear. That's what observations tell us at least. So using vectors to represent quantum states doesn't seem like a random choice (why our vectors are complex is a completely different question). As it turned out, we're able to explain all of the representational choices from these sorts of first principles. We just need a more robust (and formal) mathematical framework.
When we measure quantum states, we need to get something. No component of the vector can be hidden from view. Otherwise, when we project onto some other quantum state in the process of measurement, the information in the hidden component will never contribute. And for this same reason, we need to be able to compose and decompose states into orthonormal bases. It may not look like it immediately, but that is a completeness condition. Or at least that fact in conjunction with the superposition principle.
Say we have an infinite set of quantum states—this is the most general case, and it shows up all over the place. We should be able to add these states however we please to get a resultant superposition. And vice versa. The only condition being that the superposition vector must have a finite magnitude (the probability must be normalizable). Since the coefficients for each component exists in \(\mathbb{C}\), we require the sum of these coefficients (squared) converge, and the metric for the resultant vector generated by this orthonormal basis follows immediately from this series, therefore all Cauchy sequences in our vector space must converge.
This means that quantum states must exist in a Hilbert space. Now all the initial questions just boil down to equivalences between Hilbert spaces.
Let's recap real quick. Linearity gave us a vector space, and measurability gave us a Hilbert space. There is only one more property we'll want to impose: separability. This choice seems to be a matter of convenience. It would be nice to be able to specify a quantum state by doing a finite number of measurements along a finite number of orthogonal directions. But, as any intro quantum mechanics course elucidates, there are many systems with infini-dimensional state spaces (just think about a particle in a square well). So, we need the next best thing: a countable number of directions should specify our state uniquely. We say our Hilbert space is separable.
Now consider \(\ell^2(\mathbb{C})\), the space of square summable sequences. For any separable Hilbert space \(\mathcal{H} \) we know there is a countable basis \(\{x_i\}_{i=1}^\infty \). It is straightfoward to show that the map that takes each \(x_i\) to the corresponding unit vector in \(\ell^2(\mathbb{C})\), \(e_i=(0, \dots, 1, 0, \dots) \), is an isometric isomorphism. In the case that our state space is n-dimensional finite, \(\ell^2(\mathbb{C})\) reduces to \(\mathbb{C}^n \).
In essence, we've just shown that there is only one Hilbert space in quantum mechanics (up to dimensionality), and that our choice of representation is both valid and natural. Our choices of complex vector spaces or countably spanned wavefunctions are consequences of linearity, measurability (completeness), and separability.
Why do we represent quantum mechanical states the way we do?