In 1925 and 1926 two versions of quantum mechanics were born almost at once. Werner Heisenberg’s used infinite matrices; Erwin Schrödinger’s, a wave equation. Both gave the same results in experiments, but they seemed to speak of different things, and physicists used tricks that mathematicians found unacceptable, such as Dirac’s delta “function”.

In Göttingen, working with Hilbert, von Neumann showed that the two were the same theory written in different coordinates, and rewrote everything in a single language. The book Mathematical Foundations of Quantum Mechanics, of 1932, is still the reference for anyone who wants the theory with full rigour.

States are vectors

In von Neumann’s language, the state of a physical system is a vector of length 1 in a Hilbert space, a space with a scalar product, like the plane or three-dimensional space, but possibly with infinitely many dimensions. In a minimal example, a system that, when measured, can only give two results, “up” or “down”, has states in the plane:

$$\psi = a\,\mathbf{e}_\uparrow + b\,\mathbf{e}_\downarrow, \qquad a^2 + b^2 = 1.$$

The rule that connects the vector to the laboratory, proposed by Max Born and organised by von Neumann, is: the probability of each result is the square of the corresponding component. With $a = \frac35$ and $b = \frac45$,

$$P(\uparrow) = \left(\tfrac35\right)^2 = \tfrac{9}{25} = 36\%, \qquad P(\downarrow) = \left(\tfrac45\right)^2 = \tfrac{16}{25} = 64\%.$$

The condition $a^2 + b^2 = 1$, that the vector has length 1, is Pythagoras’s theorem saying that the probabilities add up to 100%.

Measuring is projecting

After the measurement, the system does not stay in $\psi$: it passes into the state that corresponds to the result obtained, $\mathbf{e}_\uparrow$ or $\mathbf{e}_\downarrow$. Von Neumann wrote this as a projection of the vector onto an axis, and called the quantities that are measured, such as energy or position, operators. The values a measurement can give are the eigenvalues of these operators. It was the first time laboratory physics had been described entirely with infinite-dimensional linear algebra, and it was for this that von Neumann created much of the operator theory taught in graduate mathematics today.

Exercise: normalising a state

A state is proportional to $\mathbf{e}_\uparrow + \mathbf{e}_\downarrow$. Write it with length 1 and compute the probabilities.

The length of $(1, 1)$ is $\sqrt{2}$, so the state is $\psi = \frac{1}{\sqrt2}\mathbf{e}_\uparrow + \frac{1}{\sqrt2}\mathbf{e}_\downarrow$. The probabilities are $\left(\frac{1}{\sqrt2}\right)^2 = \frac12$ for each result: a fair quantum coin.

Exercise: the angle and the probability

If the state makes an angle $\theta$ with the axis $\mathbf{e}_\uparrow$, that is, $\psi = \cos\theta\,\mathbf{e}_\uparrow + \sin\theta\,\mathbf{e}_\downarrow$, what is $P(\uparrow)$? For what angle is it $75\%$?

$P(\uparrow) = \cos^2\theta$. For $\cos^2\theta = \frac34$, $\cos\theta = \frac{\sqrt3}{2}$, so $\theta = 30°$. The fundamental identity $\cos^2\theta + \sin^2\theta = 1$ guarantees, once again, that the two probabilities add up to 1.

Sources: John von Neumann, Mathematische Grundlagen der Quantenmechanik (1932) · Grete Hermann, “Die naturphilosophischen Grundlagen der Quantenmechanik” (1935) · John S. Bell, “On the problem of hidden variables in quantum mechanics”, Reviews of Modern Physics (1966).

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