Angular Momentum of Electron

Any object’s rotation and revolution are always measured in relation to its axis. In the case of electrons, the orbital and spin motion axes are not linear, but angular, and the related momentum is angular momentum, which causes the electron to behave like a tiny magnet.

Momentum

The quantity of motion which is possessed by an object is referred to as momentum in physics. The momentum of an object refers to the state of the object when it is in motion. Because all objects have mass, they have momentum when they move because their mass is moving. The amount of substance moving and the speed at which it is travelling are the two parameters that define how much momentum an item has. Momentum is influenced by mass and velocity. An object’s momentum which is determined in the form of an equation is equal to its mass multiplied by its velocity.

momentum = mass × velocity

p=m×v

Angular momentum is conserved because it is a meaningful quantity. According to physicists, angular momentum is the rotational equivalent of linear momentum. As a result, total angular momentum in a closed loop/structure is regarded as constant.

Momentum of Electron

Bohr’s atomic model developed a number of postulates about how electrons are arranged in distinct orbits around the nucleus. In Bohr’s atomic model, the angular momentum of electrons orbiting around nucleus is quantized. He went on to claim that electrons can only move in orbits with an integral multiple of h/2 angular momentum. This notion about the quantization of an electron’s rotational momentum was later articulated by Louis de Broglie. According to him, a travelling electron in a circular orbit behaves like a particle wave.

The angular momentum of an electron is expressed by Bohr as mvr or nh2π. Here, v denotes the velocity, n denotes the orbit in which the electron is located, m denotes the electron’s mass, and r is the radius of the nth orbit).

In a closed domain, angular momentum is a property of the mass in motion around a fixed axis which is conserved. Angular momentum is the rotational analogue of linear momentum (also considered as moment of momentum or the rotational momentum). It is a significant quantity in physics since it is a conserved quantity, meaning that the total angular momentum of a closed system remains constant.

In three dimensions, the angular momentum of a point particle is a pseudovector r p, which is the cross product of particle’s position vector r (related to some origin) and its momentum vector, which is p = mv in Newtonian physics. Angular momentum, unlike momentum, is reliant on where the origin is chosen since the particle’s position is determined from the origin.

L= m×v×r

Angular Momentum of Electron Equation

De Broglie came up with a theory to explain why angular momentum could be quantized in the way Bohr thought it could be. When you take the wavelength of an electron and suppose that an integral number of wavelengths must fit within the perimeter of an orbit, you get the same quantized angular momenta as Bohr.

The circumference of a circular orbit has to be an integer number of wavelengths. As a result, the total distance travelled by an electron in the k circular orbit of radius rk is equal to the circle’s diameter, 2rk.

Therefore, the orbital angular momentum is

2πrk= kλ  ——– (1)

Here, 

= de Broglie Wavelength

Form de Broglie equation, we have

λ=hp 

Here,

h = Planck’s constant

p = momentum of an electron

therefore, 

λ=hmvk ——– (2)

Here,

mvk = momentum of an electron revolving in the k orbit.

Now, put the value of in equation (1) then we get;

2πrk=khmvk 

The angular momentum of electron equation is given as

kh2π= m×vkrk 

Angular Momentum of Electron Formula

The formulas for angular momentum of electron are as follows;

L= m×v×r 

ΔE = h x v 

m×v×r=(n×h)/(2×π)  

Quantization of Angular Momentum of Electron

When the angular momentum is quantized, the orbit’s radius and energy are also quantized. Transitions of an electron from one allowed orbit/energy to another, according to Bohr, created the discrete lines in the hydrogen atom’s spectrum. He also believed that, as Einstein stated, energy for a transition is obtained or released in the form of a photon, and that energy is obtained or released in the form of a photon.

∆E=h×v 

Rutherford proposed that electrons orbit the nucleus of an atom. One problem with this concept is that in classical physics, circling electrons undergo centripetal acceleration, and accelerating charges lose energy by radiating, therefore a stable electronic orbit is impossible. Despite this, Bohr proposed stable electronic orbits, calculating electronic angular momentum as

L= m×v×r 

Conclusion

The term “momentum” refers to the state of an object when it is in motion (moving).

momentum = mass × velocity

p=m×v

The angular momentum of an electron is expressed by Bohr as mvr or nh2.

The formulas for angular momentum of electron are as follows;

L= m×v×r 

ΔE = h x v 

m×v×r=(n×h)/(2×π)