The de Broglie wavelength is an important concept while studying quantum mechanics. The wavelength (λ) that is associated with an object in relation to its momentum and mass is known as the de Broglie wavelength. A particle’s de Broglie wavelength is usually inversely proportional to its force.
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de Broglie Waves
It is said that matter has a dual nature of wave particles. de Broglie waves, named after the discoverer Louis de Broglie, is the property of a material object that varies in time or space while behaving similarly to waves. It is also called matter-waves. It holds great similarity to the dual nature of light, which behaves as a particle and wave that has been proven experimentally.
Also Read: Photoelectric Effect
The physicist Louis de Broglie suggested that particles might have both wave properties and particle properties. The wave nature of electrons was also detected experimentally to substantiate the suggestion of Louis de Broglie.
The objects which we see in day-to-day life have wavelengths which are very small and invisible; hence, we do not experience them as waves. However, de Broglie wavelengths are quite visible in the case of subatomic particles.
de Broglie Wavelength for Electrons
In the case of electrons going in circles around the nuclei in atoms, the de Broglie waves exist as a closed loop, such that they can exist only as standing waves and fit evenly around the loop. Because of this requirement, the electrons in atoms circle the nucleus in particular configurations or states, which are called stationary orbits.
de Broglie Wavelength Formula and Derivation
de Broglie reasoned that matter also could show wave-particle duality, just like light, since light can behave both as a wave (it can be diffracted and it has a wavelength) and as a particle (it contains packets of energy hν). He also reasoned that matter would follow the same equation for wavelength as light, namely,
λ = h / p
Where p is the linear momentum, as shown by Einstein.
Derivation
de Broglie derived the above relationship as follows:
1) E = hν for a photon and λν = c for an electromagnetic wave.
2) E = mc2, means λ = h/mc, which is equivalent to λ = h/p.
Note: m is the relativistic mass and not the rest mass since the rest mass of a photon is zero.
Now, if a particle is moving with a velocity v, the momentum p = mv, and hence λ = h / mv
Therefore, the de Broglie wavelength formula is expressed as:
λ = h / mv
Applications of de Broglie Waves
1. As the wave properties of matter are only observable for very small objects, the de Broglie wavelength of a double-slit interference pattern is produced by using electrons as the source. 10 eV electrons (which is the typical energy of an electron in an electron microscope): de Broglie wavelength = 3.9 x 10-10 m.
This is comparable to the spacing between atoms. Therefore, a crystal acts as a diffraction grating for electrons. The diffraction pattern allows the crystal structure to be determined.
2. In a microscope, the size of the smallest features we can see is limited by the wavelength used. With visible light, the smallest wavelength is 400 nm = 4 x 10-7 m. Typical electron microscopes use wavelengths 1000 times smaller and can be used to study very fine details.
Thermal de Broglie Wavelength
The thermal de Broglie wavelength (λth) is approximately the average de Broglie wavelength of the gas particles in an ideal gas at the specified temperature.
The thermal de Broglie wavelength is given by the expression:
λD = h / √ 2 π m kBT
where,
h = Planck constant
m = mass of a gas particle
kB = Boltzmann constant
T = temperature of the gas
λD = λth = thermal de Broglie wavelength of the gas particles
Also Read:
| Value of Planck’s Constant |
| Boltzmann Constant |
Bohr’s Model for Hydrogen
The electrons are in the form of disk-shaped clouds, and they move in circular orbits around the nucleus in atoms. In the hydrogen atom, the electron in the ground state with the minimum energy can be modelled by a rotating disk, the inner edge of which has a radius of ½ rB(1) and the outer edge has a radius of 3/2 rB (2) where rB is the Bohr radius.
If we assume that the electron’s orbit in the atom includes ‘n’ of de Broglie wavelengths, then in case of a circular orbit with the radius, for the circle perimeter and the angular momentum L of the electron, we will obtain the following:
2 πr = n λB,
L = rp = nh / 2π
λB = h / p
This is exactly the postulate of Bohr’s model for the Hydrogen atom. According to the postulate, the angular momentum of the hydrogen atom is quantized and proportional to the number of the orbit ‘n’ and the Planck constant.
Solved Problems
Question 1: An electron and a photon have the same wavelength. If p is the momentum of the electron and E is the energy of the photon, the magnitude of p/E in SI unit is
(a) 3.0108 (b) 3.3310-9
(c) 9.110-31 (d) 6.6410-34
Answer: b
As we know, for an electron, λ = h/p
Or
p = h/λ
And for photon E = hc / λ
Thus, p / E = 1 / c = 1 / (3 x 108 m/s) = 0. 33 x 10-8 s/m
Question 2: What is the energy and wavelength of a thermal neutron?
