What is Orbital Angular Momentum?

Momentum explains  the multiplication of velocity and mass can give rise to the momentum of a body. When a body is in motion with a certain amount of mass, it possesses momentum. Well, angular momentum and momentum both are different by their different functionality.

Electrons are always revolving around the nucleus. It possesses angular momentum. So, the angular momentum of an electron is the rotation or the spinning of the electron around the nucleus.

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The product of mass and the velocity of the object is Momentum. Any object moving with mass is known to possess momentum, and the only difference in angular momentum is that it deals with rotating or spinning objects.

Angular Momentum is a vector quantity, which means the direction is also considered, along with magnitude.

Take this example: when you try to get on a bicycle you will first try to get a balance without a kickstand and will fall.  Once you start pedalling, the wheels pick up angular momentum,  and you see that they will resist change. Thus, the balancing becomes easier.

Angular momentum direction is given by the right-hand rule. The right-hand rule states that:

i) Position your right hand such that the fingers are in the direction of r.

ii) Next, curl them around your palm in such a way that they point towards the direction of Linear momentum(p).

iii) Finally, the outstretched thumb gives the direction of angular momentum(L).

Angular Momentum Quantum Number

It was Bohr who put forward the formula for the calculation of the angular momentum of an electron.

According to Bohr, the formula is mvr or nh / 2π 

Here, 

• v = the velocity

• n = the orbit in which electron is present

• m = mass of the electron 

• r = the radius of the nth orbit

Orbital Angular Momentum Quantum Number

Bohr’s atomic model is responsible for the generation of different postulates that are based on the arrangement of electrons.

Those arrangements of electrons can vary as they are in different orbits around the nucleus. 

Bohr’s atomic model has some specific comments on the angular momentum of electrons. Bohr suggested that the electrons that are revolving around the nucleus are quantized. 

He had some additional concepts over the orbiting of electrons around the nucleus. He made a statement that electrons can travel to the selected orbits only where an electron’s angular momentum is an integral product of h/2. 

Bohr's statement was not good enough to postulate the reason for the angular momentum and orbiting of electrons around the nucleus.

Later, the quantization of the angular momentum of an electron was perfectly analysed and put forward by Louis de Broglie. He suggested that an electron in motion within its circular orbit acts like a particle wave.

Example: 

Q. What is the Angular Quantum Number of  d electrons?

Ans: The orbital angular momentum is (L) = \[\sqrt{l(l+1)}\frac{h}{2 \pi}\]

 Here, the orbital d is used. So, the value of l = 2

Therefore, the orbital angular momentum can be represented as \[\sqrt{6}\frac{h}{2 \pi}\]

De Broglie’s Explanation 

De Broglie had mentioned a statement that was based on the quantization of the Angular Momentum of Electrons. The particle waves have certain behaviour that can be observed analogously. This procedure helped him understand the motion of the waves that are roaming on a string. 

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Particle waves are the types of waves that lead towards the standing waves. These waves are apprehended under different resonant conditions. When you try to pluck a stationary string, then a certain amount of wavelengths is getting excited. 

When we consider the fact differently, we will recognize that certain wavelengths only survive to give rise to angular momentum. 

All of these standing waves have nodes at the ends.

When standing waves are prepared, the total distance covered by a wave is an integral number of wavelengths. 

So, if we calculate the total distance covered by the electron when it is moving in the kth circular orbit having the radius r_k can be written as:

2πr_k = kλ ….1

Here, λ = de Broglie wavelength

As per de Broglie wavelength, the equation is

λ = h/p

Here, h = Planck’s constant

p = electron’s momentum

So, λ = h/mv_k ….2

Here, mvk = the momentum of an electron orbiting around the kth orbit.

So, if we put together two equations 1 and 2, we will find:

• 2πr_k = kh / mv_k

• mv_kr_k= kh / 2π

In the above process, de Broglie’s hypothesis has proven the second postulate of Bohr successfully. The hypothesis from de Broglie has made its start from the quantization of angular momentum of the revolving electron. Students can gain another conclusion about the quantized electron orbits and energy states. Both of them are present because of the wave nature of the electron.

How to Calculate Orbital Angular Momentum of P Electrons?

The answer to the question is very simple. Here is the solution:

We know that orbital angular momentum can be = \[\sqrt{l(l+1)}\frac{h}{2 \pi}\]

If the orbit is p, then l = 1

So, the orbital angular momentum of P electron =\[\sqrt{l(l+1)}\frac{h}{2 \pi}\]= \[\frac{h}{2 \pi}\]

Therefore, \[\frac{h}{2 \pi}\] is the answer that stands correct for the calculation of orbital angular momentum of P electron.

Applications of Angular Momentum of Electrons

Electron vortex beams are mostly used in different industries. Their applications include mapping of magnetization, identification of crystal chirality and studying chiral molecules and chiral Plasmon resonances.

We use two methods for measuring orbital angular momentum; they are:

• Wavefront flattening

• Cylindrically symmetric Stern-Gerlach-like measurement

Conclusion

This article helps derive Orbital Angular Momentum and its nuances. It is highly beneficial for students from exam and competition point of view. They can refer to this article or download the PDF to study.