Introduction

After our relatively detailed historic introduction above, we only shortly touch the important other findings that lead to quantum mechanics. The first is the photoelectric effect, discovered by Hertz in 1887 in tin plates that got positively charged when irradiated with UV light. Electrons are emitted from a metal surface only if the frequency is above a certain limit. Also, experiments by Philipp Lenard showed that the kinetic energy $(1/2)mv^2$ is independent of the intensity of the radiation. Einstein explained this effect in 1905 by introducing discrete quanta of light, i.e. photons, with energy

$$E=h\nu,$$ (11)

which are absorbed in order to kick an electron out of the metal. Energy conservation requires $E= E_{\rm kin} + W$, where $W$ the `work function', i.e. the energy to get the electrons out of the metal, and $E_{\rm kin}=\frac{1}{2}m_ev^2$ is the additional kinetic energy of the electron (mass $m_e$) when it has a finite velocity $v$. On the other hand, the general expression for the total energy of a particle is

$$E^2=(mc^2)^2+p^2c^2,$$ (12)

and for photons that move with the velocity of light $c$ the mass (rest mass) $m$ must be zero. This yields

$$pc=h\nu\leadsto p=h/\lambda.$$ (13)

Introducing the wave vector ${\bf k}$ with $\vert{\bf k}\vert=2\pi/\lambda$, pointing in the same direction as ${\bf p}$, this establishes the relation between momentum and wave vector,

$${\bf p}=\hbar {\bf k} \quad {\mbox{\rm for photons}}.$$ (14)

In particular, the photoelectric effect showed that light has a particle aspect. On the other hand, it was known that interference etc. were consequences of the wave properties of light.

In 1923, this particle-wave duality was extended to matter, i.e. massive objects, by de Broglie. Wave properties of matter had already been discussed in the 19th century by Hamilton. It was known that geometrical optics could be derived from the wave theory of light (Eikonal equation). In a similar way, there was an Eikonal equation in a branch of theoretical mechanics called Hamilton-Jacobi theory. By this one could speculate that classical mechanics had to be a limiting case of some more complete theory (quantum mechanics), in the same spirit as geometrical optics is the limiting case of wave theory.

A non-relativistic, freely moving particle of mass $m$ and momentum $p$ has a kinetic energy $E=p^2/2m$. If the particle-wave duality can be extended from photons to massive objects, this particle also can be considered as a wave, and one could postulate the same relation between momentum and wave vector as for photons,

$${\bf p}=\hbar {\bf k} \quad {\mbox{\rm for massive particles}}.$$ (15)

Further experimental hints stem from experiments where electrons are scattered at crystal surfaces and behave like waves (Davisson, Germer).

The de-Broglie relation means that a particle can be described as a wave with wave vector ${\bf k}$ and angular frequency $\omega$. The simplest form of such a wave is a plane wave

$$\Psi({\bf x},t)= A e^{i({\bf kx}-\omega t)},$$ (16)

but how should this quantity (which is a complex and not a real number)describe a particle? One could form real superpositions into $\sin$ and $\cos$, but even then this `particle' would be extended from minus to plus infinity which seems absolutely awkward. Intuition tells one that a particle should be localized in space; at any fixed time $t_0$ it should be at some point ${\bf x}_0$ somewhere in space. Still, going ahead with the wave concept of matter, one can form superpositions of plane waves.