Answer:
KE = (3/2) kT = (3/2) (1.38 × 10–23) (293) = 607 × 10–23 J
λ = h / p
λ = h / √2m0 (KE)
λ = 6 .63 × 10–34 / √2 (1.67 × 10–27) (607 × 10–23)
λ = 0.147 nm
Question 3: A particle of mass m is confined to a narrow tube of length L. Find
(a) The wavelengths of the de-Broglie wave, which will resonate in the tube
(b) The corresponding particle momenta
(c) The corresponding energies
Answer:
(a) The de Broglie waves will resonate with a node at each end of the tube.
A few of the possible resonance forms are listed below:
λn = (2L / n); = 1, 2, 3, …
(b) Since de-Broglie wavelengths are
λn = h / pn
∴ pn = h / λn = nh / 2L, n = 1, 2, 3, …
(c) (KE) n = pn2 / 2m = n2h2/8L2m, n = 1, 2, 3, …
Question 4: What is the wavelength of an electron moving at 5.31 x 106 m/sec?
Given: mass of electron = 9.11 x 10-31 kg h = 6.626 x 10-34 J·s
Answer:
de Broglie’s equation is
λ = h/mv
λ = 6.626 x 10-34 J·s/ 9.11 x 10-31 kg x 5.31 x 106 m/sec
λ = 6.626 x 10-34 J·s/4.84 x 10-24 kg·m/sec
λ = 1.37 x 10-10 m
λ = 1.37 Å
The wavelength of an electron moving 5.31 x 106 m/sec is 1.37 x 10-10 m or 1.37 Å.
Question 5: Which of the following is called non-mechanical waves?
a) Magnetic waves
b) Electromagnetic waves
c) Electrical waves
d) Matter waves
Answer: b
The waves which travel in the form of oscillating electric and magnetic waves are called electromagnetic waves. Such waves do not require any material for their propagation and are called non-mechanical waves.
Question 6: Which of the following is associated with an electron microscope?
a) Matter waves
b) Electrical waves
c) Magnetic waves
d) Electromagnetic waves
Answer: a
The waves associated with microscopic particles when they are in motion are called matter waves. An electron microscope makes use of the matter waves associated with fast-moving electrons.
Question 7: Calculate the de-Broglie wavelength of an electron that has been accelerated from rest on application of a potential of 400 volts.
a) 0.1653 Å
b) 0.5125 Å
c) 0.6135 Å
d) 0.2514 Å
Answer: c
de-Broglie wavelength = h/√ (2×m×e×V)
de-Broglie wavelength = (6.625×10-14)/√(2×9.11×10-31×1.6×10-17×400)
Wavelength = 0.6135 Å.
Question 8: The de Broglie wavelength of a particle is the same as the wavelength of a photon. Then, the photon’s energy is:
(a) Equal to the kinetic energy of the particle
(b) Less than the kinetic energy of the particle
(c) Greater than the kinetic energy of the particle
(d) Nothing can be specified
Answer: c
The photon’s energy depends only on the frequency of the photon. Hence, the photon’s energy is greater than the kinetic energy of the electron.
Question. 9: An electron and a proton have the same de Broglie wavelength. Then, the kinetic energy of the electron is
(a) Zero
(b) Infinity
(c) Equal to the kinetic energy of the proton
(d) Greater than the kinetic energy of the proton
Answer: d
The electron and the proton have the same de Broglie wavelength, which means their momenta are the same. Since the mass of the proton is greater than the mass of the electron, its speed is less than the speed of the electron. Hence, KE = (½) PV, the kinetic energy of the electron is greater than the kinetic energy of the proton.
Question. 10: A nucleus of mass M at rest emits an α-particle of mass m. The de Broglie wavelengths of the α-particle and residual nucleus will be in the ratio
(a) m : M (b) (M+m) : m
(c) M : m (d) 1 : 1
Answer: d
The nucleus is initially at rest. By the conservation of momentum principle, we can assume that the momenta of the α-particle of mass m and the residual nucleus will be equal and opposite. The de Broglie wavelength is inversely proportional to momentum. Hence, the de Broglie wavelengths of the α-particle and residual nucleus will be in the ratio = 1:1.
Question 11: The operation of the electron microscope depends on the:
(a) Very short wavelength of X-rays
(b) Wave nature of electrons
(c) Electromagnetic theory of waves
(d) Photoelectric effect
Answer: b
The wave nature of the electron particles is utilised in the electron microscope.
Frequently Asked Questions on de Broglie Wavelength
What is the formula for de Broglie wavelength?
According to de Broglie, the wavelength associated with a particle of mass m moving with a velocity v is given by
λ = h/p = h/mv
Can wave-like properties associated with microscopic particles be observed experimentally?
The de Broglie wavelength associated with macroscopic particles is very small, and hence cannot be detected.
Can the de Broglie wavelength of microscopic particles be detected?
The wavelength of microscopic particles like electrons and protons can be observed experimentally.
What will happen to the value of de Broglie wavelength if the velocity of the proton is increased?
The de Broglie wavelength will decrease if the velocity of the photon increases.
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