MATTER WAVES
Although waves behave like particles in some cases, no one suggested before 1924 that particles could behave like waves in some cases. In 1924, the French physicist Louis de Broglie stated that even a stream of particles has the potential to behave like a wave. There were some reasons behind his decision,
- Nature loves symmetry. Therefore, the two entities of nature—matter and energy must be in mutual symmetry.
- If energy (radiation) is in wave form and particle form, then matter will also be in particle form and wave form.
- As we know, a beam of light (which is a wave) can transfer energy and momentum to different points in an object; Again, a particle stream can also transfer energy and momentum to different points in an object. Hence, the current of these particles will be matter waves.
DUAL NATURE OF RADIATION • WAVE PARTICLE DUALITY
Regarding electromagnetic radiation as a stream of photon particles explains the photoelectric effect, black matter radiation, atomic spectra, etc.; However, this theory cannot analyze light phenomena such as deviation, inversion, convergence etc. On the other hand, the wave theory of radiation can accurately explain phenomena such as diffraction, inversion, convergence etc. Hence, experimentally, according to modern theory, radiation sometimes behaves like a wave and sometimes like a stream of particles. That is, there are two forms of radiation—wave form and particle form. Thus, wave theory and particle theory are not mutually exclusive, but complementary like the reverse side of the same coin. The same wave is called particle duality.
de BROGLIE'S HYPOTHESIS
Matter also behaves like waves. We know that the energy of a photon if the frequency of radiation is v,
E = h𝛎 or, E = hc/λ [c = 𝛎 λ]
or, λ = h/(E/c) = h/p ………(i)
where, p is the momentum of the photon. According to de Broglie’s hypothesis, equation (i) no applies to any electron or particle. Where λ is the wavelength of the electron (or particle) with momentum p, known as the de Broglie wavelength. That is, the length obtained by substituting the values of Planck constant h and momentum p of the particle in equation (i) is the de Broglie wavelength of the moving particle.
Complete Statement of de Broglie's Hypothesis
Any current of particles behaves like waves known as matter waves. The wavelength of these matter waves,
λ = h/p = h/mv ……..(ii)
Where, m = mass of particle, v = velocity of particle, p = mv = momentum of particle.
From the above relationship involving wavelength (a property of waves) and momentum (a property of particle) we conclude that,
- If v = 0, then λ = ∞; This means that only moving particles are involved in waves.
- The de Broglie wavelength does not depend on whether the particle is charged or not. This means that matter-waves are not electromagnetic waves because electromagnetic waves originate from accelerated charges.
- If either m or v is too large, the de Broglie wavelength of the particle is too short.
- As the angular momentum (p) increases, the wavelength (λ) decreases.
de BROGLIE WAVELENGTH OF MOVING ELECTION
An electron charge value, e = 1.6 X 10-19 C, has a rest-mass, m0 = 9.11 X 10-31 kg. If the speed of the electron is much less than the speed of light c then its effective mass m = m0. If the electron is accelerated by a positive potential of V volts, its acquired kinetic energy,
1/2mv² = eV [v = Accelerated velocity of the electron]
or, m²v² = 2meV or, mv = √2meV
Hence, the de Broglie wavelength of the moving electron is,
λ = h/mv = h/√2meV
= (6.625 X 10-34) / (2 X 9.11 X 10-31 X 1.6 X 10-19) X √V = (1.227 X 10-9)/√V m = 12.27/V Å
Discussion of The Value of The Wavelength of Matter Waves
1. Let the velocity of an electron (mass, m = 9.1 X 10-31 kg) be v = 10⁷ m.s-1. Then the de Broglie wavelength of the electron is,
λ = h/mv = (6.63 X 10-34) / (9.1 X 10-31 X 10⁷) = 7.3 X 10-11 m = 0.73 Å (appx)
This wavelength is equivalent to the X-ray wavelength of electromagnetic radiation.
2. Let a moving marble have mass, m = 10 g = 0.01 kg and velocity, v = 10 m.s-1. Then the de Broglie wavelength of the marble is,
λ = h/mv = (6.63 X 10-34) / (0.01 X 10) = 6.63 X 10-33 m
This value of wavelength is so small that no effective method of observing or measuring it exists. Neither electromagnetic radiation nor any other real wave is known to have such a short wavelength. From the above discussion it can be concluded that the de Broglie hypothesis has no practical importance for the macroscopic objects we observe in our daily life. The de Broglie matter-wave concept is important only for particles at the atomic level.
NATURE OF MATTER WAVES
According to de Broglie’s hypothesis, any moving particle can be guided by a matter-wave. Obviously, this requires matter-waves to obey certain conditions:
- Corresponding to the moving particles, the matter-wave must also be a progressive wave.
- The velocity at which the particle moves must be equal to the velocity of the matter waves.
- At any instant the particle in motion is at a fixed point; Therefore, the matter-wave must also be such that it can indicate the position of the particle at that moment.
Matter waves cannot be represented by pure sine waves. That is, the real nature of matter-waves is not well understood.
Let us consider a traveling wave moving along the x axis direction, Ψ = a sin(ωt – kx + δ), where a = wave amplitude, ω = angular velocity of the wave, k = wave number and δ = phase difference.
(I) This wave extends from x = – to x = +∞, with no attenuation of the wave anywhere in between. Hence it cannot in any way indicate the instantaneous position of the moving particle.
(II) Assuming that matter waves resemble pure sine waves, it can be shown that the de Broglie wave velocity exceeds the speed of light in zero, which contradicts Einstein’s theory of relativity.
Besides, a little thought shows that pure sine waves do not exist in nature. No real wave can span from -∞ to +∞ without attenuation. So, in reality we always observe the wave group.
NATURE OF MATTER WAVES
The waveform changes when multiple sine waves of different frequencies are superimposed. Thus, the resulting wave formed by the superposition of a large number of sine-waves of gradually varying frequency is called a wave group or wave pocket. Its characteristics are:
- It is also a wave moving in a certain direction (in this case along the x axis direction).
- It is a wave confined to a short range; Hence this wave can indicate the instantaneous potential position of a moving particle.
- Mathematical analysis shows that the velocity of such a wave, vg = dω/dk, this vg is called the group velocity. From proper calculations, vg = v; That is, this group velocity vg corresponds to the velocity v of the moving particle.
Photon Wave
Before discussing particle-wave duality of matter, we saw that electromagnetic radiation is also duality; Sometimes radiation is directed through waves and sometimes through currents. Obviously, it is also possible to indicate a photon moving like a particle with a corresponding waveform.
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