Can an operator acting on a wave function be equal to taking a measurement?

Question

I'm aware in general when an operator acts on a quantum wave function it is not the same as taking a measurement ie the hamiltonian acting on a superposition state will not give a constant multiplied by that state as taking an energy measurement on a superposition state will instead collapse it down to one energy eigenfunction probabalistically and will give the energy eigenvalue associated with that state. I am also aware that not all operators are hermitian so won't give real observables. However, would i be able to say that a hermitian operator acting on one of its corresponding eigenfunctions is the same as taking a measurement?

Answer 1

Mathematically, such an operator in that particular situation would have the same effect as an (ideal) measurement of the eigenized observable, i.e. it leaves the state unchanged, for the "state" is not the vector but rather the associated Hilbert space ray, and the introduced eigenvalue factor doesn't change that. But I would caution against reading too much into this - if anything, it's more of a "mathematical coincidence" than anything helpful about thinking of how measurements work in quantum theory.

Instead, a "better" way to think of Hermitian operators in the classic formulation of quantum theory is they serve as the analogues of physical quantities, but have the right mathematical structure to deal with the fundamental informational tradeoffs present at the quantum level. Hence there is not really a lot of direct meaning to the action of an observable operator $\hat{O}$ upon a state vector $|\psi\rangle$, i.e.

$$\hat{O} |\psi\rangle$$

defies any "intuitive" interpretation, and you should not seek one.

That said, I will admit that this is a bit of an unsatisfactory approach to quantum theory, because as you point out, say, the Hamiltonian actually represents an "action" being performed, namely the effect of dynamics, and the term "operator" also carries as kind of connotation of action, and so we might like to have a formalism where that carries to measurements too. And in fact, you can come up with such a formalism, and here's how.

The idea is basically to note that we can consider "measurement" of an observable to really be "answering a question" regarding its value or, that is to say, procurement of the truth or falsity of a proposition thereabout. For example, when it comes to the classic "spinning electron" problem with its "spin up" vs. "spin down" states, we are asking the question "what is the $z$-component of the spin of the electron?" which then in turn can be broken down into two yes-no questions, viz. "is the relevant component spin-up?" and "is the relevant component spin-down?" To each of those two questions, we can then assign a projector operator (or what I call an "answer operator") $\hat{\Pi}_{S_z}(S_z)$ which has the effect of changing a state vector $|\psi\rangle$ to another which is basically that one but with the information that the component had the value $S_z$, i.e. the post-measurement state you are thinking of. Hence, we can represent the observable as the total collection of such yes/no questions about whether individual outcomes did or did not obtain, and thus mathematically, a function that returns projection operators for each possible outcome - much like how a probability mass function assigns a probability to each possible outcome.

And likewise, in the most general case so we can also accommodate continuously-valued observables, we can describe these as analogous to probability measure functions, i.e.

$$P(X \in E)$$

the probability that the measurement of the variable $X$ lies within the set $E$, i.e. an event. Only here, what want is something that returns a projector operator that represents the action that happens, as you say, when a measurement reveals that the variable does indeed lie in that set:

$$\hat{\Pi}_O(O \in E).$$

Such a construction is generally called a Projection-Valued Measure or PVM, and you can, in fact, build QM on PVMs without ever needing to touch the classic "eigen" Hermitian-operator stuff. It's just that then more mathematical work is required to develop things like how that, say, the algebra of such PVMs, when understood as standing for observable quantities, would have to work. Personally, I actually would prefer the PVM approach because it lets you talk right off the bat about measurements with limited precision (i.e. "between which two ticks on the ruler does it lie?") and is more intuitive for just the reason you suggest - the application of the above to a state vector does exactly what you are trying to say it does, i.e. it makes it into what it would be after the measurement.

Moreover, a generalization of a PVM, called a POVM, or positive operator-valued measure, provides the most realistic model of the results of actual measurements in quantum theory. In particular, they are precisely what you get when you overlay the same kind of classical statistics that we would use to talk about errors in measurements at a classical-physics or general experimental science level, atop the PVM formalism just described.

Answer 2

No. I am aware of only two uses of operators acting on states:

1. Generating time, space translations and rotations using Schrodinger-like equations

2. $\langle \psi |A| \psi \rangle$ is the expected value of the operator in a large number of measurements. This is the closest thing to what you want.

Answer 3

No, operators are not the same thing as "taking a real measurement". At the end of the day a quantum measurement is always an irreversible energy, momentum, angular momentum and charge transfer but in an experiment this can become a frequency, a distance measurement, a time difference or even a temperature measurement (among a plethora of other things). If we take many such quantum measurements, however, and do a bit of post-processing, then we can recover probability distributions that are similar to what we get from some of the measurement operators in von Neumann's Hilbert space formalism. The details of this post-processing depend very much on the physical system we are investigating. It is usually done by experimentalists rather than theorists (or theorists who are working on the experimental side) and the details can be found in textbooks about atomic, nuclear, high energy and solid state physics, among others. They are rarely mentioned in theoretical textbooks about quantum mechanics because there is no simple, one-size-fits-all kind of explanation for what real measurements are. Most of the details of the translation process depend on how experimentalists manage to use the limited tools that nature provides us with (vs. the unlimited mathematical expressions that theorists have access to).

A simple example of the necessary intermediate step are optical line spectra in atomic and molecular physics. The formalism gives us energy levels but in an experiment we can only measure differences between energy levels and even then we don't know a priori which levels they belong to. That has to be reconstructed from other data, e.g. how the wavelengths shift with an external electric and magnetic field. That may not sound like a big deal, but it does give you a hint that the math that we use to solve the Schroedinger equation is not quite the same as the physics of actual quantum systems.