Quantisation of Angular Momentum - Explained Visually

Añadido: 2026-05-15

[00:00:00]The old model of the atom, an electron revolving around a nucleus, had one obvious thing, angular momentum. And this is a very important physical quantity because for rotating systems, angular momentum is conserved in nature.

[00:00:15]But now, in modern physics, this electron revolving around a nucleus has been replaced by a stationary electron cloud model of the atom.

[00:00:25]This is because quantum mechanics can predict the probability density of where the electron is most likely to be found in the atom. Now, in this model, the angular momentum quantity may not be very obvious, but it is still ever-present. And more importantly, in atomic physics, it is quantized.

[00:00:42]So, the question is what is quantization of angular momentum?

[00:00:46]You see, in the classical model of a particle revolving around a nucleus, there is no restriction on the magnitude or the direction of the angular momentum. Depending upon the speed, radial distance, or the plane of revolution, the angular momentum vector can take any direction or magnitude in classical physics.

[00:01:06]But that is not true in quantum physics because when we talk about quantum systems, angular momentum can only take those values which is allowed by the theory of quantum mechanics.

[00:01:18]In fact, the magnitude and the direction of angular momentum can only take very specific [music] values, which is known as the quantization of angular momentum.

[00:01:33]You see, in the quantum mechanical framework, this quantity is associated with four distinct operators that can give us some meaningful information about the system. So, Lx, for example, is the operator associated with the x component of angular momentum vector. Ly and Lz are the operators corresponding to the y and z component of this angular momentum vector. When we combine these operators to create the magnitude, we end up getting the L squared operator.

[00:02:00]Now, in theory, together these operators can give us all the information about angular momentum vector, but the problem is in quantum mechanics, we have something called the uncertainty principle.

[00:02:12]You must have all heard of the position and the linear momentum uncertainty principle that for a moving particle, you cannot measure the position and the linear velocity at any given point in time simultaneously with absolute accuracy. Similar uncertainty relationships also exist for the angular momentum, which says that you cannot measure the components of angular momentum LX, LY, and LZ simultaneously with absolute accuracy for a given system. In fact, there is a limit given by these uncertainty relations, beyond which you cannot accurately measure them in a given system. Now, these kinds of uncertainty relations goes back to commutator algebra of the quantum mechanical framework. You see, whenever two operators do not commute, they have a corresponding uncertainty relationship for them. And this is true for LX, LY, and LZ. However, what is interesting is that this is not necessarily true for L squared operator. So, if we find the L squared commutator with either LX or LY or LZ separately, then we find that they do commute, which means we can measure L squared and LX together or L squared and LY together or L squared and LZ together. So, that means we have to make a choice. And by convention in the physics community, we choose L squared and LZ representation. And therefore, the theory of quantum mechanics can give us precise information about L squared, the magnitude of angular momentum, and LZ, the Z component of angular momentum for a given system. But, this is an information that we can obtain only from the wave function solution of the system. So, coming to the wave function solution, it is a solution of the Schrödinger equation when we try to solve for central potentials like the Coulombic interaction of an atom. And because of spherical symmetry, we write this wave function in terms of spherical coordinates r theta phi. And when we do that, the wave function can be written in three distinct parts. The radial solution, as the name suggests, gives us that part of the wave function solution which varies with respect to the radial distance from the nucleus. While the angular solution gives us that part of the wave function solution which varies as we go from north to south. And the azimuthal solution gives us that part of the wave function solution as you go from west to east along the equator or along a latitude. And the various boundary conditions associated with these solutions lead to three distinct quantum numbers n, l, and m. Now, n is related to energy of the system, so we are not concerned with that in today's video. L and m, however, are very much responsible for the angular momentum of the system. In fact, if we combine the angular solution and the azimuthal solution, we get what is called spherical harmonics which are effectively the eigen states of angular momentum vector. So, if we apply these operators L squared and LZ onto the spherical harmonics, we get two very beautiful solutions. In fact, these equations are known as the eigenvalue equations for angular momentum operator.

[00:05:30]And these solutions, or the eigenvalues corresponding to L squared and LZ, they depend on the quantum numbers L and M.

[00:05:39]I'll try to show you an intuitive way of how these quantum numbers are decided.

[00:05:43]So, first the azimuthal solutions, which are nothing but oscillatory solutions given by e to the power i m phi. Phi being the angle from west to east if you go along a latitude. Now, we can look at the behavior of cos m phi which is similar to that of sin m phi although separated by a phase difference of pi by 2. So, for m is equal to 2, you end up getting this kind of an oscillatory solution. Now, these kinds of solutions are easy to understand because we are very much used to oscillatory solutions along the x-axis. But, what if I try to represent the same oscillation in a polar plot because that is a much better representation of the azimuthal nature of the solutions. So, in this plot, the radial distance represents the functional value of the oscillation and wherever the function goes to zero, the radius becomes zero and the plot looks something like this. I can do the same for other values of m, m equals 0, 1, 2, 3 and we end up getting more and more number of oscillations and as a result, more and more lobes in the polar plot.

