I’ve been reading a lot about massive stellar objects, degenerate matter, and how the Pauli exclusion principle works at that scale. One thing I don’t understand is what it means for two particles to occupy the same quantum state, or what a quantum state really is.
My background in computers probably isn’t helping either. When I think of what “state” means, I imagine a class or a structure. It has a spin field, an energy_level field, and whatever else is required by the model. Two such instances would be indistinguishable if all of their properties were equal. Is this in any way relevant to what a quantum state is, or should I completely abandon this idea?
How many properties does it take to describe, for example, an electron? What kind of precision does it take to tell whether the two states are identical?
Is it even possible to explain it in an intuitive manner?
Is this calculated by assuming the wavefunction is static? Like, maybe a steady-state eigenfunction of the system’s evolution with an eigenvalue that’s 1, or another root of unity.
Typically sorta? The way the Schrödinger equation is typically solved is by taking linear combinations of eigenfunctions (of the Hamiltonian) and making them time-dependent with a time-dependent phase out front.
The eigenfunctions are otherwise time-independent since you can usually make the Hamiltonian be time independent.
If the problem is easier to think about with a time-dependent Hamiltonian, you can use the Heisenberg formulation of quantum mechanics, which makes the wavefunctions static and lets the operators evolve in time. This can be helpful in a number of situations—typically involving light.
I assume you mean eigenfunction of the Hamiltonian here, but the eigenvalue associated with that eigenfunction would be the energy of the state, so you can’t really make it be a root of unity (it must, in fact, be fully real since energy is an observable)
I’ll admit, I only have a fuzzy understanding of even the basics of Hamiltonian mechanics. I understand quantum computing, though, and that evolution of a circuit is a unitary (linear) operator/matrix. So, wouldn’t continuous evolution be a one-parameter Lie subgroup of the unitary operators over your Hilbert space? Any eigenvalue would have to be a root of unity, with the exact one corresponding to rate of change in phase, because otherwise you end up with probabilities not summing to 1.
I think it would be analogous to the normal modes for a classical standing wave, which are also used as examples of an eigenfunction.
Maybe the more relevant question is if nonequilibrium, dynamical quantum systems can also be said to be quantised in the same way. Can they?
That sounds wild!
“I should really get back around to responding to that comment about time-evolution in quantum systems”
“hmm…”
Well, better late than never.
I am in quite the opposite situation; my experience is in the raw physics without much of the application to quantum computing. From what I understand, though, I think this is largely correct.
In general, observables (such as the Hamiltonian) are Hermitian (self-adjoint), which is neither a superset nor a subset of unitary operators. You are not, of course, restricted to only applying observables to your quantum state (In fact, you could apply any operator you want to your quantum state; the physical meaning behind most operators involves fundamentally changing the system, but it’s not strictly forbidden to do this). We require observables to be represented by Hermitian operators because the eigenvalues of Hermitian matrices are always real (since the value you observe is an eigenvalue of the observables’ matrix representation, you don’t want any of the eigenvalues to be non-physical complex numbers).
I had to look up why specifically quantum circuits require unitary operators, and I found this Stack Exchange response, which describes how unitary operators are used to find the time-varying component of the wavefunction that solves Schrödinger’s equation. I think we were kinda describing the same thing, ultimately: continuous evolution of a quantum system is dictated by the Hamiltonian (as shown by it’s presence in Schrödinger’s equation), and the time-varying component of the solution is unitary (the non-time varying component is a linear combination of eigenstates of the Hamiltonian whose associated eigenvalues need not be a root of unity).
Basically, when you say:
… the answer is ‘yes,’ but the Hilbert space itself is defined by the eigenstates of the Hamiltonian, which itself could be changing in time,† meaning that a complete description of the time evolution requires slightly more careful consideration.
In your example of a standing wave on a string, unitary operators would take you from one mode to another, but if the length of string is changing, those unitary operators are changing too.
†A time-varying Hamiltonian implies a time-varying energy in the system. This sounds like breaking energy conservation (and it kinda is), but it is used to describe any system that is being pumped from the outside.