De Broglie Equation and Bohr’s Hypothesis of Atom

The two theories, de Broglie equation and Bohr’s hypothesis of atoms, have a great impact on quantum mechanics. De Broglie equation was the work of Louis de Broglie, a French physicist, and Bohr’s hypothesis of atoms belongs to Neil Henrik Bohr is a great scientist who gave lots of contributions to quantum mechanics. In this article, we will be discussing the relation between the de Broglie equation and Bohr’s hypothesis of atoms by deriving their equations. The equations of de Broglie were a relief for the scientific world, more importantly for Neil Henrik Bohr because he had postulated in his theory that angular momentum is equal to integral multiplied by h/2π.

By equating de Broglie equation and Bohr’s hypothesis of atoms together, the postulate of Bohr’s theory can be proved. We will discuss vastly in this article about both theories and understand the meaning of Bohr’s hypothesis.

De Broglie Equation

In 1929, Louis de Broglie, a French physicist, was acknowledged with the Nobel Prize for his great work in quantum mechanics. Through his works, he showed the world how mathematically subatomic particles can share wave properties as well. All of his mathematical theories were later proved by lots of experiments.

Here, he stated that all the particles that move can sometimes function as a wave, and as particles. De Broglie wave or matter-wave is termed for the waves related to the moving particles, and the wave wavelength shown here is called Broglie wavelength.

Wave-particle duality is the term given for the particles that exhibit particle and wave nature. Wave-particle duality was firstly observed from electromagnetic radiation (photons or electromagnetic waves).

Equations of the de Broglie Hypothesis

de Broglie described his idea that any matter can show wavelike properties. Without the de Broglie Hypothesis, scientists could not understand the nature of the smallest scales in quantum mechanics because these are the basic understanding of quantum mechanics.

In quantum theory, the most noticeable thing is the wave nature of matter. We can find it from the de Broglie Hypothesis. The wavelength of an electron was mathematically determined by de Broglie using Planck’s equation (E = hf) and mass-energy equivalence equation of Albert Einstein’s  (E = mc²), momentum in a series of substitutions, and the wave speed equation (v = λf ). By computing Planck’s equation and the mass-energy equivalence equation of Albert Einstein, we get:

E = mc²=hf

Here E= energy

m = mass

f = frequency

H = Planck’s constant

We know that massive particles don’t travel at the speed of light; thus, we are substituting c with v (velocity of the particle):

              mv²=hf

 v/λ (taken from wave speed-equation) lambda is wavelength here, replacing f. Now, we can simplify it and get:

            λ =h/mv

At last, as we know, mass m times velocity v is equal to momentum p :

              λ  =h/p

Now, this is termed as de Broglie Equation Broglie wavelength is measured in the standard units metres (m).

Bohr’s quantum hypothesis

Neil Henrik Bohr is a great scientist who gave lots of contributions to quantum mechanics. He received the Nobel prize in 1922. He was also a scientific researcher and a philosopher Neil Henrik Bohr observed that by obeying Maxwellian electromagnetism and Newtonian mechanism, all the electrons in the atom revolve around the nucleus of the atom. He included the following into the Rutherford model hypotheses.

• An atom absorbs or emits radiation when it makes a transition from one state(stationary) to another state, where the frequency is represented by frequency conditions.

• Electrons can stay only in specific orbits of an atom(special States). He called these orbits (special States) stationary states.

• In Special States (stationary states) the electrons move by only obeying (Maxwellian electromagnetism and Newtonian mechanism all the electrons) classic rules.

Relation Between de Broglie Equation and Bohr’s Hypothesis of Atom

As per the de Broglie hypothesis, electrons are standing waves that extend around the nucleus of the orbit of an atom, and they are not the particles[solids] revolving around the nucleus. From Bohr atomic structure, then the angular momentum of electrons in moving state is: mvr=nh/2π

Here, m = mass of electron

r = orbit radius of an atom

n = Quantum principle number (1, 2, 3, 4….)

And, as per Broglie’s equation,

mvr=nh/2π

Or, mv=h/λ

Here,λ = moving electron’s wavelength

Now, when we combine both Bohr’s Hypothesis of Atom and Broglie Equation, we will get: 2rπ=nλ

In an orbit of Bohr, an integral number of wavelengths must fit in. When we analyse this statement from the final formula received here. Standing waves should be generated from an integral number of wavelengths. A standing wave could not travel away from the allowed space. It should produce a fixed profile or a stationary wave.

Conclusion

In this article, we discussed the Relation Between the de Broglie Equation and Bohr’s Hypothesis of Atom Bohr’s quantum hypothesis and De Broglie Equation. Both theories of de Broglie Equation and Bohr’s Hypothesis of Atom have a great impact on the quantum mechanics of modern science.

The equations of de Broglie were a relief for the scientific world, more importantly for Neil Henrik Bohr because he had postulated in his theory about angular momentum is equal to integral multiplied by h2n or mvr=nh2. After finding the relationship between the two theories, we conclude that: in an orbit of Bohr, an integral number of wavelengths must fit in. When we analyse this statement from the final formula received. Standing waves should be generated from an integral number of wavelengths. 

A standing wave could not travel away from the allowed space. It should produce a fixed profile or a stationary wave.