[00:06:50]So, this gives you a very beautiful visual idea of what m really represents.

[00:06:57]It represents the oscillations of the wave function around the azimuthal direction and with greater and greater value of m, you end up getting more oscillations which by the way corresponds to a greater value of angular momentum because with more oscillations, the effective wavelength decreases and we know that wavelength and angular momentum or momentum in general have an inverse proportionality.

[00:07:20]But, you may ask why integral values of m? This is because when we undergo one complete revolution, I want to come back to the same point with the same value of the function. And if I try to do that for let's suppose m is equal to 2.5, then that doesn't happen.

[00:07:38]If you notice the polar plot, the wave does not close in on itself. And these kinds of values are therefore not allowed. The wave function must, at the end of the day, have the same value at the same location, even though you took one complete revolution and came back to the same point. So, this boundary condition effectively restricts the value of the quantum number m to only integral values. You can have 0 1 2 3 4 like that, or the negative values, because even the negative values are allowed. The positive and the negative of m simply changes the direction of the angular momentum vector.

[00:08:17]Now, if we come to the angular solutions, that part of the Schrödinger equation which is responsible for theta, then we effectively get something called associated Legendre functions. The associated Legendre functions gives us how the wave function varies as we go from north to south pole.

[00:08:39]I know the mathematical expressions are quite complicated here, but notice a few essential details. The Legendre functions are mth order derivatives of what is known as a Legendre polynomial.

[00:08:51]The Legendre polynomial is given by the Rodrigues formula. Students who are familiar with mathematical physics may have seen these expressions before. Now, the way to solve this kind of a differential equation corresponding to theta is to essentially employ what is known as the power series method. But the power series method does not really give us finite solutions for all cases.

[00:09:12]It only gives us finite solutions when the power series terminates after a particular series number. So, the short answer is to get a finite wave function solution, we must terminate the power series solution that leads to very specific integral values of l. l essentially represents the number of terms present in the power series solution. So, L therefore is now restricted to values like 0 1 2 3 like that. But, if you also look at the connection between associated Legendre function and the Rodrigues formula, the Legendre polynomial is a polynomial of the order of L. And if you take a derivative of a polynomial of the order of L, you cannot do the derivative more than L number of times because if you do that, you'll end up getting zero. Which means that M is now therefore restricted to all the values less than L. So, for example, if L equals 0, then M can only have a zero value. But, if L is equal to 1, M can have values of -1 0 or +1. And then you can take it forward for L is equal to 2 3 and further. So, given these quantum numbers, I can write down the mathematical expressions for each of them.

[00:10:25]And I can in fact represent them in a normal 2D plane graph. You can clearly see the oscillatory nature of these solutions. What's even interesting is if I try to plot them in a polar plot with respect to theta, then suddenly we have these beautiful diagrams, these lobe and petal-like shapes. In fact, if we combine the azimuthal solutions that you saw earlier and these associated Legendre polynomial plots, we effectively get an idea about the shape of the orbitals and why orbitals have those unique shapes of lobes and petals etc. But, a detailed discussion on that is probably a topic for the next video.

[00:11:05]Today, I want to focus on angular momentum.

[00:11:11]So, coming back to those equations, we can now see how the various values of L and M quantum number influences what is going to be the angular momentum magnitude and the angular momentum direction.

[00:11:24]So, for example, if we take L is equal to 0, the S orbital, it's clear that the magnitude of angular momentum is zero and the Z component of angular momentum is zero.

[00:11:35]That means the S orbital has no angular momentum at all.

[00:11:43]And this is kind of the easiest example to understand. But if we go to L is equal to 1, what happens then? Here, the magnitude of angular momentum is root 2 H cut. But the Z component can have three distinct values of minus H cut, zero, and plus H cut. How do we represent them in a diagram, for example?

