3.4 The Orthodox Statistical Interpretation

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.

3.4.1 Only eigenvalues

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 Hamiltonian.

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, “measurement” causes the wave function $\Psi$ to “collapse” into 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 $\psi_n$ of the total energy Hamiltonian. The energy that is measured is the corre­sponding eigenvalue:

$$\left. \begin{array}{l} \Psi = c_1 \psi_1 + c_2 \psi_2... ...ergy} = E_n \end{array} \right. \mbox{for {\em some} }n$$

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.

Key Points

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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.

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The measured value is the corre­sponding eigenvalue.

3.4.2 Statistical selection

There is another hot potato besides the collapse itself; it is the selection of the eigen­function to collapse to. If the wave function before a “measurement” is a combin­ation of many different eigen­functions, then what eigen­function will the measurement produce? Will it be $\psi_1$? $\psi_2$? $\psi_{10}$?

The answer of the orthodox inter­pretation is that nature contains a mysterious random number generator. If the wave function $\Psi$ before the “measurement” equals, in terms of the eigen­functions,

$$\Psi = c_1 \psi_1 + c_2 \psi_2 + c_3 \psi_3 + \ldots$$

then this random number generator will, in Einstein's words, “throw the dice” and select one of the eigen­functions based on the result. It will collapse the wave function to eigen­function $\psi_1$ in on average a fraction $\vert c_1\vert^2$ of the cases, it will collapse the wave function to $\psi_2$ in a fraction $\vert c_2\vert^2$ 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.

Key Points

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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.

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The chances of a given eigenvalue being picked are propor­tional to the square magnitude of the coefficient of the corre­sponding eigen­function in the combin­ation.