I Waves are particles, particles are waves

Compton scattering occurs when X-rays [an electromagnetic (EM) wave of a very small wave length $\lambda$] hit an electron at rest (e.g., an electron bound in a solid). In the scattering event the wave is deflected and the electron is accelerated to a large velocity $𝐯$, i.e., it acquires a large final momentum ${𝐩}_e$. (We use boldface letters to denote vectors in three-dimensional space.) One observes that the wave changes its wave length to ${\lambda }^{\prime }>\lambda$, and that this change depends on the deflection angle $\varphi$ (i.e., the position of the detector which collects the scattered wave).

Classical electromagnetism does not permit such a change of wave length. In this theory, a plane wave with fixed propagation direction $𝐧$ has space and time dependence

$$\exp (i𝐤\cdot 𝐫-i\omega t),$$

(1)

where $\omega =2\pi f$ is the angular frequency (with $f$ the usual frequency), while $𝐤=k𝐧$ is the wave vector, which points into the propagation direction and has length $|𝐤|=k=2\pi /\lambda$. The scalar $k$ is called the wave number, and is strictly related to the angular frequency via the relation $\omega /k=c$, where $c$ is the speed of light. Upon scattering off a static object, the frequency of the wave does not change, and so its wave length cannot change as well.

However, a phenomenological explanation of the observations can be given when one assumes that the X-rays are composed of particles, called photons, which carry specific amounts (quanta) of energy

$$E=\mathrm{\hslash }\omega$$

(2)

and momentum

$$𝐩=\mathrm{\hslash }𝐤,$$

(3)

where

$$\mathrm{\hslash }=\mathrm{1.054\hspace{0.17em}571\hspace{0.17em}726}(47)\times {10}^{-34}\text{Js}$$

(4)

is the reduced Planck’s constant (Planck’s constant itself is defined as $h=2\pi \mathrm{\hslash }$). Equation (2) is known as the Planck relation, while Eq. (3) is known as the de Broglie relation.

These photons move at a velocity given by the speed of light $c=\lambda f=\omega /k$. According to the kinematic relations of special relativity, they hence carry no mass and obey $E=cp$. The electron has a finite mass $m$ and rest energy $E_e=m{c}^2$ before the collision, which changes to $E_e^{\prime }=\sqrt{m^2{c}^4+p_e^2{c}^2}$ after the collision. From the conservation laws for energy $\mathrm{\hslash }\omega +E_e=\mathrm{\hslash }{\omega }^{\prime }+E_e^{\prime }$ and momentum $\mathrm{\hslash }𝐤=\mathrm{\hslash }{𝐤}^{\prime }+{𝐩}_e$ it follows¹¹ 1 using $\omega =kc$, $k=2\pi /\lambda$, and $\mathrm{\hslash }kc+E_e=\mathrm{\hslash }{k}^{\prime }c+E_e^{\prime }\Rightarrow$
${[\mathrm{\hslash }(k-k^{\prime })+mc]}^2=m^2{c}^2+\underset{p_e^2,\text{ since }\mathrm{\hslash }𝐤=\mathrm{\hslash }{𝐤}^{\prime }+{𝐩}_e}{\underbrace{{\mathrm{\hslash }}^2(k^2+k^{\prime 2}-2k{k}^{\prime }\cos \varphi )}}$ that

$${\lambda }^{\prime }-\lambda =\frac{2\pi \mathrm{\hslash }}{mc}(1-\cos \varphi ).$$

(5)

This indeed recovers the experimentally observed change of the wave length.
Points to remember

• •

  De-Broglie relation $𝐩=\mathrm{\hslash }𝐤$

• •

  Planck relation $E=\mathrm{\hslash }\omega$

When small particles (electrons, $\alpha$-particles, or even C${}_{60}$ ‘bucky balls’) are sent through small slits, they randomly change direction in violation to classical mechanics. Over many experimental runs, one can identify a probability $P(𝐫)d𝐫$ that a particle arrives in a some small region $d𝐫$ around a point $𝐫$, and this probability looks similar to the intensity distribution of a diffracted wave.

This can be explained by de Broglie’s phenomenological concept of ‘matter waves’

$$\mathrm{\Psi }(𝐫;t)=A\exp (i\text{𝐤}\cdot \text{𝐫}-i\omega t),$$

(6)

where the wave properties are related to the known kinetic properties of the particle in exactly the same manner as for the photons — the angular frequency is $\omega =E/\mathrm{\hslash }$ and the wave vector is $𝐤=𝐩/\mathrm{\hslash }$. The experimentally determined probability $P(𝐫)$ is then found to be proportional to the intensity ${|\mathrm{\Psi }(𝐫;t)|}^2$ of the matter wave. We call $\mathrm{\Psi }(𝐫;t)$ the wave function.
Points to remember

• •

  All information about the state of a quantum mechanical system is encoded in a wave function, which in the examples above is of the form $\mathrm{\Psi }(𝐫;t)$.

• •

  The nature of this information is probabilistic. E.g., the probability density of a particle at position $𝐫$ is given by $P(𝐫)={|\mathrm{\Psi }(𝐫;t)|}^2$.