UNIT VII

MODERN PHYSICS AND QUANTUM MECHANICS

Chapter-22

de Broglie Matter-Waves and Wave–Particle Duality

de Broglie Matter-Waves and Wave–Particle Duality

WAVE–PARTICLE DUALITY

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• Mass: A particle must have definite mass.
• Velocity: A particle can move from one place to another with a certain velocity.
• Position: A particle may be located at some definite place or point.
• Momentum: A particle having mass and velocity possesses momentum during its motion.
• Energy: A particle has energy in different forms in different situations such as potential energy, kinetic energy, rest-mass energy.

• Lack of position: A wave is always realised as a disturbance; it is spread out over a relatively large region of space. It cannot be located at some definite place or point.
• Mass: It is very difficult to think about the mass being associated with a wave.
• Frequency and wavelength: A wave or disturbance, which advances in a medium, has a certain frequency and wavelength.
• Phase of wave velocity: Phase gives an idea about the instantaneous position and direction of a wave.
• Amplitude: The amplitude of a wave gives an idea of the intensity of disturbance in the medium.

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concept of a particle

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concept of a wave

de BROGLIE HYPOTHESIS OF MATTER-WAVES

According to this theory, light may be considered as a stream of photons (particles) having mass (hν/c²), energy (hν), velocity (c), and momentum (hν /c). However, quantum theory (photon theory) could not explain the phenomena like interference, diffraction, and polarisation. It indicates that the various phenomena of light (radiation) can be made only on the basis of the dual nature of light (radiation).

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Thus, light has dual nature, i.e., it possesses both particle and wave nature. It is important to remember that wave and particle nature can never appear together.

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Extending the idea of wave–particle duality of radiation (light), Louis de Broglie in 1924 suggested that this duality is true not only for radiation but it is also true for all the moving material particles of the universe. It means that like radiation, matter also have wave–particle duality. The wavelength of the matter-wave is given by

de BROGLIE HYPOTHESIS OF MATTER-WAVES Contd….

EXPRESSION FOR WAVELENGTH OF MATTER-WAVE (de BROGLIE WAVELENGTH)

PROPERTIES OF MATTER-WAVES

• The lighter particles have greater wavelength than the heavier particles.
• The smaller the velocity of the particle, the greater is the wavelength λ=h/mv associated with it.
• From the expression of the de Broglie wavelength, i.e., λ = (h/mv, if v = 0, then λ = ∞, whereas if v = ∞, then λ = 0. This shows that the matter-waves are generated only when the particle is in motion.
• The matter-waves are independent of the charge. Thus, they are produced by both charged and uncharged particles. This shows that the matter-waves are not electromagnetic waves; they are entirely different waves.
• The velocity of the matter-waves is not constant. It depends on the velocity of the particle, while the velocity of the electromagnetic waves is constant.
• The velocity of the matter-waves may be greater than the velocity of light.

Wave velocity of a matter-wave is given in terms of group velocity as

PROPERTIES OF MATTER-WAVES Contd..

where u _group is equal to the particle velocity (see Section 22.15). Since a particle cannot travel with the velocity more than the velocity of light, the velocity of matter-wave will be greater than c.

• Wave–particle duality (wave nature of matter) introduces the concept of uncertainty. This concept suggests that if the particle nature of matter becomes certain, the wave nature will be uncertain and vice versa.

NATURE OF ELECTRON

At the same time, there are some experimental evidences which prove that the electron has a wave nature as well, i.e., a wave is associated with the electron during its motion. The following are some experiments which support the wave nature of electrons:

Davisson and Germer diffraction experiment

G.P. Thomson experiment

Before 1924, the electron was exclusively regarded as a particle but after the suggestion of de Broglie, the electron was given the wave–particle duality. There are many experimental evidences which prove that the electron is a particle, as it has a definite mass, charge, energy, and momentum. Additionally, the impact of the electron on the screen of zinc sulphide proves its identity as a particle.

DAVISSON AND GERMER EXPERIMENT FOR MATTER-WAVES

Principle: If an electron is accelerated through a potential difference of V volts, then the electron acquires the kinetic energy equivalent to qV joule, where q is the charge on the electron (in coulomb). The total energy E of the electron becomes :

With above expressions de Broglie hypothesis gives us

……..(22.11)

EXPERIMENTAL SET UP

The experimental results are discussed on the basis of different curves obtained between scattering angle ф and the intensity of scattered beam of electrons, corresponding to different accelerating voltages V. It is observed that intensity is maximum at 54 volt corresponding to the scattering angle of 50⁰ .

Results and discussion:

Bragg’s atomic planes

In order to test the theoretical result experimentally we use Bragg’s law

Using Bragg’s law we get ѳ= 65⁰ . Interatomic spacing d=0.9 A⁰

WAVE VELOCITY OR PHASE VELOCITY

The velocity of propagation of planes of constant phase through a medium is known as wave velocity, or phase velocity.

The velocity of advancement of a monochromatic wave (i.e., a wave of single frequency and wavelength) in a medium is known as wave velocity, or phase velocity.

NEED FOR WAVE PACKET REPRESENTATION

For the material particle (including electron and proton), u is always less than c. It means that, according to v=c²/u, the phase velocity of the wave associated with the material particle is always greater than c. Now, it can be concluded that the particle and its corresponding de Broglie wave cannot travel together. Hence, the particle should be left behind to its de Broglie wave.

From the earlier discussion, it seems that the particle will not be able to keep pace with the associated de Broglie wave. However, it is not so. In order to explain the conflicting idea of velocity relationship between material particle and its de Broglie wave, Schrödinger himself introduced the idea of wave packet. According to the idea of wave packet, a moving material particle is equivalent to a wave packet (number of waves) instead of a single wave.

NEED FOR WAVE PACKET REPRESENTATION contd..

WAVE PACKET

A wave packet is the resultant of a group of waves, slightly differing in velocity and wavelength, with such phase and amplitude that they interfere constructively over a small region of space where the particle can be located. Outside this space, they interfere destructively so that the amplitude reduces to zero.

GROUP VELOCITY

The velocity with which a wave packet (or the group of waves) associated with the moving particle travels is called the group velocity, whereas the velocity with which the individual waves comprising the wave packet travel is referred to as wave velocity, or phase velocity.

EXPRESSION FOR GROUP VELOCITY

RELATION BETWEEN GROUP VELOCITY AND WAVE VELOCITY

RELATION BETWEEN GROUP VELOCITY AND PARTICLE VELOCITY

GROUP VELOCITY OF de BROGLIE WAVES

According to the relativistic consideration,