WAVE MECHANICS
De Broglie hypothesis of matter waves
INTRODUCTION
Radiation is found to exhibit dual character. Some phenomena such as interference,
diffraction, polarization etc., are due to wave character of radiation. On the other hand,
phenomena such as photoelectric effect, Compton effect etc., could not be explained on the basis
of a wave theory. These latter phenomena exhibit the particle-like character of radiation. Thus
radiation exhibits a dual character, namely, a wave character and a particle – like character.
In Compton effect, monochromatic X-rays of wavelength falling on a scatterer are
scattered into a different direction. At the same time, an electron ejected from the scattering
atom recoils in some other direction. It is found that the scattered X-ray has a wavelength ‟
which is longer than the incident wavelength . ‟ > .
Scattering atom
Scattered X-ray, ‟
P‟ = h /‟
Incident X-ray,
P = h /
Recoiling electron,
P = P‟ + Pe pe
Compton assumed that X-ray photons have certain momentum and he treated the scattering of X-
rays as due to the inelastic collision between two particles – an X-ray photon and a scattering
atom.
The photon energy of a photon of frequency is E = h ………….. (1)
According to Einstein‟s mass-energy relation, E = mc2 ………….(2)
where m is assumed to be the “rest mass” of the photon and c is the velocity of light.
From (1) and (2), we have, h = mc2 = mc.c ……… (3)
Treating mc as the “momentum” p of the photon, ie., mc = p, we have
h p.c ……….. (4)
, 2
c c h
Using , in eqn. (4), we get h p.c . Or p …..(5). (on cancelling c )
Thus, a photon of wavelength has an associated momentum p equal to h/ . When the
incident X-ray photon „collides‟ with an atom in the scattering target, it transfers some
momentum to an electron of the target atom. The electron recoils with momentum. pe. This
results in a decrease of momentum p of the X-ray photon on scattering. According to eqn.(5),
is inversely proportional to p. So, there is an increase of the wavelength of scattered x-ray
photon.
De Broglie’s matter wave hypothesis
Considering the dual nature of radiation, de Broglie proposed that material particles may
also have a dual character similar to radiation. In other words, in addition to the discrete particle
behaviour, matter can have some wave-like character also. Such waves associated with moving
particles are called as “Matter Waves” or “de broglie waves”.
In support of his hypothesis, de broglie argued that
i) Nature loves symmetry.
ii) Matter and Radiation are the two great entities of the Universe. Radiation
possesses a dual character. Similarly the material particles may also have some wave
character in addition to their particle behaviour.
iii) By comparing the motion of a particle and the propagation of a wave, he stressed
that their behaviour is analogous to each other. De Broglie argued that Principle of least
action in mechanics and Fermat‟s principle of least transit time in optics are analogous to
each other.
Consider a particle moving from one point to another. According to Principle of least action, a
moving particle chooses a path in which the mechanical action is minimum‟
Consider a light ray moving from one point to another. According to principleof least transit
time, a light ray chooses a path for which the transit time is minimum‟
Thus, if radiation of wavelength is assumed to have an associated momentum given by the
relation p = h / . , then the particle of mass m moving with a velocity v and momentum p
= mv has an associated matter waves of wavelength
= h / p = h/mv. This wavelength of „matter waves‟ is also known as „de broglie
wavelength’.
, 3
Wavelength of a free electron accelerated by an electrostatic field of potential V volts:
When a free electron of mass m is accelerated by an electric field of potential V volts, its
total energy which is purely kinetic is given by, ½ mv2 = eV ……(i)
Where e is the charge on the electron of mass m and v is its velocity.
Multiplying both sides of(i) by 2m, we get, m2v2 = p2 = 2meV (or) p = (2meV).
h h
Hence using de Broglie‟s eqn., we have, ………. (ii).
p 2meV
For small accelerating potentials, v << c and hence mass of the electron will be equal to its rest
h h
mass, mo and hence , the de Broglie wavelength is …….(iii).
p 2m0 eV
12.25
Substituting the values for h, mo ,and e, we have, AU ……… (iv)
V
For very large accelerating potentials (i.e., V is large), the velocity v of electron will be
m0
comparable to the velocity of light, c, and hence its relativistic mass m is to be used in
v2
1 2
c
eqn.(i. The wavelength of electrons accelerated by a potential of about 100V is about
1.2 AU (using (iv)). This is comparable to X-ray wavelengths. Hence it was predicted that
similar to X-rays, electrons also would undergo diffraction by crystals since this wavelength is
comparable to the interatomic distances in crystals. Electron diffraction by crystals was first
observed by Davisson and Germer and later by GP Thomson. As diffraction experiments could
be explained only on the basis of a wave theory, this implied that the de Broglie‟s hypothesis on
dual nature of matter was verified experimentally.
deBroglie‟s dual nature of matter is not applicable to all material particles. It is relevant only to
microscopic paricles such as electrons, protons, atoms etc. It is irrelevant to macroscopic objects.
