Undoubtedly the most effective way of teaching the mathematical formalism of quantum mechanics is the axiomatic approach. Philosophically, however, this has its dangers. Axioms are supposed to be clear and compelling. The standard axioms of quantum mechanics are neither. Because they lack a convincing physical motivation, students — but not only students — tend to accept them as ultimate encapsulations of the way things are.
- The first standard axiom typically tells us that the state of a system S is (or is represented by) a normalized element v of a Hilbert space H — in other words, a unit vector in a complex vector space that is complete (in a technical sense that those who need to know it will know).
- The next axiom usually states that observables — measurable quantities — are represented by self-adjoint linear operators acting on the elements of H, and that the possible outcomes of a measurement of an observable O are the eigenvalues of O. (If Ov=cv, where c is a number, then v is an eigenvector of O and c is the corresponding eigenvalue.)
- Then comes an axiom (or a couple of axioms) concerning the (time) evolution of states. Between measurements (if not always), states are said to evolve according to unitary transformations, whereas at the time of a measurement, they are said to evolve (or appear to evolve) as stipulated by the so-called projection postulate: if O is measured, the subsequent state of S is the eigenvector corresponding to the outcome, regardless of the previous state of S.
- A further axiom stipulates that the state of a composite system is (or is represented by) a vector in the direct product of the respective Hilbert spaces of the component systems.
- Finally there are a couple of axioms concerning probabilities. According to the first, if S is “in” the state v, and if we do an experiment to see if it has the property represented by the subspace containing w, then the probability of a positive outcome is given by Born’s rule. According to the second, if S is “in” the state v, then the expected value of an observable O is <v|Ov>.
There is much here that is perplexing if not simply wrong.
To begin with, what is the physical meaning of saying that the state of a system is (or is represented by) a normalized vector in a Hilbert space? The reason why this question seems virtually unanswerable is that probabilities are introduced almost as an afterthought. It ought to be stated at the outset that the mathematical formalism of quantum mechanics is a probability calculus. It provides us with algorithms for calculating the probabilities of measurement outcomes.
If the phase space formalism of classical physics and the Hilbert space formalism of quantum physics are both understood as tools for calculating the probabilities of measurement outcomes, the transition from a 0-dimensional point in a phase space to a 1-dimensional subspace in a Hilbert space is readily understood as a straightforward way of making room for the nontrivial probabilities that we need to deal with (and even to define) fuzzy physical quantities (which in turn is needed for the stability of “ordinary” material objects).
Because the probabilities assigned by the points of a phase space are trivial, the classical formalism admits of an alternative interpretation: we may think of (classical) states as collections of possessed properties. Because the probabilities assigned by the rays of a Hilbert space are nontrivial, the quantum formalism does not admit of such an interpretation: we may not think of (quantum) states as collections of possessed properties.
Saying that the state of a quantum system is (or is represented by) a vector (in lieu of a 1-dimensional subspace) in a Hilbert space, is therefore seriously misleading.
There are two kinds of things that can be represented by a vector (or a 1-dimensional subspace) in a Hilbert space: possible measurement outcomes and actual measurement outcomes. If a possible measurement outcome is thus represented, it is for the purpose of calculating its probability. If an actual measurement outcome is thus represented, it is for the purpose of assigning probabilities to the possible outcomes of whichever measurement is made next. If v represents the outcome of a maximal test and if w represents a possible outcome of the measurement that is made next, then the probability of that outcome is |<w|v>|2. (If the Hamiltonian is not zero, this probability is |<w|Uv>|2, U being the unitary operator that takes care of the time difference between the two measurements.)
It is essential to understand that any statement about a quantum system between measurements is “not even wrong” in Wolfgang Pauli’s famous phrase, inasmuch as such a statement is neither verifiable nor falsifiable. This bears on the third axiom (or couple of axioms), according to which quantum states evolve (or appear to evolve) unitarily between measurements, which then implies that they “collapse” (or appear to do so) at the time of a measurement.
All that can safely be asserted about the time t on which a quantum state functionally depends is that it refers to the time of a measurement — either the measurement to the possible outcomes probabilities are assigned, or the measurement on the basis of whose outcome probabilities are assigned. What cannot be asserted without metaphysically embroidering the axioms of quantum mechanics is that v(t) is (or represents) an instantaneous state of affairs of some kind, which evolves from earlier to later times. As Asher Peres pointedly observed, “there is no interpolating wave function giving the ‘state of the system’ between measurements”.
Again, what could be the physical meaning of saying that observables are (or are represented by) self-adjoint operators? We are left in the dark until we get to the last couple of axioms, at which point we learn that the expected value of an observable O “in” the state v is <v|Ov>. The expected value of a measurable quantity is defined as the sum of the possible outcomes of a measurement of this quantity each multiplied (“weighted”) by its (Born) probability, and a self-adjoint operator O can be defined so that this weighted sum takes the form <v|Ov>. That’s all there is to observables “being” self-adjoint operators.
And finally, why would the state of a composite system be (represented by) a vector in the direct product of the Hilbert spaces of the component systems? Once again the answer is self-evident if quantum states are seen for what they are — tools for assigning probabilities to the possible outcomes of measurements.
There is a widely held if not always explicitly stated assumption, which for many has the status of an additional axiom. This is the so-called eigenstate-eigenvalue link, according to which a system “in” an eigenstate of an observable O — that is, a system associated with an eigenvector of O — possesses the corresponding eigenvalue even O is not, in fact, measured. Because the time-dependence of a quantum state is not the continuous dependence on time ofbf an evolving state but a dependence on the time of a measurement, we must reject this assumption. All that Ov(t) = ov(t) implies is that a (successful) measurement of O made at the time t is certain to yield the outcome o. In other words, probability 1 is not sufficient for “is” or “has.”
If a system’s being in an eigenstate of an observable is not sufficient for the possession, by the system or the observable, of the corresponding eigenvalue, then what is?
We came across several experimental arrangements that warranted the following conclusion: measurements do not reveal pre-existent values; they create their outcomes. If so, the only sufficient condition for the existence of a value o of an observable O is a measurement of O. Observables have values only if, only when, and only to the extent that they are measured. In short, to be is to be measured. This was the insight that Niels Bohr tried to convey when he kept insisting that, out of relation to experimental arrangements, the properties of quantum systems are undefined.[2,3]
1. [↑] Peres, A. (1984). What is a state vector? American Journal of Physics 52, 644–650.
2. [↑] Jammer, M. (1974). The Philosophy of Quantum Mechanics, Wiley, pp. 68–69.
3. [↑] Petersen, A. (1968). Quantum Physics and the Philosophical Tradition, MIT Press.