Welcome back to the quantum physics series. In my previous article, I covered photoelectric effect, failure of the wave theory, and Einstein’s photon theory. If you haven’t read the previous article on quantum mechanics, I urge you to read all my articles one by one. All my articles are very useful and I have explained them in simple words.
In my today’s article I am going to give simple explanation on Bohr’s theory, de Broglie hypothesis and wave particle duality.
What is Bohr’s atomic model?
In 1913 Bohr gave the idea of atomic structure to explain the stability of the atom and the emission of sharp spectral lines. He modified the Rutherford model and proposed the following assumptions.
The electrons revolve around the nucleus in certain descreate orbits. While revolving in these allowed orbits the electrons do not emit or absorb electromagnetic radiation, even though they are in accelerated motion. Therefore, the atom is stable.
It will be possible for an electron to revolve only in an orbit for which the orbital angular momentum is an integer multiple of h/ 2π. If L (mvr) is the angular momentum then we can write-
mvr= nh/ 2π
n= 1,2,3,….. and h= Planck’s Constant
When an electron jumps from one allowed orbit to another the atom emits radiation.
The radiation is emitted or absorbed in the form of a single photon whose energy hv is equal to the difference in the energies “ΔE” of the electron in the two orbits.
Thus, ΔE = hv, v= frequency
How did Bohr’s theory explain the hydrogen spectrum?
The spectrum of the hydrogen atom is consist of many lines. These are divided into a number of series. Such as Lyman, Balmer, Paschen, Brackett, and Pfund series. These sharp spectral lines are the hydrogen spectrum and cannot be explained by classical theory. Bohr’s explained by applying Planck’s quantum hypothesis to Rutherford’s atomic model.
Bohr’s proposed two theories:
An electron can move only in orbits for which the angular momentum ‘L’ of the electron is an integer multiple of h/2π.
Absorption of radiation by an atom occurs when the electron jumps from one allowed orbit to another. The radiation is emitted or absorbed as a single photon whose energy ‘hv’ is equal to the difference between the energies of the electron in the two orbits.
That is, hv= E – e,
E= energy of the initial orbit and e= energy of the final orbit
When the hydrogen atom gets enough energy from outside, the electron moves from the inner orbit of lower energy to the outer orbit of higher energy. After 10⁻³ seconds the electron jumps back to the inner orbit by emitting radiation of wavelength which will be equal to.
1/ λ = R( 1/m² – 1/n²)
For,
m = 1 and n = 2,3,4…. We obtain Lyman series
m = 2 and n = 3,4,5…. We obtain Balmer series
m = 3 and n = 4,5,6…. We obtain Paschen series
m = 4 and n = 5,6,7…. We obtain Brackett series
m = 5 and n = 6,7,8….. We obtain Pfund series
Energy level diagram of hydrogen:
What were the shortcomings of Bohr’s theory?
- Bohr’s theory succeeded in explaining hydrogen spectrum and also gave information about atomic structure but it has some limitations which I am going to discuss one by one.
- An individual line in the hydrogen spectrum consists of many blurred lines, which is called the fine structure of hydrogen. Bohr’s theory was unable to explain this. When individual lines of the hydrogen spectrum were examined, it was found that they had lines very close to each other. This is called the fine structure of hydrogen. Bohr’s theory was unable to explain this.
- Bohr’s theory was unable to explain the variation in the intensity of the spectral lines of an element.
- This theory only applies to atoms with only one electron, such as the hydrogen atom, isotopes of hydrogen (deuterium), and ions with only one electron, such as singly ionized helium (He⁺) or doubly ionized lithium (Li²⁺). It does not explain the spectra of complex atoms.
- This theory could not give a satisfactory explanation about the distribution of electrons in atoms.
The De Broglie hypothesis explains Bohr’s theory’s limitations by introducing the concept of wave-particle duality for matter,
What is de Broglie’s hypothesis of matter wave?
The phenomena of interference, diffraction and polarization of light can be explained on the basis of the wave nature of light. However, there are some phenomena which could not be explained on this wave theory of radiation. For example, photoelectric effect, Compton effect etc. These effects can only be explained on the basis of the particle nature of radiation. Thus, the dual nature of wave and particle became associated with light. In 1924 Louis de Broglie put forward the suggestion that matter exhibits wave like properties.
The De- Broglie’s hypothesis explains Bohr’s theory’s limitations by introducing the concept of wave-particle duality for matter. De-Broglie’s hypothesis for matter waves, also known as wave-particle duality, states that all moving particles, such as electrons, protons and all matter, have both particle-like and wave-like properties. Just as light can act as both a wave and a particle (photon).
What is wave–particle duality?
Wave particle duality is that all matter exhibits both wave like and particle like properties. in other words we can say they show wave like and particle like behavior. Wave-particle duality means that all matter exhibits both wave-like and particle-like properties. In other words, they show both wave-like and particle-like behavior. This applies to all objects, even microscopic ones.
Bohr’s model of the atom shows two features:
Electron is considered as a wave and light is considered as a particle (photon).
But this concept contradicts the earlier phenomenon that an electron behaves as a particle and light as a wave, which is provided by the relation,
(λ = h/mass × speed), that connects the mass and speed of an electron to the wavelength of the corresponding wave and E= h × frequency, which links the frequency of a light wave to the energy of the corresponding photon.
Now the question arises:
- what is an electron? Is it a particle or a wave?
- What is light? Is it a wave or a photon? The answer is wave-particle duality.
The answer is wave-particle duality, which states that all objects sometimes exhibit wave-like behavior and sometimes particle-like behavior depending on the experiment and interaction.
What is the wavelength of a matter wave? ( de Broglie wavelength)
The wavelength of a matter wave, or de Broglie wavelength, is equals to to λ = h/p.
λ is the wavelength, h is Planck’s constant (a fundamental constant of nature), and p is the momentum of the particle. Momentum (p) is equal to the product of the particle’s mass (m) and velocity (v). Therefore, we can write it as λ = h/mv.
This equation shows that the wavelength of a matter wave is inversely proportional to the particle’s momentum. This means that heavier particles or faster-moving particles will have shorter de Broglie wavelengths. Higher velocity or larger mass will result in larger momentum and, consequently, shorter wavelength.
If you have any doubts about any topic related to quantum mechanics, please let me know in the comments section. I will definitely write an article on it.
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