WAVE NATURE OF MATTER -

The physical universe is made of two great entities energy and matter.

According to Quantum theory of radiation, radiant energy have dual nature i.e it some times behaves as a wave and some times as particle (Photon)

In 1924, Louis de Broglie, suggest that like radiation, moving material particles also exhibit dual character i.e sometimes they behave as particles and some times as wave.

His suggestion was based on the reasoning that since nature loves symmetry, the two physical quantities means, energy and matter must be mutually symmetrical.

According to de Broglie, the moving particle behaves as a particle when interacting with matter and it behaves like a wave when spreading unobserved through space.

The wave associated with a moving particle is called matter-wave or de-Broglie wave and its wavelength is given by

$\fn_jvn \left [ \lambda=\frac{h}{mv} \right ]$

where h is the Plank’s constant

According to de- Broglie, the above relation is applicable to both the photon of radiation and other material particles.

NOTE:-

(a) $\fn_jvn if \;v=0,\lambda\rightarrow \infty \;and\;if\;v\rightarrow \infty,\lambda=0$

it means that matter waves are associated with material particles only if they are in motion.

(b) The de- Broglie wavelength of a moving particle is independent of the charge and nature of the particle.

(c) The greater the momentum of the particle, the smaller is the wavelength of the wave associated with it and vice versa.

(d) The de- Broglie wave, though often referred to as matter waves, are not composed of matter.

(e) Matter waves are not electromagnetic waves.

DE- BROGLIE WAVELENGTH OF AN ELECTRON

Suppose an electron at rest has been accelerated through a potential difference of V volts and gain a velocity v. If m and e are the mass and charge of electron respectively, then,

Lose in potential energy= gain in kinetic energy

$\fn_jvn i.e \;\;\;\;\; eV=\frac{1}{2}mv^2$

$\fn_jvn \therefore v=\sqrt{\frac{2eV}{m}}$

The de Broglie wavelength associated with this electron is

$\fn_jvn \lambda=\frac{h}{mv}$

$\fn_jvn \lambda=\frac{h}{m\sqrt\frac{2eV}{m}}$

$\fn_jvn \lambda=\frac{h}{\sqrt{2emV}}$

we know that

$\fn_jvn h=6.62\times10^{-34}js$

$\fn_jvn e=1.6\times10^{-19}C$

$\fn_jvn m=9\times10^{-31}kg$

Putting these value, we get

$\fn_jvn \left [ \lambda=\frac{12.27}{\sqrt V}A^0 \right ]$

for $\fn_jvn V=120V, \;\lambda=0.112nm$ . 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 crystal diffraction experiments analogous to X-ray diffraction.

The relation between wavelength and temperature:

Now, We know that

$\fn_jvn \frac{1}{2}mv^2=\frac{3}{2}kT$

where k is Boltazment constant ( $\fn_jvn 1.38\times10^{-23}j/k$ )

$\fn_jvn m^2v^2=3mkT$

$\fn_jvn \left [ \lambda=\frac{h}{\sqrt{3mkT}} \right ]\;\;(\because \lambda=\frac{h}{mv})$

HEISENBERG’S UNCERTAINTY PRINCIPLE

According to this 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 ( $\fn_jvn \Delta X$ ) in the position and some uncertainty ( $\fn_jvn \Delta P$ ) in the momentum. The product of ( $\fn_jvn \Delta X$ ) and ( $\fn_jvn \Delta P$ ) is given by

$\fn_jvn \left [ \Delta X.\Delta P\geqslant \frac{h}{4\pi} \right ]$

Here $\fn_jvn if\; \Delta X\rightarrow 0\, \;then\;\Delta P \rightarrow \infty$

and $\fn_jvn if\; \Delta P\rightarrow 0\, \;then\;\Delta X \rightarrow \infty$

Now, if an electron has a definite momentum P ( $\fn_jvn i.e\;\;\Delta P=0$ ) i.e its wavelength is definite $\fn_jvn \left ( \lambda=\frac{h}{p} \right )$ , Then a wave is definite (single wave) and its wavelength extends all over space.

This means(By Born’s probability interpretation), that the electron is not localised in any finite region of space. i.e its position uncertainty is $\fn_jvn \infty\;\;i.e \;( \Delta X\rightarrow \infty)$

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. (Localised wave)

In this case $\fn_jvn \Delta X$ is not $\fn_jvn \infty$ , but has some finite value depending on the extension of the wave packet.

NOTE:- Wave packet of finite extension is formed due to many waves of different wavelengths and this wave packet is spread around some central wavelength.

PROBABILITY INTERPRETATION TO MATTER WAVES

According to Max Born (1882-1970), the Intensity(Square of the Amplitude) of the matter-wave at a point determines the probability density of the particle at that point.

Probability density means a probability per unit value. Thus, if A is the amplitude of the wave at a point, $\fn_jvn \left | A \right |^2 \Delta V$ is the probability of the particle being found in a small volume $\fn_jvn \Delta V$ around that point.

Thus, if the intensity of matter-wave is large in a certain region, there is a greater probability of the particle being found there, compared to where the intensity is small.

IMPORTANT LINKS OF DUAL NATURE OF RADIATION AND MATTER WAVE
Introduction	Electron Emission
Photoelectric Effect	Experimental Study of Photoelectric Effect
Photoelectric Effect and Wave Theory of Light	Einstein’s Photoelectric Equation: Energy Quantum of Radiation
Particle Nature of Light: The Photon	Wave Nature of Matter
Davisson and Germer Experiment