Wave Particle Duality and de Broglie’s Matter Wave

The quantum theory provided the convincing explanations of Blackbody radiation (1901), photoelectric effect (1905), line spectra of atoms by Bohr’s theory (1913), Compton effect (1924) etc. On the other hand the concept of electromagnetic wave nature of radiation provided a sound explanation of interference, diffraction and polarisation. Thus, the dual character, i.e. wave nature and particle nature, of the radiation is experimentally established in different phenomena.

Thus the concept of wave-particle dualism of radiation gave explanations of different phenomena of interaction of radiation with matter. The origin of the basic postulates (i.e. idea of quantum restriction for the stationary orbits) of Bohr-Sommerfeld model of atom which provided the explanation of line spectra appeared arbitrarily chosen. As a matter of fact, this apparently inexplicable postulate, (i.e., mur = nh/2π) became the great headache to the scientists. At that time, Einstein’s mass-energy equivalence (E = mC²) was found true in many cases. On this background, the French scientist, de Broglie proposed in 1924 that the wave particle dualism is not only confined to the photons but also to other matter. This concept of matter wave when applied to subatomic particles, it yielded results in conformity with the observed ones. This new concept paved the way in the new direction called wave mechanics which have solved many problems in the microscopic world successfully.

de Broglie Wavelength for Matter Waves

In quantum theory, the energy (E) of a photon of frequency ν is given by:

E = hν

According to Einstein’s law of mass-energy equivalence, the photon must have a finite mass; and its mass given by m is related with its energy E as follows: E = mC² (where c stands for the velocity of light). Comparing the above two equations, we get:

mC² = hν;

or, mC =h (ν/C)

Or, mC = hλ, (ν = C/λ)

Thus, the momentum of the photon (mC = P) is given by P = h/λ, (Where λ is the wavelength of the photon).

In other words, the fundamental law of relativity gives the relation E =√(P²C² +m₀²C⁴)

For photons, the rest mass (m₀) is zero. Hence,

P = E/C = h/λ

The French scientist, de Broglie extended this idea to all other particles travelling with a finite velocity. The new idea helped the scientists to interpret a number of phenomena in the microscopic world and de Broglie was rightly rewarded Nobel prize in 1929. Thus the wavelength of a particle of mass m moving with a velocity u is given by:

λ = h/mu

This wave is called matter wave and the wavelength is called the de Broglie wavelength. More correctly, m is given by m = m₀(1-u²/C²)^-1/2

Wavelength of A Moving Sub-Atomic Particle

Let us consider the electron in a hydrogen atom to move with a velocity, u = 2.3 × 10⁶ mS^-1. Thus, the λ is given by:

λ =h/mu = 6.625 × 10^-34 J S /9.1 × 10^-31 kg × 2.3 × 10⁶ mS^-1

= 0.316 × 10^-9 m (J = Kgm²S^-2)

=3.16 × 10-8 cm

=3.16 Å

The value of this wavelength is comparable to that of X-rays and it is measurable as well as believable.

Wavelength of a Moving Macroscopic Particle

Let us take a marble ball of mass 1g (i.e. 10^-3 kg) to move with a velocity, u = 2 mS^-1.  Thus, the corresponding λ is given by:

λ = h/mu = 6.625 × 10^-34 J S/ 10^-3 kg ×2 mS^-1 = 3.312 × 10^-31 m = 3.312 × 10^-21 Å

The value of this wavelength is exceedingly small and it is not measurable. Besides, due to this exceedingly high frequency the energy of the ball (E = hν) becomes exceedingly high and here it is around 6 × 10⁵ J. It indicates that when de-Broglie’s hypothesis is applied to any macroscopic body, It is found to get associated with a tremendously high energy wave which is unbelievable in our sense. This is why; the de Broglie’s wave equation has no significance for the macroscopic particles.

 Difference between De Broglie Matter Wave and Electromagnetic Wave

Though the nature of matter waves is quite different from that of electromagnetic waves, both of them follow the same mathematics, i.e. they are mathematically identical. Thus the waves from a vibrating string or from water ripples in a pond, or sound waves and the matter waves follow the same mathematical equations. The concept of matter waves works well in the world of atomic particles but when applied to the microscopic world, it leads to unconvincing conclusions. To tell the fact, except the mathematics followed by the matter waves, the nature of matter waves lies really beyond our sense.

The major differences between the electromagnetic and matter waves are given below:

(i) The electromagnetic wave radiated from its source ultimately dissipates away in space. Never gets separated from its source. Thus though the electromagnetic waves can be absorbed or emitted, it does not occur so for the matter waves.

