The Difference Between Quantum Measurements and Properties

I gave this answer on Physics Stack Exchange to the following question:

How do we measure a quantum particle properties? (length, mass and time)

and the answer, slightly edited, is as follows:

This seems like a vague question, but on reflexion, there is really only one way to answer it that makes sense to me, so it’s actually not nearly as vague as it seems.

I’ll make a strict distinction between properties and quantum measurements. You should refer to, say, the description of the Stern-Gerlach experiment in the first chapters of Volume 1 of the Feynman Lectures on Physics (I don’t have it before me). This is a good experimental prototype for the whole quantum framework because it illustrates most of the physical content of quantum theory and it is very simple: the state space is only two-dimensional (spin-up / spin-down). Using the Stern Gerlach experiment as context and illustration, I define:

  1. A quantum system is modelled by a quantum state, which is a vector living in a Hilbert space whose components represent “probability amplitudes” for the entity to be in a certain eigenstate (read here: unit basis vector of the Hilbert space). In the Stern-Gerlach experiment, it is a two-dimensional complex value vector of the form $\left(\begin{array}{c}\psi_{up}\\\psi_{down}\end{array}\right)$ holding the probability amplitudes for an electron to be spin-up or spin-down;
  2. We model quantum measurements by observables, which are Hermitian operators on the Hilbert space together with a special recipe that tells us how to interpret these operators as measurements;
  3. Time, in non-relativistic quantum theories, is a definite, real valued parameter – in theory there is no uncertainty in it: it’s just the reading on your clock when you do the quantum measurement;
  4. Other properties (as opposed to measurements) of the system are also parameters, but they are parameters in, say, Schrödinger’s equation, which models how the quantum system evolves with time. They, like time, are assumed to have no uncertainty, and they often postulated by theoretical models and can be adjusted to fit a theoretical model to results of *quantum measurements* gathered over many experiments. Your “length” and “mass” fit into this category.

For examples of properties, the Klein Gordon equation for a lone spinless particle is:

$$(\Box + \mu^2) \psi = 0$$

where $\mu = \frac{m\,c}{\hbar}$ and $m$ is the particle’s mass. This equation tells us how the particle’s quantum state evolves with time, and so its parameter $\mu$ bears on the values of quantum measurements we make. In an imaginary history, one could imagine someone postulating this model, then making many *quantum measurements* and from these fitting the appropriate $\mu$ and hence the particle mass to the model (I say “imaginary” history because most particle masses were determined historically differently, and, by the time quantum equations were written down, the values of these model parameteres were clear).

Lastly, by way of contrast to the parameters, let me define the observable. As I said, this is an Hermitian operator together with a recipe:

  1. After the measurement, the state vector $\psi$ is in one of the observable’s eigenstates and the measurement outcome is the real eigenvalue corresponding to that eigenstate;
  2. If the quantum system has quantum state $\psi$, the $m^{th}$ moment of the probability distribution $p(\lambda)$ for the measurement $\lambda$ is $\psi^\dagger \hat{E}^m \psi$ in matrix notation (or in the bra-ket notation $\left<\psi |\hat{E}^m | \psi\right>$).

One can do any unitary transformation on the Hilbert space one likes and still keep all the information about the problem (the observables undergo corresponding transformations too of course). So it is convenient, when talking about a particular measurement, to transform the Hilbert space so that the measurement’s observable becomes a diagonal matrix. In these coordinates, the probability that the state is a particular eigenvector $\psi_0$ and thus the probability to observe a measurement equal to the corresponding eigenvalue $\lambda_0$ is particularly simple, to wit $\left<\psi_0 | \psi_0\right>$.

So, in the Stern-Gerlach case, we express the quantum states in “spin” eigen co-ordinates, so that the quantum state, as said above, is represented by the matrix:

$$\psi = \left(\begin{array}{c}\psi_{up}\\\psi_{down}\end{array}\right)$$

and the spin oberservable is then:

$$\hat{S} = \left(\begin{array}{cc}+1 & 0\\0 & -1\end{array}\right)$$

that, when combined with the state as defined by the “recipe” defines the probability distribution that the spin will take on its two allowed values $\pm 1$.

I should add that I’ve never been a big lover of the name “operator” for an observable, and always use the latter name, because I like to keep reminded that there is a recipe that goes with the observable and the name “operator” for me evokes a pure mapping, as with a state transition or time evolutionary operator $\exp(i\,\hat{H}\,t)$ where $\hat{H}$ is the energy observable (Hamiltonian) that maps the quantum state at time $t=0$ to its value at the later time $t$ in the Schrödinger picture.