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De Broglie's Hypothesis-Derivation, Practice problems & FAQs

De Broglie's Hypothesis-Derivation, Practice problems & FAQs

Have you ever wondered if dogs could live underwater or if fishes could survive on land? Obviously we know, thats not possible because dogs are terrestrial animals whereas fishes are aquatic animals. But what about frogs? Or a tortoise. You would have definitely seen them on both land and water. So what do we call such animals? They are called amphibians.

On the same analogy, in the mid 1920’s, Louis de Broglie, in his doctoral dissertation, made a bold claim. His hypothesis proposed that all matter exhibits wave-like properties and relates the observed wavelength of matter to its momentum. With the development of Plank’s Quantum theory of light, the question became whether this was true only for light or whether material objects also exhibited wave-like behavior. 

De Broglie proposed that matter, like radiation, exhibits both particles as well as wave-like characteristics. Subatomic particles, like electrons, are associated with the dual nature just like the light that has both particle and wave nature.

Table of contents:

  • Derivation of de Broglie’s wavelength
  • Other forms of De-Broglie wavelength
  • Practice problems
  • Frequently Asked Questions-FAQs

Derivation of de Broglie’s wavelength

De Broglie’s wavelength can be derived by using Planck’s equation and Einstein’s mass-energy relationship as follows:

Plank’s equation stated that
E = hง----(i)
E = hc/λ

ง is the frequency of the photon, λ is the wavelength of the photon
c is velocity of light and h is Planck's constant

Einstein equation

E = mc2----(ii)
m is the mass of the particle, c is the speed of light

Equating equation (i) and (ii), 
λ = h/mc

Similarly for particles (matter),
λ = h/mv

λ = h/p----(iii)

where p is the momentum of the particle

Other forms of De-Broglie wavelength

dsfsdf

on putting values of h,me and qe, which are

h = 6.626 * 10-34 Jsec
m= 9.1 * 10-31 kg
q= 1.6 * 10-19 C

We get,

dssf

Practice problems:

Q1. What will be the wavelength of a ball of mass 10 kg moving with a velocity of 100msec-1?

A) 6.626 10-37 m
B) 2.209 10-34 m
C) 11.04 10-34 m
D) None of these

Answer: (A)

sdsf

We can see, order of the wavelength of the daily life particle is extremely small, therefore de-Broglie wavelength calculation for the daily life particle has no physical significance

Q2. The mass of an electron is 9.1 x 10-31 kg. If its K.E. is 3.5 x 10-26 J., calculate wavelength associated with it (take h=6.0 x 10-34Js)

A) 746.87 nm
B) 453.98 nm
C) 237.7 nm
D) 1234.76 nm

Answer: (C)

cxv

Q3. The approximate ratio of de-Broglie wavelength of neutron and electron moving with the same speed is about

A) 1:4
B) 1:1839 
C) 1:1 
D) 1:2 

Answer: (B)

jhbjh

Q4. Which of the following has the lowest de-Broglie wavelength (all have the equal velocity)

A) Hydrogen gas molecule
B) Electron
C) Proton
D) Nitrogen gas molecule

Answer: (b)

Solution: we know, =hmv

Species have equal velocity, wavelength is inversely proportional to their mass. So, electrons 

have the smallest mass in all given options.

Q5. Which of the following nature of graph represents the variation of particles de-Broglie wavelength with momentum

A) rectangular hyperbola 
B) parabolic
C) linear
D) can’t be predicted

Answer: (A)

Solution:

Frequently Asked Questions-FAQs

Question 1. What is the limitation of de-Broglie hypothesis?
Answer: Order of the wavelength of the daily life particle is extremely small, therefore de-Broglie wavelength calculation for the daily life particle has no physical significance

Question 2. What do you understand by the dual nature of electrons?
Answer: Electrons can be considered to possess properties of both a particle and a wave at the same time.

Question 3. Which experiment conclusiely proved de-Broglie hypothesis?
Answer: The Davisson and Germer experiment demonstrated the wave nature of the electrons, confirming the earlier hypothesis of de Broglie. Electrons exhibit diffraction when they are scattered from crystals whose atoms are spaced appropriately.

Question 4. de-Broglie was awarded the nobel prize in physics for which achievement?
Answer: de-Broglie awarded nobel prize in physics (1929) for discovery of wave nature of electrons.

Related Topic Links:

Millikan’s oil drop experiment

Discovery of neutrons

Discovery of electron

Photoelectric effect

Heisenberg's uncertainty principle

Thomson's atomic model

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