(ii) The electromagnetic wave consists of transverse vibrations produced from a combination of electric and magnetic fields perpendicular to each other, but it does not occur so for the matter waves.

(iii) The velocity of the de Broglie matter wave depends on the nature of particle under consideration while the electromagnetic wave runs with the velocity of light.

The de Broglie Electron wave

It has been observed already that when the de Broglie’s equation is applied to a moving electron, the calculated wavelength associated with the electron is quite reasonable. It is evident, that higher the velocity, smaller is the de Broglie wavelength. By subjecting a suitable potential gradient (V), the velocity (u) of the electron can be monitored as follows:

Kinetic energy = ½ mu² = eV

Or, u = √(2Ve/m)

Or, mu = √2emV

We can have the value of wavelength λ as follows:

λ = h/mu = h/√2emV

Or, λ = h/√2em₀V (taking, m = m₀)

By using the values of h, m₀, e and V (in volts) in SI system, we get:

λ = 6.625 × 10^-34 JS/√(2 × 9.11 × 10^-31 kg × 1.6 × 10^-19 C × V)

  = (12.27 /√V) × 10^-10 m

In CGS system, λ = 7.08 × 10^-9 /√V cm, where V is in statvolt (1 statvolt = 300 volts)

From the above equation, It is seen that when V = 150 volts, the electron beam produces wavelength around 1Å. Here it is worth noting that to have the X rays of wavelength around 1 Å, it requires about 12000 volts while for the electron wave of the same wavelength, only about 150 volts are required. To have a wavelength of the order of 0.05 Å, around 50 kilovolts are required. Thus it is seen that by selecting a suitable potential difference the electron beam can be made to have a wavelength matching with that of X-rays.

Relativistic Correction in the de Broglie Electron Wave

When the potential difference is made signifying high (50 kV) to have the electron beam of the wavelength of the order of 0.05 Å, the velocity of the electron is very high. Under the circumstances, the approximation m = m₀ is not valid and the consideration becomes necessary. It is treated as follows:

The relativistic formula for the kinetic energy (T) is given by:

T = (m – m₀)C²;

Or, eV = (m –m₀)C²;

Or, m = m₀ +eV/C²

Or, m = m₀( 1+ eV/m₀C²)

Or, √m = √m₀(1+eV/m₀C²)^1/2 Å

Now, by submitting the value of m, we get:

λ = (12.27/√V)(1+eV/m₀C²)^-1/2 Å

The expression of relativistic kinetic energy (T) is obtained as follows:

E (total energy) of a particle = mC² = m₀C²(1-u²/C²)^-1/2    

m = moving mass of particle having velocity u; m₀ =rest mass of the particle

In terms of kinetic energy (T) we can write:

E = T +m₀C²;

Or, T = E - m₀C² = E – m₀C²(1-u²/C²)-1/2 - m₀C²

Or, T = (m – m₀)C²

If the rest mass-energy (m₀C²) is very small, then E = T = mC² = PC (i.e. m >> m₀)

It may be noted that classical kinetic energy is given by P²/2m where P = momentum of the particle.

Bohr’s Quantum Restriction from de Broglie Concept

According to Bohr’s theory, the stationary orbits are defined by the relation, mur = n(h/2π). Now in terms of wave-particle dualism, the revolving electron in a particular stationary orbit can be considered as a stationary wave, i.e. the position of its maxima (i.e. crests) and minima (i.e. troughs) do not change with time. Such stationary waves can be obtained, if the two ends of an electron meet in the same phrase to produce a regular series of crests and troughs. To satisfy this condition, the circumstances (=2πr) of the Bohr’s circular orbit must be an integral multiple of the wavelength (λ). i.e.

2πr = nλ

Shapes of electronic waves in a Bohr's electronic orbit (a: stationary waves, b: non-stationary waves)

If the above condition is not maintained, the position of crest and troughs will be changing with time and the electron wave will be out of phase giving rise to a destructive interference. Such electron wave will be out of phase giving rise to a destructive interference. Such electron waves are not stationary.

Now by considering the de Broglie’s equation we get

2πr = n(h/P)

Or, 2πr = n(h/mu)

Or, mur = nh/2π

This is Bohr’s quantum restriction to define a stationary orbit. Therefore the quantum restriction required to define a stationary orbit does not find any support in the classical theory and it gets a sound basis in the concept of wave-particle duality as proposed by de Broglie.