[00:12:05]It can only have a fixed length, so it can only be found on the surface of a sphere. Any value above that or any value below that is not allowed. So for a P orbital, the angular momentum vector will lie only on the surface of the sphere. Now, what if we include the Z component? The Z component is effectively the component of this L vector onto the Z axis. Now, all the angular momentum vectors that will have a very specific Z component lies along the conical surface, which intersects with the sphere and creates the circular shape. Now, this diagram visually demonstrates beautifully what are the magnitudes and the directions of the angular momentum vector for the P orbital. As far as magnitude is concerned, only one value is possible, root 2 H cut. No other value is allowed. But as far as the direction is concerned, the angular momentum vector can lie on any one of these conical or circular surfaces. Now, at this point in time, there are a few questions that may have come up in your mind. First of all, when we earlier talked about the convention of L squared and L Z, I specifically mentioned that these are the only two quantities that we can precisely know. But from the diagram, you may say that wait, the choice of coordinate axis is ours, right? So, why don't we choose the Z axis to be along the direction of the angular momentum vector? Now, if you notice, if I do make that choice, if I choose the Z axis to be the in the direction of the angular momentum vector, then Ly and Lz will become precisely equal to zero. Now, that is not allowed in quantum mechanics. It goes back to the uncertainty principle.

[00:13:52]So, therefore, this is the only reasonable explanation. And even by the way here, as the angular momentum vector can take any orientation on the conical shape, if you look at its rotation at each point along the circle, it projects different values on the XY plane, which is perpendicular to the Z axis. And because it projects different values on the XY plane, the components of Lx and Ly is constantly changing. In fact, the average of Lx and average of Ly comes out to be zero because they can take positive and negative values here. So, the theory can only tell us what Lz and L magnitude is. It cannot tell us what Lx and Ly are. And one more misconception that may arise here in this diagram is is the angular momentum vector precessing around the Z axis?

[00:14:43]Even though I've shown the animation in this manner to create a visual representation, there is no precession involved. If I look at all these three distinct cases separately, what these shapes actually mean is that the angular momentum vector can take any direction lying on the inverted cone, the circle, and the cone. So, even specifying the angular momentum vector with an arrow is kind of misleading because it can be anywhere in this particular shape. It is only the magnitude and the Z component that we are pretty much sure of. The exact direction is still kind of smeared along the cone or along the circular surface.

[00:15:27]We can do the same thing for D orbital for quantum number L equals to two.

[00:15:33]If I do that, we will see that the magnitude here comes out to be root six H cut and the possible Z components comes out to be plus two H cut, plus H cut, zero, minus H cut and minus two H cut.

[00:15:45]In a very similar manner, we can represent them in this beautiful diagram. The angular vector has a magnitude which is root six H cut, which is fixed by the radius of a sphere. And their Z components leads to these kinds of conical and circular surfaces where the angular momentum vector is effectively smeared across those surfaces.

[00:16:15]Now, till this point in time, we have only talked about the orbital angular momentum of an electron in the presence of a nucleus. However, the electron also has its own distinct spin angular momentum. This is an intrinsic property of the angular momentum that an electron has and as it turns out, the eigen value equations for the spin operator is also somewhat similar.

[00:16:41]The only difference is that the quantum number S can only take values of half.

[00:16:46]So, if we represent that visually, we get this kind of a shape.

[00:16:53]The electron's intrinsic angular momentum can only have plus H cross by two in the positive Z axis or minus H cross by two in the negative Z [music] axis.

[00:17:07]Now, this is something that is verified by what is known as the Stern-Gerlach experiment. So, in the Stern-Gerlach experiment, we pass a beam of electrons through a non-uniform magnetic field.

[00:17:20]And when we do that, because the non-uniform magnetic field interacts slightly differently with the spin up and the spin down, so the beam splits into two parts. And this result is an actual proof that the electron has an intrinsic spin. Now, we can perform a similar experiment for the orbital angular momentum case. So, for example, if we take S orbital, L is equal to zero, you'll end up getting a scenario in which the beam does not split because the S orbital has no angular momentum in the first place. But, for L is equal to one, there are three distinct orientations, so the beam will split into three spots. While for L is equal to two, there are five distinct orientations, so the beam will split into five distinct spots. Now, this is something that I've only shown for a visual understanding perspective because the Stern-Gerlach experiment is a little bit hard to perform for orbital angular momentum because usually in atoms, the orbital angular momentum and the spin angular momentum couple together to create a sort of an effective angular momentum of the system.

[00:18:20]Nonetheless, both the spin angular momentum and the orbital angular momentum in an atom are quantized. They can only have very specific magnitudes and specific directions, which is allowed by the quantum mechanical theory. And this is one of the ways in which a quantum system is so vastly different from our classical understanding of angular momentum. I have made a lot of effort in creating these visualizations to give you a better understanding of the topic. If this is something that you are interested in, then please make a comment in the video and I'll try to create more such animated and visual perspective of understanding various topics in physics. I'm Divyendu Das.

[00:19:00]This is For the Love of Physics. Thank you so much. Take care. Bye-bye.