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In 1924, the French physicist De Broglie conducted a series of experiments to show that electrons also have particle and wave nature, just as photons or light. According to him, every particle exhibits dual characteristics. Furthermore, he indicated that the path of electrons is wavy, similar to the light having a definite frequency. In 1929, De Broglie received the Nobel Prize for his theory. So let us learn more about the derivation of De Broglie’s equation.
De Broglie Equation
The experiment of cathode-ray diffraction by George Paget Thomson and Davisson and the Germer experiment that is explicitly applied to the electrons conformed to De Broglie’s equation.
Thus, the derivation of De Broglie Equation questions the matter’s wave properties, mainly electrons.
λ = h/mv
In this equation,
λ = wavelength,
h = Planck’s constant
m = particle’s mass
v = velocity of the particle
Derivation of De Broglie Equation
For the derivation of De Broglie’s equation, we need to follow the two equations (theories):
Einstein’s equation of matter and energy
E = mc2
Where
E = energy,
m = mass,
c = speed of light in vacuum
Planck’s equation indicating energy from waves
E = hv
Where,
E = energy,
h = Planck’s constant whose value is 6.62607 x 10-34 Js
v = frequency
De Broglie considered the above two energies equal as he believed that the particles and waves display similar traits. Based on his hypothesis
mc2 = hν
The actual particles don’t travel at the speed of light. Hence, De Broglie submitted velocity (v) for the speed of light (c).
i.e, v = c/λ
Hence, mc2 = h x c/λ
λ = h/mc
The equation is known as De Broglie’s equation. It is also known as the matter-wave equation.
(Here, h is Planck’s constant = 6.62607 x 10-34 J s, m is the mass of the photon, c is the velocity of light, i.e., 3 x 108 ms-1, λ is the wavelength of the photon.)
As we know, mc is also known as momentum (p). Hence, we can also write De Broglie’s Equation as
λ = h/p
p = mc (momentum of the photon)
The Experiment of the De Broglie Hypothesis
In the quest to explore the derivation of De Broglie Equation questions, Physicists Clinton Davisson and Lester Germer of Bell Labs experimented with the hypothesis by firing electrons at a crystalline nickel target. The firing resulted in a diffraction pattern matching the predictions of the De Broglie Wavelength. Proving De Broglie’s hypothesis landed Davisson/Germer a Nobel Prize in 1937.
Experiments such as the quantum variants of double-slit also support the De Broglie hypothesis.
Importance of De Broglie Equation
The De Broglie hypothesis does not work on objects of regular size because their associated wavelength is too small to detect.
The following examples can verify this:
- Suppose an electron of mass 7 x 10-31 kg moves with a velocity of 105 ms-1. The De Broglie wavelength of the electron can be calculated as follows:
λ = h/mv
= [(6.626 x 10-34 kgm2s-1) / (7 x 10-31 kg x 105 ms-1)]
= 0.9465 x 10-8 m
= 9.465 x 10-9 m
Such small values of λ can be measured only by the method used to determine X-rays’ wavelength.
2.Now, a ball of mass 10-2 kg moves with a velocity of 102 ms-1. Its De Broglie wavelength will be
λ = h/mv
= [(6.626 x 10-34 kgm2s-1) / (102 kg ×102 ms-1)]
= 6.62 x 10-38 m
Such small wavelength values are difficult to measure. Thus, we can conclude that the De Broglie relation has no impact on massive objects. However, the idea of the electron’s dual nature seemed reasonable to wonder about.
Application of De Broglie’s Equation
The De Broglie equation finds application in calculating the wavelength of moving particles such as bullets, balls, or electrons. The derivation of De Broglie Equation indicates how two different theories when combined can yield the velocity of the moving particle.
A particle of mass (m) moves with velocity (v). So then, De Broglie’s equation for such a particle would be λ = h / mv.
Everyday objects have lower wavelength values than electrons. Therefore, they mostly exhibit particle nature, and it becomes difficult to detect their wave nature.
De Broglie’s hypothesis plays an important role only in the domain of subatomic particles.
De Broglie Equation Examples
These examples will help you understand how to calculate the wavelength of a moving electron using the De Broglie Equation.
- What is the wavelength of an electron moving at 6.31 x 106 m/sec?
Given: mass of electron = 7.91 x 10-31 kg
h = 6.626 x 10-34 kgm2s-1
λ = h/mv
= [(6.626 x 10-34 kgm2s-1)/7.91 X 10-31 X 6.31 x 106 mkgs-1)
= 6.626 x 10-34/49.91 x 10-25 m
= 0.1327 x 10-9 m
= 1.327 Å
Thus, the wavelength of an electron moving at 6.31 x 106 m/sec is 1.327 Å.
2. Find the wavelength value of an electron that is travelling at the speed of 2.0×106 m/s (mass of one electron: me=8.109×10-31 kg).
λ = h/mv
λ = [(6.626 x 10-34 kgm2s-1)/8.109×10-31 kg x 2.0×106 m/s]
λ = 0.4085 x 10-9 m
λ = 4.085×10-10 m
= 4.1 Å.
Thus, the wavelength of an electron moving at 2.0×106m/s is 4.1 Å.
Conclusion
Historic experiments suggest De Broglie’s concept applies only to sub-microscopic objects of the range of atoms, molecules, or smaller subatomic particles. The derivation of the De Broglie equation notes suggests that the phenomena have no influence on macroscopic particles.
However, diffraction experiments in 1999 confirmed the De Broglie wavelength for the behaviour of molecules such as buckyballs. Here, we learned about the meaning of the De Broglie equation as well as explored its derivation. The article has covered the importance and applications of the equation to express its value in the science field.
