• De-Broglie hypothesis
• De-Broglie Wave Velocity or Phase velocity
• Group velocity
• Calculation of De-Broglie Wavelength of an electron
• Relation between Group velocity and particle velocity
• Relation between Group velocity and phase velocity
• De-Broglie Wavelength in terms of energy and
temperature
• A proton and a deuteron have the same kinetic energy
which has longer wave length.
• Why are wave properties of particle normally observed
only when we study very small particle
De-Broglie Wave
The phenomena of interference, diffraction, polarization of light could be
explained on the wave nature of light. However there are certain phenomena which
could not be explained on this wave theory of radiation. For example, photoelectric
effect, Compton-effect etc. could be explained only through the corpuscular nature
of radiation. Thus a dual nature, wave and particle came to be associated with light.
In 1924 Louis de-Broglie suggested that matted also had a dual character
(particle and wave). According to him electrons and protons which ordinarily
behave like particle, under certain conditions behave like a train of waves. The
wavelength of this wave depends upon the momentum which in turn depends upon
the mass and velocity of particles.
De-Broglie hypothesis
According to quantum theory, radiation consists of quanta or photons,
* Each of energy E= h υ where h is plank’s constant the value of which is 6.62x10-
32
joule-sec.
* Mass of the photon, m = E/c2 = h υ/c2
* Momentum of the photon, p = m. c = E.c/c2 = c h υ/c2 = h υ/c =h/λ,
where c is the speed of the photon and λ is the wavelength of the radiation of
frequency υ
By this analogy , De- Broglie suggested that a moving particle is associated with a
wave.
The frequency of the wave is taken to be υ = E/ h = mc 2/h , where m is the mass of
the particle. The wavelength of the wave λ = p/h =h/mv, where v is the velocity of
the particle.
Thus a particle of mass m moving with a velocity v has an associated wavelength
λ = h/p = h/mv -------------------------(1)
This is known as de-Broglie wave equation. The associated wave is termed as de-
Broglie wave, and λ is called as de-Broglie wavelength.
De-Broglie Wave Velocity or Phase velocity
According to de –Broglie’s view each particle of matter (like e - , p etc) may be
regarded as consisting of a group of waves or a wave packet. Each component wave
propagates with a definite velocity called phase velocity v p .
Phase velocity is the velocity that travel through the medium of any particular point
in a given phase in a wave of single frequency.
Vp= υ .λ, [E=h υ, υ=E/h=mc2/h and λ=h/p=h/mv]
Vp= mc2/h . h/mv
Vp= c2 /v --------------(2)
Since v is less then the velocity of light. Phase velocity V p must be greater then the
velocity of light. It has no physical significance. Phase velocity can not observed.
Group velocity
A group consists of a number of waves of different frequencies superimposed
upon each other. For example, while light consists of a continuous visible
wavelength spectrum ranging from about 3000A in the violet to about 7000A in the
red region. The group of waves is called a packet and with time the packet moves
forward in the medium with a velocity called the group velocity.
Consider a group of waves consisting of only two components of equal
amplitude and having angular frequencies ω 1 and ω2 differing by a small amount
and represented by the equations
y1 = a cos(ω1t – k1x)
y2 = a cos(ω 2t – k2x) [ω = 2π υ and k = 2π/λ]
ω1/k1 and ω2/k2 represent their respective phase velocity. Supposed ω 1/k1 ≠ ω2/k2
i.e. it is a dispersive medium.
The resultant amplitude is given by
Y = y1+y2
Y = a [cos(ω1 t – k1x) + cos(ω 2t – k2x)]
Y= 2a cos[(ω1 +ω2)t/2 – (k1+k2)x/2 ] cos [(ω1 -ω2)t/2 – (k1-k2)x/2]
Y = 2a cos (ωt –kx )cos [(∆ωt/2 – ∆kx/2)]
[ω= (ω1 + ω2) /2, k= (k1+k2 )/2]
∆ ω = ω 1 - ω 2 , ∆ k = k 1+ k 2
The resultant wave has two parts.
(1) A wave of frequency ω , propagation constant k and velocity
Vp = ω/k = 2πυ/2π/ λ = υ λ
This is the phase velocity or wave velocity
(2) A 2nd wave of frequency ∆ω/2 , propagation constant ∆k/2 and velocity
Vg = ∆ω/∆k , This is the group velocity
Calculation of De-Broglie Wavelength of an electron, p, alpha,n
Let v be the velocity acquired by the electron when it is accelerated through
potential difference V.
