Applications of quantum mechanics

Decay of the kaon

The kaon (also called the K⁰ meson), discovered in 1947, is produced in high-energy collisions between nuclei and other particles. It has zero electric charge, and its mass is about one-half the mass of the proton. It is unstable and, once formed, rapidly decays into either 2 or 3 pi-mesons. The average lifetime of the kaon is about 10⁻¹⁰ second.

In spite of the fact that the kaon is uncharged, quantum theory predicts the existence of an antiparticle with the same mass, decay products, and average lifetime; the antiparticle is denoted by K⁰. During the early 1950s, several physicists questioned the justification for postulating the existence of two particles with such similar properties. In 1955, however, Murray Gell-Mann and Abraham Pais made an interesting prediction about the decay of the kaon. Their reasoning provides an excellent illustration of the quantum mechanical axiom that the wave function Ψ can be a superposition of states; in this case, there are two states, the K⁰ and K⁰ mesons themselves.

A K⁰ meson may be represented formally by writing the wave function as Ψ = K⁰; similarly Ψ = K⁰ represents a K⁰ meson. From the two states, K⁰ and K⁰, the following two new states are constructed:

From these two equations it follows that

The reason for defining the two states K₁ and K₂ is that, according to quantum theory, when the K⁰ decays, it does not do so as an isolated particle; instead, it combines with its antiparticle to form the states K₁ and K₂. The state K₁ (called the K-short [K⁰_S]) decays into two pi-mesons with a very short lifetime (about 9 × 10⁻¹¹ second), while K₂ (called the K-long [K⁰_L]) decays into three pi-mesons with a longer lifetime (about 5 × 10⁻⁸ second).

The physical consequences of these results may be demonstrated in the following experiment. K⁰ particles are produced in a nuclear reaction at the point A (Figure 7). They move to the right in the figure and start to decay. At point A, the wave function is Ψ = K⁰, which, from equation (16), can be expressed as the sum of K₁ and K₂. As the particles move to the right, the K₁ state begins to decay rapidly. If the particles reach point B in about 10⁻⁸ second, nearly all the K₁ component has decayed, although hardly any of the K₂ component has done so. Thus, at point B, the beam has changed from one of pure K⁰ to one of almost pure K₂, which equation (15) shows is an equal mixture of K⁰ and K⁰. In other words, K⁰ particles appear in the beam simply because K₁ and K₂ decay at different rates. At point B, the beam enters a block of absorbing material. Both the K⁰ and K⁰ are absorbed by the nuclei in the block, but the K⁰ are absorbed more strongly. As a result, even though the beam is an equal mixture of K⁰ and K⁰ when it enters the absorber, it is almost pure K⁰ when it exits at point C. The beam thus begins and ends as K⁰.

Gell-Mann and Pais predicted all this, and experiments subsequently verified it. The experimental observations are that the decay products are primarily two pi-mesons with a short decay time near A, three pi-mesons with longer decay time near B, and two pi-mesons again near C. (This account exaggerates the changes in the K₁ and K₂ components between A and B and in the K⁰ and K⁰ components between B and C; the argument, however, is unchanged.) The phenomenon of generating the K⁰ and regenerating the K₁ decay is purely quantum. It rests on the quantum axiom of the superposition of states and has no classical counterpart.