Wave Nature of Matter: Concept, Proof, Applications

Wave Nature of Matter: The wave nature of matter is one of the most thought-provoking theories in Physics. Phenomena like diffraction, polarization, and Interference explain that light is a wave. Like any other wave, light waves can also be specified by frequency, wavelength, amplitude, and intensity. In comparison, phenomena like blackbody radiation and the photoelectric effect justify the particle nature of light. A particle is specified by its mass, velocity, momentum, and energy.

Thus the wave-particle duality for light waves was established. The argument about wave-particle duality of light was extended to matter by Louis de Broglie in \(1924\). According to De Broglie, nature would be symmetric if matter also had both particle and wave properties. The matter is made of atoms and molecules. All atoms are further made of sub-atomic particles like neutrons, protons, and electrons. As suggested by de Broglie, the particles such as electrons being emitted from matter should also exhibit wave-like character under appropriate circumstances.

It is worth noting that this wavelength of matter waves can be associated with matter particles only if they are in motion. Also, the wavelength of a moving particle is independent of the charge and nature of the particle. An interesting thing to note is that de Broglie had no actual experimental evidence to his hypothesis when he proposed the same. In \(1927\), Davisson and Germer shot electrons onto a nickel crystal and established the wave nature of electrons. 

This article deals with the study of wave nature of matter, De Broglie hypothesis, De Broglie equation, and its applications.

De Broglie Hypothesis

According to the de-Broglie hypothesis, a moving material particle sometimes acts as a wave and sometimes as a particle. A wave is associated with every moving material particle.

The invisible wave associated with a moving particle is called de-Broglie wave or matter-wave, and it propagates in the form of wave packets with the group velocity.

Who was Louis-Victor de Broglie?

In \(1923\), a physics graduate from France named Louis-Victor de Broglie \((1892–1987)\) made a phenomenal proposal hoping that nature is symmetric. If light has particle and wave properties, matter also has particle and wave properties; then nature would be symmetric. This suggestion, made as part of his doctorate thesis, was greeted with some scepticism. When a copy of his thesis was sent to Albert Einstein, he said it was not only correct, but it might be of global importance. With the guidance of Einstein and a few other famous physicists, de Broglie was awarded his doctorate.

Fig – Louis-Victor De Broglie

De Broglie’s hypothesis of wave nature for all particles led to the foundation of quantum mechanics. An Austrian physicist Erwin Schrödinger \((1887–1961)\), published four papers in \(1926\), in which the wave nature of particles was treated explicitly with some wave equations. 

De Broglie’s work was a benchmark for the development of quantum mechanics. Louis De Broglie was awarded the Nobel Prize in \(1929\) for his discovery. Also, Davisson and G. P. Thomson were awarded the Nobel Prize in \(1937\) for their experimental verification of de Broglie’s hypothesis.

Mathematical Analysis of Wave Nature of Matter

De-Broglie Wavelength: According to the de-Broglie hypothesis, the wavelength of a matter wave is given by: 

\(\lambda=\frac{h}{p}=\frac{h}{m v}=\frac{h}{\sqrt{2 m E}} \quad \Rightarrow \lambda \propto \frac{1}{p} \propto \frac{1}{v} \propto \frac{1}{\sqrt{E}}\)

Where, 

\(h=\) Planck’s constant

\(m=\) Mass of the particle

\(v=\) Speed of the particle

\(E=\) Kinetic Energy of the particle

The smallest wavelength whose measurement is possible is that of \(\gamma\)-rays.

The wavelength of matter waves associated with the microscopic particles like electron, proton, neutron, and \(\alpha\)-particle, etc. is of the order of \(10^{-10} \mathrm{~m}\).

Different Forms of the De-Broglie Equation

Consider a charged particle accelerated through potential difference \(V\). The kinetic energy of a charged is \(E=\frac{1}{2} m v^{2}=q V\).

Hence de-Broglie wavelength \(\lambda=\frac{h}{p}=\frac{h}{\sqrt{2 m E}}=\frac{h}{\sqrt{2 m q V}}\)

\(\lambda_{\text {electron }}=\frac{12.27}{\sqrt{V}} \stackrel{\circ}{A}\)

\(\lambda_{\text {proton }}=\frac{0.286}{\sqrt{V}} \stackrel{\circ}{A}\)

\(\lambda_{\text {deutron }}=\frac{0.202 \times 10^{-10}}{\sqrt{V}} \stackrel{\circ}{A}\),

\(\lambda_{\alpha-\text { particle }}=\frac{0.101}{\sqrt{V}} \stackrel{\circ}{A}\),

Note: 

1. For uncharged particles like Neutron, \(\lambda_{\text {Neutron }}=\frac{0.286 \times 10^{-10}}{\sqrt{E(\operatorname{ineV})}} m=\frac{0.286}{\sqrt{E(\text { ineV })}}\)