According to the principle of matter wave hypothesis, a free electron of mass 9.1x10-31Kg accelerated by
100Volts has a de broglie wavelength of about 1.2AU. Their wave nature could be experimentally verified
by electron diffraction by crystals. However, if we consider any moving macroscopic object (such as
cricket ball, bullet etc) its de broglie wavelength will be of the order of 10-30 AU or even lesser. This
wavelength is too small (0) to be experienced in any experiment. Hence the wave nature of matter is
irrelevant as far as macroscopic objects are concerned.
, 4
HEISENBERG‟S UNCERTAINTY PRINCIPLE
Statement: The Total minimum uncertainty involved in the simultaneous measurement of a
pair of canonically conjugate variables cannot be less than ħ/2 (= h/4), where h is the
Planck”s constant.
Explanation: Consider a pair of canonically conjugate variables such as position x and x
component of momentum px . If x and px are the uncertainties in the individual measurements
of position x and momentum px , then the total uncertainty involved in their simultaneous
measurement is x . px .
Then according to Heisenberg‟s uncertainty principle, x . px ħ/2 where ħ = h/2 .
Similarly Heisenberg‟s uncertainty principle relating y and py is y . py h/4 and so on.
Heisenberg‟s uncertainty principle related to other similar pairs of variables are,
E . t h/4.
where E is the uncertainty in the measurement of energy of an atom in an excited state and t
is the lifetime of the atom in that excited state. Similarly,
J . h/4
where J is the total angular momentum of a particle and is its angular position.
Proof for Heisenberg’s uncertainty Principle: 1. Gamma ray microscope expt.
Suppose it is desired to simultaneously determine both the position x and linear
momentum px of a free electron moving along x-direction. Let us imagine that a microscope with
a very high resolving power is used to accurately locate the position of the electron. The space
under the microscope objective is illuminated by appropriate
A B
+(h/)sin
-(h/)sin
Electron x
O
h/
-rays
radiation photons. When the electron moves directly under the microscope, it will scatter some
of the photons. The scattered photons which enter into the microscope anywhere between OA
De Broglie hypothesis of matter waves
INTRODUCTION
Radiation is found to exhibit dual character. Some phenomena such as interference,
diffraction, polarization etc., are due to wave character of radiation. On the other hand,
phenomena such as photoelectric effect, Compton effect etc., could not be explained on the basis
of a wave theory. These latter phenomena exhibit the particle-like character of radiation. Thus
radiation exhibits a dual character, namely, a wave character and a particle – like character.
In Compton effect, monochromatic X-rays of wavelength falling on a scatterer are
scattered into a different direction. At the same time, an electron ejected from the scattering
atom recoils in some other direction. It is found that the scattered X-ray has a wavelength ‟
which is longer than the incident wavelength . ‟ > .
Scattering atom
Scattered X-ray, ‟
P‟ = h /‟
Incident X-ray,
P = h /
Recoiling electron,
P = P‟ + Pe pe
Compton assumed that X-ray photons have certain momentum and he treated the scattering of X-
rays as due to the inelastic collision between two particles – an X-ray photon and a scattering
atom.
The photon energy of a photon of frequency is E = h ………….. (1)
According to Einstein‟s mass-energy relation, E = mc2 ………….(2)
where m is assumed to be the “rest mass” of the photon and c is the velocity of light.
From (1) and (2), we have, h = mc2 = mc.c ……… (3)
Treating mc as the “momentum” p of the photon, ie., mc = p, we have
h p.c ……….. (4)
, 2
c c h
Using , in eqn. (4), we get h p.c . Or p …..(5). (on cancelling c )
Thus, a photon of wavelength has an associated momentum p equal to h/ . When the
incident X-ray photon „collides‟ with an atom in the scattering target, it transfers some
momentum to an electron of the target atom. The electron recoils with momentum. pe. This
results in a decrease of momentum p of the X-ray photon on scattering. According to eqn.(5),
is inversely proportional to p. So, there is an increase of the wavelength of scattered x-ray
photon.
De Broglie’s matter wave hypothesis
Considering the dual nature of radiation, de Broglie proposed that material particles may
also have a dual character similar to radiation. In other words, in addition to the discrete particle
behaviour, matter can have some wave-like character also. Such waves associated with moving
particles are called as “Matter Waves” or “de broglie waves”.
