Quantisation of Angular Momentum - Explained (Visually)

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.

But now, in modern physics, this

electron revolving around a nucleus has

been replaced by a stationary electron

cloud model of the atom.

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.

So, the question is what is quantization

of angular momentum?

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.

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.

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.

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.

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.

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.

And these solutions, or the eigenvalues

corresponding to L squared and LZ, they

depend on the quantum numbers L and M.

I'll try to show you an intuitive way of

how these quantum numbers are decided.

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.

So, this gives you a very beautiful

visual idea of what m really represents.

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.

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.

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.

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.

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.

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.

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.

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.

Today, I want to focus on angular

momentum.

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.

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.

That means the S orbital has no angular

momentum at all.

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?

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.

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?

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.

We can do the same thing for D orbital

for quantum number L equals to two.

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.

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.

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.

The only difference is that the quantum

number S can only take values of half.

So, if we represent that visually, we

get this kind of a shape.

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.

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.

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.

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.

This is For the Love of Physics. Thank

you so much. Take care. Bye-bye.