1.1 The measurement problem
Our journey to the higher reality begins with the quantum wave function, often written as the Greek letter, psi: . The wave function is a mathematical description of the quantum state of a system, and it satisfies Schrödinger’s equation. Schrödinger’s equation is an expression of the conservation of energy in quantum terms for a given system.
The utility of the wave function is that it can tell you the probability that a specific event or outcome will happen in the given system. (You find this by taking the absolute square of the wave function: . The reason for taking the absolute square instead of, say, just the square, is that wave functions are generally expressed as complex numbers. Taking the absolute square yields an ordinary positive number, which, of course, probabilities have to be.)
A difficulty with the concept of the wave function was graphically illustrated by Einstein during a discussion at the now legendary 1927 Solvay Conference in Brussels (see figure).
Einstein’s thought experiment
In the figure, a beam of electrons is fired at a plate with a small hole. If the hole is small enough, the wave function for the electron emerging from the hole is hemispherical, so that it will reach every part of the hemispherical screen at the same moment. Suppose that the electron is detected on the hemispherical screen well to the left of the screen. Just before that event, the wave function has a value at every point of the screen. Therefore, it has a value at a point well to the right of the screen, just before the electron is detected over to the left. So, what is it that tells the wave function at the right of the screen to change its value immediately to zero when the electron is detected on the left?
When the electron actually appears at a point on the screen, the probability of it simultaneously appearing anywhere else on the screen has to be zero – it mustn’t appear in two different places at the same time! (Just for starters, energy wouldn’t be conserved.) So, the wave function must now instantly change its value to zero everywhere except at the point where the electron appeared. If it didn’t instantly drop to zero – if there is any delay at all after the electron has been detected – then the non-zero value of the wave function elsewhere would allow the possibility of the electron appearing at other places on the screen as well.
But how can the wave function collapse to zero everywhere instantaneously? There would be no time for “news” of the electron’s arrival at one point on the screen to spread out to the rest of the wave function in order to reduce its value everywhere else to zero. As Einstein put it, this interpretation of the wave function “implies to my mind a contradiction with the postulate of relativity”.
What Einstein meant was that the news would have to travel instantly to all parts of the wave function, exceeding the speed of light, the universal speed limit postulated in his very own theory, the special theory of relativity. Technically, we say that such behaviour in a wave function would be nonlocal. Einstein was bothered by the nonlocality requirement of quantum mechanics for the rest of his life, as we shall see.
Einstein rounded off his presentation saying that the only way that he could see his way out of the dilemma was to accept that Schrödinger’s wave function described the probability of finding the electron but that the particle must nevertheless exist as a localized particle following a distinct path during its journey to the screen. However, this flew directly in the face of Heisenberg’s Uncertainty Principle, which says, in effect, that, in its journey to the screen, the electron does not even possess both a definite position and momentum. Heisenberg maintained that “the path comes into being only because we observe it”.
The root of the tension between Einstein and the Copenhageners (an informal name for the followers of Heisenberg and Bohr – Copenhagen was where Bohr had established his Institute for Theoretical Physics in 1921) was that Einstein believed that the universe is ultimately predictable (so, if you knew the precise values of the hidden variables of a particle, then you could predict with certainty what it would do next) and the Copenhageners claimed that the universe is fundamentally unpredictable – you can give probabilities of things happening, but you cannot be certain that they will happen. To put it simply, Einstein believed in a deterministic universe and the Copenhageners believed in an indeterministic universe. It was this tension that Everett dissolved with his Many Worlds Interpretation, as we shall see later in the website.
The other question prompted by Einstein’s thought experiment is what is special about the point on the screen where the electron is detected – why that point and not a different one?
These questions – why a particular point on the screen is chosen and how the rest of the wavefunction can collapse to zero simultaneously – comprise the measurement problem.
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