de Broglie hypothesis for matter waves in quantum theory

Hypothesis is a predicted theory that satisfies other phenomena but isn’t verified experimentally. Scientist de Broglie gave a such type of theory for the matter particles which plays important roles in Quantum mechanics. In this article, we are going to discuss de Broglie hypothesis for matter waves in quantum mechanics, its explanation and formula and some numerical problems based on de Broglie’s hypothesis formula.

Contents of this article:

  • 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 matter particles behave like waves when they are in motion. It can be observed at microscopic level only. It’s difficult to observe the wave nature of macroscopic particles due to their heavy mass. 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 the wavelength of a matter particle is the equation of de Broglie’s hypothesis.

If m is 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 Planck’s constant and p is the momentum of the matter particle.

Explanation of de Broglie hypothesis

de Broglie hypothesis helps people to know the wave and particle duality nature of matter particles. 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, matter particles have another name 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 at the same speed. The electron has a 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 the wavelength of matter waves is inversely proportional to the mass of the particle. So, the electron will have a very large wavelength than the tennis ball. This is why the wave nature of the matter particles like electrons, protons, atoms, molecules, etc. are more easily visible than that of 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.

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

Velocity of the electron in the 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 electrons 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 this article on de Broglie hypothesis for matter waves in quantum mechanics. If you have any doubts on this topic you can ask me in the comment section.

Thank you!

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