de Broglie hypothesis for matter waves in quantum theory

The de Broglie’s hypothesis is very important in Quantum mechanics. A hypothesis is a predicted theory that satisfies with other phenomena but not verified experimentally. Scientist de Broglie gave a this type of theory for matter particles. In this article, we are going to discuss de Broglie hypothesis for matter waves in quantum mechanics, its expression or formula, explanation and some numerical problems on de Broglie’s hypothesis formula.

• Statement of de Broglie hypothesis
• Formula of de Broglie hypothesis
• Explanation of de Broglie’s hypothesis
• Numerical problems on de Broglie hypothesis

Statement of de Broglie hypothesis

The de Broglie hypothesis states that all the matter particles behaves like waves when they are in motion. It can be observed easily in microscopic level and it is difficult to observe that in macroscopic level where the mass of the particle is very large. Sometimes, this statement is called the definition of de Broglie hypothesis.

Equation of de Broglie hypothesis

If a matter particle behaves like a wave then it must have a wavelength. The expression for wavelength of matter particle is the equation of de Broglie’s hypothesis.

If m be the mass of the matter particle which is moving with a speed v, then de Broglie hypothesis gives the equation for the wavelength of the matter wave as $\small {\color{Blue} \lambda =\frac{h}{mv}}$

or, $\small {\color{Blue} \lambda =\frac{h}{p}}$

Where, h is the Planck’s constant and p is the momentum of the matter particle.

Explanation of de Broglie hypothesis

People became to know the wave and particle duality nature of matter particles after the de Broglie hypothesis of matter waves in quantum mechanics. Scientist de Broglie predicts that all the matter particles like electron, proton, atom, molecules, etc. which have very very small masses behave like waves. Therefore, the matter particles have another name as matter waves.

The de Broglie hypothesis is associated with matter particles. Here, we take two particles – an electron and a tennis ball, both are moving with same speed. The electron has its mass of 9.11×10-31 kg which is very very smaller than the mass of the tennis ball of 100 grams or 0.1 kg. Now, in the formula for de Broglie wavelength we can see that wavelength of matter waves is inversely proportional to the mass of the particle. So, the electron will have very large wavelength than the tennis ball. This is why the wave nature of the matter particles like electron, proton, atom, molecules, etc. are easily visible than that of the macroscopic particles of greater masses. Remember that matter particles will behave like waves only when they are in motion.

Some questions and numerical problems on de Broglie’s hypothesis

1. Find de Broglie wavelength of an electron moving in the first orbit of hydrogen atom.

Mass of the electron is m=9.11×10-31 kg

Velocity of the electron in first orbit of a hydrogen atom is V=2.18×106 m/s

Then the de Broglie wavelength of the electron is, $\small {\color{Blue} \lambda =\frac{h}{mv}}$

or, $\small {\color{Blue} \lambda =\frac{6.625\times 10^{-34}}{9.11\times 10^{-31}\times 2.18\times 10^{6}}}$

or, the de Broglie wavelength of the electron is 3.33×10-10 meter or 3.33 Angstrom.

2. Which experiment confirms de Broglie hypothesis experimentally?

Davison-Germer experiment gives the experimental proof of de Broglie’s hypothesis. In this experiment, the wavelength of electron has been observed on the interference pattern.

3. What is the expression for de Broglie wavelength of a Photon?

de Broglie wavelength for a photon particle is, $\small {\color{Blue} \lambda =\frac{h}{p}}$

or, $\small {\color{Blue} \lambda =\frac{hc}{E}}$

Where, c is the speed of light and E is the energy of photon and E=pc

This is all from the de Broglie hypothesis for matter waves in the quantum mechanics. If you have any doubt on tis topic you can ask me in the comment section.

Thank you!

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