11.8 Wave Nature of Matter
The dual (wave-particle) nature of light (electromagnetic radiation, in general) comes out clearly from what we have learnt in this and the preceding chapters. The wave nature of light shows up in the phenomena of interference, diffraction and polarisation. On the other hand, in photoelectric effect and Compton effect which involve energy and momentum transfer, radiation behaves as if it is made up of a bunch of particles – the photons. Whether a particle or wave description is best suited for understanding an experiment depends on the nature of the experiment. For example, in the familiar phenomenon of seeing an object by our eye, both descriptions are important. The gathering and focussing mechanism of light by the eye-lens is well described in the wave picture. But its absorption by the rods and cones (of the retina) requires the photon picture of light.
A natural question arises: If radiation has a dual (wave-particle) nature, might not the particles of nature (the electrons, protons, etc.) also exhibit wave-like character? In 1924, the French physicist Louis Victor de Broglie (pronounced as de Broy) (1892-1987) put forward the bold hypothesis that moving particles of matter should display wave-like properties under suitable conditions. He reasoned that nature was symmetrical and that the two basic physical entities – matter and energy, must have symmetrical character. If radiation shows dual aspects, so should matter. De Broglie proposed that the wave length λ associated with a particle of momentum p is given as
\[ \lambda=\frac{h}{p}=\frac{h}{m v} \]
where m is the mass of the particle and v its speed. Equation (11.5) is known as the de Broglie relation and the wavelength λ of the matter wave is called de Broglie wavelength. The dual aspect of matter is evident in the de Broglie relation. On the left hand side of Eq. (11.5), λ is the attribute of a wave while on the right hand side the momentum p is a typical attribute of a particle. Planck’s constant h relates the two attributes.
Equation (11.5) for a material particle is basically a hypothesis whose validity can be tested only by experiment. However, it is interesting to see that it is satisfied also by a photon. For a photon, as we have seen,
\[ p=h v / c \] Therefore, \[ \frac{h}{p}=\frac{c}{v}=\lambda \]
That is, the de Broglie wavelength of a photon given by Eq. (11.5) equals the wavelength of electromagnetic radiation of which the photon is a quantum of energy and momentum.
Clearly, from Eq. (11.5 ), λ is smaller for a heavier particle (large m) or more energetic particle (large v). For example, the de Broglie wavelength of a ball of mass 0.12 kg moving with a speed of 20 m \(s^{–1}\) is easily calculated: \[ \begin{array}{l} p=m v=0.12 \mathrm{~kg} \times 20 \mathrm{~m} \mathrm{~s}^{-1}=2.40 \mathrm{~kg} \mathrm{~m} \mathrm{~s}^{-1} \\ \lambda=\frac{h}{p}=\frac{6.63 \times 10^{-34} \mathrm{~J} \mathrm{~s}}{2.40 \mathrm{~kg} \mathrm{~m} \mathrm{~s}^{-1}}=2.76 \times 10^{-34} \mathrm{~m} \end{array} \]
This wavelength is so small that it is beyond any measurement. This is the reason why macroscopic objects in our daily life do not show wave-like properties. On the other hand, in the sub-atomic domain, the wave character of particles is significant and measurable.
Consider an electron (mass m, charge e) accelerated from rest through a potential V. The kinetic energy K of the electron equals the work done (eV ) on it by the electric field:
\[ K=e V \]
\[ \begin{aligned} &\text { Now, } K=\frac{1}{2} m v^{2}=\frac{p^{2}}{2 m}, \text { so that }\\ &p=\sqrt{2 m K}=\sqrt{2 m e V} \end{aligned} \]
The de Broglie wavelength λ of the electron is then
\[ \(\lambda=\frac{h}{p}=\frac{h}{\sqrt{2 m K}}=\frac{h}{\sqrt{2 m e V}}\) Substituting the numerical values of \(h, m, e,\) we get \(\lambda=\frac{1.227}{\sqrt{V}} \mathrm{nm}\) \]
where V is the magnitude of accelerating potential in volts. For a 120 V accelerating potential, Eq. (11.11) gives λ = 0.112 nm. This wavelength is of the same order as the spacing between the atomic planes in crystals. This suggests that matter waves associated with an electron could be verified by crystal diffraction experiments analogous to X-ray diffraction. We describe the experimental verification of the de Broglie hypothesis in the next section. In 1929, de Broglie was awarded the Nobel Prize in Physics for his discovery of the wave nature of electrons.
The matter–wave picture elegantly incorporated the Heisenberg’s uncertainty principle. According to the principle, it is not possible to measure both the position and momentum of an electron (or any other particle) at the same time exactly. There is always some uncertainty (\(\Delta x\)) in the specification of position and some uncertainty (\(\Delta p\)) in the specification of momentum. The product of \(\Delta x\) and \(\Delta p\) is of the order of ħ * (with \(\hbar=h / 2 \pi\)), i.e.,
\[ \Delta x \Delta p \approx \hbar \]
Equation (11.12) allows the possibility that ∆x is zero; but then ∆p must be infinite in order that the product is non-zero. Similarly, if ∆p is zero, ∆x must be infinite. Ordinarily, both ∆x and ∆p are non-zero such that their product is of the order of ħ.
Now, if an electron has a definite momentum p, (i.e. ∆p = 0), by the de Broglie relation, it has a definite wavelength λ . A wave of definite (single) wavelength extends all over space. By Born’s probability interpretation this means that the electron is not localised in any finite region of space. That is, its position uncertainty is infinite \((\Delta x \rightarrow \infty)\), which is consistent with the uncertainty principle.
In general, the matter wave associated with the electron is not extended all over space. It is a wave packet extending over some finite region of space. In that case ∆x is not infinite but has some finite value depending on the extension of the wave packet. Also, you must appreciate that a wave packet of finite extension does not have a single wavelength. It is built up of wavelengths spread around some central wavelength.
By de Broglie’s relation, then, the momentum of the electron will also have a spread – an uncertainty \(\Delta p\). This is as expected from the uncertainty principle. It can be shown that the wave packet description together with de Broglie relation and Born’s probability interpretation reproduce the Heisenberg’s uncertainty principle exactly.
In Chapter 12, the de Broglie relation will be seen to justify Bohr’s postulate on quantisation of angular momentum of electron in an atom.
Figure 11.6 shows a schematic diagram of (a) a localised wave packet, and (b) an extended wave with fixed wavelength
(a) The wave packet description of an electron. The wave packet corresponds to a spread of wavelength around some central wavelength (and hence by de Broglie relation, a spread in momentum). Consequently, it is associated with an uncertainty in position (∆x) and an uncertainty in momentum (∆p). (b) The matter wave corresponding to a definite momentum of an electron extends all over space. In this case, ∆p = 0 and ∆ x → ∞.