In support of his hypothesis, de broglie argued that
i) Nature loves symmetry.
ii) Matter and Radiation are the two great entities of the Universe. Radiation
possesses a dual character. Similarly the material particles may also have some wave
character in addition to their particle behaviour.
iii) By comparing the motion of a particle and the propagation of a wave, he stressed
that their behaviour is analogous to each other. De Broglie argued that Principle of least
action in mechanics and Fermat‟s principle of least transit time in optics are analogous to
each other.
Consider a particle moving from one point to another. According to Principle of least action, a
moving particle chooses a path in which the mechanical action is minimum‟
Consider a light ray moving from one point to another. According to principleof least transit
time, a light ray chooses a path for which the transit time is minimum‟
Thus, if radiation of wavelength is assumed to have an associated momentum given by the
relation p = h / . , then the particle of mass m moving with a velocity v and momentum p
= mv has an associated matter waves of wavelength
= h / p = h/mv. This wavelength of „matter waves‟ is also known as „de broglie
wavelength’.
, 3
Wavelength of a free electron accelerated by an electrostatic field of potential V volts:
When a free electron of mass m is accelerated by an electric field of potential V volts, its
total energy which is purely kinetic is given by, ½ mv2 = eV ……(i)
Where e is the charge on the electron of mass m and v is its velocity.
Multiplying both sides of(i) by 2m, we get, m2v2 = p2 = 2meV (or) p = (2meV).
h h
Hence using de Broglie‟s eqn., we have, ………. (ii).
p 2meV
For small accelerating potentials, v << c and hence mass of the electron will be equal to its rest
h h
mass, mo and hence , the de Broglie wavelength is …….(iii).
p 2m0 eV
12.25
Substituting the values for h, mo ,and e, we have, AU ……… (iv)
V
For very large accelerating potentials (i.e., V is large), the velocity v of electron will be
m0
comparable to the velocity of light, c, and hence its relativistic mass m is to be used in
v2
1 2
c
eqn.(i. The wavelength of electrons accelerated by a potential of about 100V is about
1.2 AU (using (iv)). This is comparable to X-ray wavelengths. Hence it was predicted that
similar to X-rays, electrons also would undergo diffraction by crystals since this wavelength is
comparable to the interatomic distances in crystals. Electron diffraction by crystals was first
observed by Davisson and Germer and later by GP Thomson. As diffraction experiments could
be explained only on the basis of a wave theory, this implied that the de Broglie‟s hypothesis on
dual nature of matter was verified experimentally.
deBroglie‟s dual nature of matter is not applicable to all material particles. It is relevant only to
microscopic paricles such as electrons, protons, atoms etc. It is irrelevant to macroscopic objects.
According to the principle of matter wave hypothesis, a free electron of mass 9.1x10-31Kg accelerated by
100Volts has a de broglie wavelength of about 1.2AU. Their wave nature could be experimentally verified
by electron diffraction by crystals. However, if we consider any moving macroscopic object (such as
cricket ball, bullet etc) its de broglie wavelength will be of the order of 10-30 AU or even lesser. This
wavelength is too small (0) to be experienced in any experiment. Hence the wave nature of matter is
irrelevant as far as macroscopic objects are concerned.
, 4
HEISENBERG‟S UNCERTAINTY PRINCIPLE
Statement: The Total minimum uncertainty involved in the simultaneous measurement of a
pair of canonically conjugate variables cannot be less than ħ/2 (= h/4), where h is the
Planck”s constant.
Explanation: Consider a pair of canonically conjugate variables such as position x and x
component of momentum px . If x and px are the uncertainties in the individual measurements
of position x and momentum px , then the total uncertainty involved in their simultaneous
measurement is x . px .
Then according to Heisenberg‟s uncertainty principle, x . px ħ/2 where ħ = h/2 .
Similarly Heisenberg‟s uncertainty principle relating y and py is y . py h/4 and so on.
Heisenberg‟s uncertainty principle related to other similar pairs of variables are,
E . t h/4.
where E is the uncertainty in the measurement of energy of an atom in an excited state and t
is the lifetime of the atom in that excited state. Similarly,
J . h/4
where J is the total angular momentum of a particle and is its angular position.
Proof for Heisenberg’s uncertainty Principle: 1. Gamma ray microscope expt.
Suppose it is desired to simultaneously determine both the position x and linear
momentum px of a free electron moving along x-direction. Let us imagine that a microscope with
a very high resolving power is used to accurately locate the position of the electron. The space
under the microscope objective is illuminated by appropriate
A B
+(h/)sin
-(h/)sin
Electron x
O
h/
-rays
radiation photons. When the electron moves directly under the microscope, it will scatter some
of the photons. The scattered photons which enter into the microscope anywhere between OA