Then work done on the electron =eV
K.E. gained by the electron =1/2 mv2
1/2mv2 =eV
v = √2eV/m
The wavelength associated with the electron
λ = h/mv =h/m√2eV/m = h/ √2eVm
In SI unit λ = 12.26 / √V Ǻ
[for alpha particle q=2e, m=4e]
Relation between Group velocity and particle velocity
A particle moving with a velocity v is supposed to consists of a group of waves
according to de-Broglie hypothesis. For a material particle of rest mass m 0 moving
with a velocity v and having an effective mass m the total energy E is given by
E = mc2 = m0c2/ √( 1-v2/c2)
P = mv = m0 v / √( 1-v2/c2)
The frequency of the associated de-Broglie wave
υ = E/h = m0c2/ h √( 1-v2/c2)
ω =2π υ = 2π m0c2/ h √( 1-v2/c2)
dω = 2π m0 v dv / h ( 1-v2/c2)3/2 ----------------------(1)
The wavelength of the associated de-Broglie wave
λ = h/p = h (1- v2/c2)1/2/m0 v
k = 2 π/ λ = 2 π m0 v / h ( 1-v2/c2)1/2
dk = 2 π m0 / h [ ( 1-v2/c2)-1/2dv + v2 /c2 ( 1-v2/c2)-3/2dv ]
dk = 2 π m0 / h( 1-v2/c2)3/2 [dv - v2 /c2 dv +v2 /c2 dv]
dk = 2 π m0 dv / h( 1-v2/c2)3/2 ---------------------(2)
Dividing (1) by (2) we have
Group velocity vg = dω/dk =v= the particle velocity
Thus the de-Broglie wave group associated with a particle travels with the same
velocity as the particle it self.
Relation between Group velocity and phase velocity
Vp = ω/k or ω = Vp k
Vg = dω/dk = d (Vp k) /dk
Vg = Vp + k . d Vp / dk [k= 2π/λ or dk= -2π/λ2 d λ or k/dk = - λ/d λ]
Vg = Vp - λ. d Vp / d λ
De-Broglie Wavelength in terms of energy and temperature
If E is the kinetic energy of the moving particle then E= ½ mv2 = ½ m2v2/m = ½
p2/m
P= √2Em
According to kinetic theory of gases, average K. E. of the materials particle is given
by
E= ½ mv2 = 3/2 KT [K= Boltzmann’s constant and T = absolute
temperature]
We know de-Broglie wavelength is λ=h/p=h/mv
λ=h/√2Em = h√3KTm
Problem
1. A proton and a deuteron have the same kinetic energy which has longer wave
length.
KE of proton =1/2 m vp2 , KE of deuteron = ½ . 2m vd2
½ . 2m vd2 =1/2 m vp2
vd =1/√2 vp
λd / λp = (h/2m vd )/( h/m vp)
λd / λp = 1/ √2
λp = √2 λd
So proton has a higher de Broglie wavelength
2. Why are wave properties of particle normally observed only when we study
very small particle
de Broglie wavelength for macroscopic particle
λ=h/p = 6.6 x10 -34 /10-3kgm/s. 10-2 m/s = 6.6 x 10-29 m (so small for
measurable )
de Broglie wavelength for electron, proton
λ=h/p = 6.6 x 10 -34 / 9 x10-31. 107 = 0.73 x 10-10 m
Problems
1. What is the De-Broglie wavelength of an electron whose speed is 9x109cm/sec.
2. Find the De-Broglie wavelength of 15 kev electron.
3. Find the wavelength of a 100 gm object whose speed is 100cm/sec.
4. What is the kinetic energy of an electron whose De-Broglie wavelength is 5000A0?
5. An electron has a speed of 104 cm/sec (accurate to 0.1%) . What is the uncertainty in
the position of this electron.
6. If an electron remains in an exist state in an atom for 10-8 sec. What is the uncertainty
in the energy of that state.
7. An electron beam is accelerated by a potential difference of 150 volts. Calculate the
De-Broglie wavelength associated with the beam.
8. Find the De-Broglie wavelength of (a ) 46 g golf ball with a velocity of 30 m/s and (b)
an electron with a velocity of 107.
9. A measurement establish the position of a proton with an accuracy of 10-11m . Find
the uncertainty in the proton's position 1 s later. Assume <<c.
10. A hydrogen atom is 5.3 x 10-11 m in radius. Use the uncertainty principle to estimate
the minimum energy an electron can have in this electron.
What is the de Broglie equation?
Definition: The de Broglie equation is an equation used to describe the wave properties of matter,
specifically, the wave nature of the electron: λ = h/mv, where λ is wavelength, h is Planck's constant, m
is the mass of a particle, moving at a velocity v.
In 1924 a young physicist, de Broglie, speculated that nature did not single out light as being the only
matter which exhibits a wave-particle duality. He proposed that ordinary ``particles'' such as electrons,
protons, or bowling balls could also exhibit wave characteristics in certain circumstances.
Quantitatively, he associated a wavelength to a particle of mass m moving at speed v :
.
