In addition to the operators defined in the previous section, quantum mechanics requires rules on how to use them. This section gives those rules, along with a critical discussion what they really mean.
According to quantum mechanics, the only “measurable
values” of position, momentum, energy, etcetera, are the
eigenvalues of the corresponding operator. For example, if
the total energy of a particle is
measured, the only
numbers that can come out are the eigenvalues of the total energy
There is really no controversy that only the eigenvalues come out; this has been verified overwhelmingly in experiments, often to astonishingly many digits accuracy. It is the reason for the line spectra that allow the elements to be recognized, either on earth or halfway across the observable universe, for lasers, for the blackbody radiation spectrum, for the value of the speed of sound, for the accuracy of atomic clocks, for the properties of chemical bonds, for the fact that a Stern-Gerlach apparatus does not fan out a beam of atoms but splits it into discrete rays, and countless other basic properties of nature.
But the question why and how only the eigenvalues come out is much more tricky. In general the wave function that describes physics is a combination of eigenfunctions, not a single eigenfunction. (Even if the wave function was an eigenfunction of one operator, it would not be one of another operator.) If the wave function is a combination of eigenfunctions, then why is the measured value not a combination, (maybe some average), of eigenvalues, but a single eigenvalue? And what happens to the eigenvalues in the combination that do not come out? It is a question that has plagued quantum mechanics since the beginning.
The most generally given answer in the physics community is the
orthodox interpretation. It is commonly referred to as
the “Copenhagen Interpretation”, though that interpretation, as
promoted by Niels Bohr, was actually much more circumspect than what
is usually presented.
According to the orthodox interpretation,measurementcauses the wave function tocollapseinto one of the eigenfunctions of the quantity being measured.
Staying with energy measurements as the example, any total energy
measurement will cause the wave function to collapse into one of the eigenfunctions of the total energy
Hamiltonian. The energy that is measured is the corresponding
This story, of course, is nonsense. It makes a distinction between
nature (the particle, say) and the “measurement
device” supposedly producing an exact value. But the
measurement device is a part of nature too, and therefore also
uncertain. What measures the measurement device?
Worse, there is no definition at all of what “measurement” is or is not, so anything physicists, and philosophers, want to put there goes. Needless to say, theories have proliferated, many totally devoid of common sense. The more reasonable “interpretations of the interpretation” tend to identify measurements as interactions with macroscopic systems. Still, there is no indication how and when a system would be sufficiently macroscopic, and how that would produce a collapse or at least something approximating it.
If that is not bad enough, quantum mechanics already has a law, called the Schrödinger equation (chapter 7.1), that says how the wave function evolves. This equation contradicts the collapse, (chapter 8.5.)
The collapse in the orthodox interpretation is what the classical
theater world would have called
Deus ex Machina. It
is a god that appears out of thin air to make things right. A god
that has the power to distort the normal laws of nature at will. Mere
humans may not question the god. In fact, physicists tend to actually
get upset if you do.
However, it is a fact that after a real-life measurement has been made, further follow-up measurements have statistics that are consistent with a collapsed wave function, (which can be computed.) The orthodox interpretation does describe what happens practically in actual laboratory settings well. It just offers no practical help in circumstances that are not so clear cut, being phrased in terms that are essentially meaningless.
- Even if a system is initially in a combination of the eigenfunctions of a physical quantity, a measurement of that quantity pushes the measured system into a single eigenfunction.
- The measured value is the corresponding eigenvalue.
There is another hot potato besides the collapse itself; it is the
selection of the eigenfunction to collapse to. If the wave function
measurement is a combination of many
different eigenfunctions, then what eigenfunction will the
measurement produce? Will it be ? ?
The answer of the orthodox interpretation is that nature contains a
mysterious random number generator. If the wave function before
measurement equals, in terms of the
throw the diceand select one of the eigenfunctions based on the result. It will collapse the wave function to eigenfunction in on average a fraction of the cases, it will collapse the wave function to in a fraction of the cases, etc.
The orthodox interpretation says that the square magnitudes of the coefficients of the eigenfunctions give the probabilities of the corresponding eigenvalues.
This too describes very well what happens practically in laboratory experiments, but offers again no insight into why and when. And the notion that nature would somehow come with, maybe not a physical random number generator, but certainly an endless sequence of truly random numbers seemed very hard to believe even for an early pioneer of quantum mechanics like Einstein. Many have proposed that the eigenfunction selections are not truly random, but reflect unobserved “hidden variables” that merely seem random to us humans. Yet, after almost a century, none of these theories have found convincing evidence or general acceptance. Physicists still tend to insist quite forcefully on a literal random number generator. Somehow, when belief is based on faith, rather than solid facts, tolerance of alternative views is much less, even among scientists.
While the usual philosophy about the orthodox interpretation can be taken with a big grain of salt, the bottom line to remember is:
Random collapse of the wave function, with chances governed by the square magnitudes of the coefficients, is indeed the correct way for us humans to describe what happens in our observations.As explained in chapter 8.6, this is despite the fact that the wave function does not collapse: the collapse is an artifact produced by limitations in our capability to see the entire picture. We humans have no choice but to work within our limitations, and within these, the rules of the orthodox interpretation do apply.
Schrödinger gave a famous, rather cruel, example of a cat in a box to show how weird the predictions of quantum mechanics really are. It is discussed in chapter 8.1.
- If a system is initially in a combination of the eigenfunctions of a physical quantity, a measurement of that quantity picks one of the eigenvalues at random.
- The chances of a given eigenvalue being picked are proportional to the square magnitude of the coefficient of the corresponding eigenfunction in the combination.