2. The de-Broglie wavelength of thermal neutrons having energy \(E=k T\) at ordinary temperature is \(\lambda=\frac{h}{\sqrt{2 m k T}}\);

Where \(k=\) Boltzmann’s constant \(=1.38 \times 10^{-23}\) Joules/Kelvin, \(T=\) Absolute temp

\(\lambda_{\text {Thermal Neutron }}=\frac{6.62 \times 10^{-34}}{\sqrt{2 \times 1.07 \times 10^{-17} \times 1.38 \times 10^{-23} T}}=\frac{30.83}{\sqrt{T}} A\)

Graphs of De-Broglie Wavelength

Note: 

1. When a particle exhibits wave nature, a matter-wave is a wave packet rather than an ordinary wave.

Characteristics of Matter Waves

1. According to Max Born, a Matter-wave represents the probability of finding a particle in space. Max Born’s probability interpretation states that ‘The probability of finding the particle at any point in space depends on the square of amplitude (intensity) of matter-wave associated with the particle at that point.
2. Matter waves are not electromagnetic.
3. Matter-wave is associated with every moving particle (whether charged or uncharged). So, it is independent of charge.
4. Matter waves can be observed practically only when the de-Broglie wavelength is nearly equal to the size of the particles in nature.  
5. Electron microscope works based on the wave nature of matter.
6. The phase velocity of the de-Broglie waves can exceed the speed of the light.
7. Matter waves are non-mechanical waves.
8. The number of de-Broglie waves associated with the \(n^{\text {th }}\) orbital electron is \(n\).
9. Only those circular orbits around the nucleus are stable whose circumference is an integral multiple of de-Broglie wavelength associated with the orbital electron. 

Davisson and Germer Experiment 

American physicists Lester H. Germer and Clinton J. Davisson in \(1925\) and, independently, British physicist G. P. Thomson (son of J. J. Thomson, discoverer of the electron) in \(1926\) scattered electrons from crystals and found diffraction patterns. These patterns are exactly consistent with the Interference of electrons having the de Broglie wavelength and are somewhat analogous to light interacting with a diffraction grating.

This experiment studies the scattering of an electron from a solid or verifies the wave nature of electrons. A beam of electrons released by an electron gun falls on a nickel crystal cut along a cubical axis at a particular angle. \(Ni\) Crystal behaves like a \(3-D\) diffraction grating which diffracts the electron beam.

Fig: Davisson Germer experimental arrangement

The diffracted beam of electrons falls on a detector which can be positioned at any angle by rotating it about the point of incidence. The incident energy of electrons can also be varied by changing the applied potential difference across the electron gun.

At all scattering angles, the intensity of the scattered beam of electrons should be the same according to Classical physics. Davisson and Germer experimentally found that the intensity of the scattered beam of electrons was different at different scattering angles.

Intensity is maximum at \(54 \,V\) potential difference and \(50^{\circ}\) diffraction angle. If the de-Broglie waves electrons exist, then they should be diffracted as \(X\)-rays as found by Roentgen. Using Bragg’s formula \(2 d \sin \theta=n \lambda\), Where \(d=\) distance between diffracting planes, \(\theta=\frac{(180-\phi)}{2}\) = glancing angle for incident beam = Bragg’s angle, we can determine the wavelength of these waves.

The distance between diffraction planes in \(Ni\)-crystal for this experiment is \(d=0.91 \stackrel{\circ}{A}\) and the Bragg’s angle \(=65^{\circ}\). Therefore for \(n=1, \lambda=2 \times 0.91 \times 10^{-10} \sin 65^{\circ}=1.65 \stackrel{\circ}{A}\).

Now the de-Broglie wavelength can also be determined by using the formula \(\lambda=\frac{12.27}{\sqrt{V}}=\frac{12.27}{\sqrt{54}}=1.67 \stackrel{\circ}{A}\).  Thus the de-Broglie hypothesis is verified.

Heisenberg’s Uncertainty Principle

The wave nature of electrons leads to the quantization of energy levels in atomic bound systems as per Bohr’s atomic model. Only those states are allowed where matter waves interfere constructively exist. Since quantization is possible in an atom, the electron cannot spiral into the nucleus. It cannot exist closer to or inside the nucleus. The wave nature of matter is what prevents matter from collapsing and gives atoms their sizes. Because of the wave character of matter, the idea of well-defined orbits gives way to a model in which there is a cloud of probability, consistent with Heisenberg’s uncertainty principle.

According to Heisenberg’s uncertainty principle, it is impossible to measure the particle’s position and momentum simultaneously.

Let \(\Delta x\) and \(\Delta p\) be the uncertainty in the simultaneous measurement of the position and momentum of the particle, then \(\Delta x \Delta p= ħ\); where \(ħ=\frac{h}{2 \pi}\) and \(h=6.63 \times 10^{-34} \mathrm{~J-s}\) is the Planck’s constant.

