The Schrodinger Equation and Born’s Rule

In quantum mechanics, the state of a system at any given time is represented by a mathematical function called a wave function. The wave function represents the coordinates of the system’s state vector, which in turn represents the state of the system in an infinite-dimensional Hilbert space (which is just like ordinary Euclidean space, only with more dimensions) in which each dimension corresponds to a possible value for a measurable variable. For example, one dimension of the Hilbert space corresponds to a particle being located at a given point, another dimension represents the particle being located at another point, etc. This is all just a mathematical abstraction, obviously, and it isn’t entirely clear exactly how this mathematical model relates to the actual physical world it describes. We’ll return to this issue later.

The Schrödinger Equation

In this one-minute video, physicist Dominic Walliman gives a simple introduction to the Schrödinger equation.

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A fundamental law of quantum mechanics called the Schrödinger equation, formulated by Austrian physicist Erwin Schrödinger, describes how the wave function changes over time. The Schrödinger equation is analogous to the laws of classical physics: it describes or governs how things happen at the fundamental level. However, according to the traditional way of understanding quantum mechanics (the so-called “Copenhagen interpretation,” which we’ll discuss later), the Schrödinger equation is only valid so long as the system is not measured.

It isn’t easy to say exactly what sorts of interactions with a system count as “measurements,” and it’s even harder to explain why nature seems to care whether someone is measuring it or not! These are serious problems for the traditional way of interpreting quantum mechanics (the Copenhagen interpretation). We’ll return to this point later.

In order to predict the outcome of a measurement, we use Born’s rule, devised by German physicist Max Born. This rule tells us the probabilities of each possible outcome of a measurement, given the system’s present state (which is described by the wave function). In a double-slit experiment, for instance, the detector film “measures” the electron’s position after it passes through the slits. Born’s rule implies that each electron has a certain probability, or chance, of landing in any given region of the detector film. The rule specifies a greater chance of landing in regions corresponding to constructive interference, and a lower probability of landing in regions of destructive interference, thus explaining the interference pattern that appears after many electrons pass through the slits.

Together, these two laws of quantum mechanics—the Schrödinger equation and Born’s rule—have been extremely successful in predicting the behaviors of subatomic systems. Strangely, however, Born’s rule seems to imply that the physical world stops obeying the Schrödinger equation whenever we measure a system! According to the Schrödinger equation, if a system is in a superposition when it is measured, the system should remain in a superposition and the measuring device itself should go into a superposition of all the possible measurement outcomes. Obviously that doesn’t happen. We always get a definite result when we measure any property of a system, even if we are sure that the system was in a superposition beforehand. (Specific examples will be given on the next page.)

Some theorists deny that collapse actually occurs, suggesting alternative explanations for the observations traditionally associated with quantum collapse. We’ll discuss two of these “no collapse” theories later in this chapter. (Don’t get your hopes up, though. The supposed explanations sound at least as crazy as the idea of quantum collapse itself, if not crazier.)

Apparently, whenever we measure something that was in a superposition, its state suddenly jumps to a definite value (or a narrow range of superposition values) for the measured variable. This sudden change is sometimes referred to as a wave function collapse (or quantum collapse), because the wave function—which represents the state of the system—seems to “collapse” from a fuzzy, smeared out superposition of possible values to a specific value or narrow range of values. For instance, when an electron hits the detector film in the double-slit experiment, its wave function suddenly collapses from a spread-out state with no definite position to a state small enough to make just a single dot on the film. Moreover, this sudden change apparently happens in a chancy (probabilistic) way. According to the traditional view of quantum mechanics, there is no predetermined fact about where the electron will land on the film. It’s just a matter of chance. Another example of quantum chanciness, involving the polarization of light, will be discussed on the next page.