If \(\Delta x=0\) then \(\Delta p=\infty\)

And if \(\Delta p=0\) then \(\Delta x=\infty\) i.e., if we can measure the exact position of the particle (say an electron), then the uncertainty in the measurement of the linear momentum of the particle is infinite. Similarly, if we can measure the exact linear momentum of the particle, i.e., \((p=0\), then we cannot measure the particle’s exact position at that time.

Note: 

1. The wave nature of matter causes the quantization of orbital energy in atoms. Only those orbits are possible where matter waves of electrons undergo constructive Interference in the orbit, requiring an integral number of wavelengths to fit in an orbit’s circumference; that is \(n \lambda_{n}=2 \pi r_{n}(n=1,2,3, \ldots)\), where \(\lambda_{n}\) is the electron’s de Broglie wavelength.

2. Owing to the Heisenberg uncertainty principle and the wave nature of electrons, there are no well-defined orbits; rather, there are only electron clouds of probability.

3. Bohr correctly proposed that the energy and radii of the orbits of electrons in atoms are quantized, with energy for transitions between orbits given by \(\Delta E=h f=E_{i}-E_{f}\) , where \(\Delta E\) is the change in energy between the initial and final orbits and \(h f\) is the energy of an absorbed or emitted photon.

4. This concept is useful to plot an energy-level diagram of electron orbits.

5. The allowed orbits are circular and must have quantized orbital angular momentum given by , \(L=m_{e} v r_{n}=\frac{n h}{2 \pi}(n=1,2,3 \ldots)\) where \(L\) is the angular momentum, \(r_{n}\) is the radius of the orbit \(n\), and \(h\) is Planck’s constant.

Electron Microscope

An electron microscope is an application of the Wave nature of electrons. As per the Davisson Germer experiment, a potential of only \(54 \,{\rm{V}}\) can produce electrons with much smaller wavelengths than that of visible light (hundreds of nanometres). Electron microscopes are used to detect much smaller details than optical microscopes.

There are two types of electron microscopes:

1. Transmission electron microscope (TEM) 
2. Scanning electron microscope (SEM)

The transmission electron microscope (TEM) accelerates electrons emitted from a hot filament (the cathode). The electron beam is broadened and passed through a sample. A magnetic lens focuses the image onto a fluorescent screen, a photographic plate, or a light-sensitive camera, from which it is transferred to a computer for analysis. This microscope is similar to the optical microscope in design, but it requires a thin sample examined in a vacuum. It can resolve small samples of size \(0.1 \,{\text {nm}} (10^{- 10} \,{\text {m}})\) and has a magnifying power of \(100\) million times the size of the original object. It has allowed us to see individual atoms and the structure of cell nuclei.

Fig – Transmission electron microscope

The scanning electron microscope (SEM) uses secondary electrons produced by the primary beam interacting with the sample’s surface. Magnetic lenses are used to focus the beam onto the sample. It moves the beam all around electrically to scan the sample in all directions. A detector is used to process the data for each electron position. The scanning electron microscope does not require a thin sample and provides a \(3-D\) view. Its resolving power is about ten times less than a transmission electron microscope.

Fig: Scanning electron microscope

Fig – Eye of a common fruit fly as seen by a Scanning electron microscope

Summary

1. The wave associated with a moving particle is called a matter wave.
2. According to de-Broglie, the de-Broglie wavelength is given by
\(\lambda=\frac{h}{p}=\frac{h}{m v}=\frac{h}{\sqrt{2 m E}}\)
3. The de-Broglie wavelength should be of the order of the particles’ size to observe the wave nature of matter particles practically.
4. Davisson Germer’s experiment proves the wave nature of electrons.
5. An electron microscope is an application of the wave nature of electrons.

Learn De-Broglie Wavelength Formula

FAQ’s on Wave Nature of Matter

Q.1. What is the wave nature of matter?
Ans: Matter particles (microscopic or macroscopic) in motion are associated with a wave. This wave is called the matter-wave.

Q.2. Name a phenomenon that shows the wave nature of matter?
Ans: The phenomena of electron diffraction show the wave nature of matter.

Q.3. Who discovered the wave nature of matter?
Ans: Louis-Victor de Broglie discovered the wave nature of matter.

Q.4. What is the dual nature of matter?
Ans: Matter has both the properties of a particle and as well as a wave. This phenomenon is called the dual nature of matter.

Q.5. Write one application of the wave nature of electrons?
Ans: Electron microscope is an application of the wave nature of electrons.

Q.6. What is the limitation of the de Broglie hypothesis?
Ans: De Broglie’s hypothesis applies only to microscopic particles such as protons, neutrons, electrons, etc. It is not applicable for macroscopic objects like football, cricket ball, etc. Macroscopic objects in motion have matter-wave with wavelength, but it is too small that it does not have any real existence.

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