0:00
The old model of the atom, an electron
0:02
revolving around a nucleus, had one
0:04
obvious thing, angular momentum. And
0:07
this is a very important physical
0:09
quantity because for rotating systems,
0:12
angular momentum is conserved in nature.
0:15
But now, in modern physics, this
0:17
electron revolving around a nucleus has
0:19
been replaced by a stationary electron
0:23
cloud model of the atom.
0:25
This is because quantum mechanics can
0:27
predict the probability density of where
0:29
the electron is most likely to be found
0:31
in the atom. Now, in this model, the
0:34
angular momentum quantity may not be
0:35
very obvious, but it is still
0:37
ever-present. And more importantly, in
0:39
atomic physics, it is quantized.
0:42
So, the question is what is quantization
0:46
You see, in the classical model of a
0:48
particle revolving around a nucleus,
0:50
there is no restriction on the magnitude
0:53
or the direction of the angular
0:55
momentum. Depending upon the speed,
0:57
radial distance, or the plane of
0:59
revolution, the angular momentum vector
1:01
can take any direction or magnitude in
1:06
But that is not true in quantum physics
1:09
because when we talk about quantum
1:11
systems, angular momentum can only take
1:14
those values which is allowed by the
1:16
theory of quantum mechanics.
1:18
In fact, the magnitude and the direction
1:20
of angular momentum can only take very
1:23
specific [music] values,
1:25
which is known as the quantization of
1:33
You see, in the quantum mechanical
1:35
framework, this quantity is associated
1:37
with four distinct operators that can
1:39
give us some meaningful information
1:40
about the system. So, Lx, for example,
1:43
is the operator associated with the x
1:46
component of angular momentum vector. Ly
1:49
and Lz are the operators corresponding
1:51
to the y and z component of this angular
1:53
momentum vector. When we combine these
1:55
operators to create the magnitude, we
1:58
end up getting the L squared operator.
2:00
Now, in theory, together these operators
2:02
can give us all the information about
2:04
angular momentum vector, but the problem
2:07
is in quantum mechanics, we have
2:08
something called the uncertainty
2:12
You must have all heard of the position
2:14
and the linear momentum uncertainty
2:15
principle that for a moving particle,
2:18
you cannot measure the position and the
2:20
linear velocity at any given point in
2:22
time simultaneously with absolute
2:25
accuracy. Similar uncertainty
2:26
relationships also exist for the angular
2:29
momentum, which says that you cannot
2:30
measure the components of angular
2:32
momentum LX, LY, and LZ simultaneously
2:37
with absolute accuracy for a given
2:39
system. In fact, there is a limit given
2:42
by these uncertainty relations, beyond
2:44
which you cannot accurately measure them
2:47
in a given system. Now, these kinds of
2:49
uncertainty relations goes back to
2:52
commutator algebra of the quantum
2:54
mechanical framework. You see, whenever
2:55
two operators do not commute, they have
2:58
a corresponding uncertainty relationship
3:00
for them. And this is true for LX, LY,
3:03
and LZ. However, what is interesting is
3:07
that this is not necessarily true for L
3:09
squared operator. So, if we find the L
3:12
squared commutator with either LX or LY
3:16
or LZ separately, then we find that they
3:19
do commute, which means we can measure L
3:23
squared and LX together or L squared and
3:27
LY together or L squared and LZ
3:31
together. So, that means we have to make
3:33
a choice. And by convention in the
3:35
physics community, we choose L squared
3:39
and LZ representation. And therefore,
3:42
the theory of quantum mechanics can give
3:44
us precise information about L squared,
3:47
the magnitude of angular momentum, and
3:50
LZ, the Z component of angular momentum
3:53
for a given system. But, this is an
3:55
information that we can obtain only from
3:57
the wave function solution of the
3:59
system. So, coming to the wave function
4:01
solution, it is a solution of the
4:03
Schrödinger equation when we try to
4:05
solve for central potentials like the
4:07
Coulombic interaction of an atom. And
4:09
because of spherical symmetry, we write
4:11
this wave function in terms of spherical
4:13
coordinates r theta phi. And when we do
4:16
that, the wave function can be written
4:17
in three distinct parts. The radial
4:20
solution, as the name suggests, gives us
4:22
that part of the wave function solution
4:24
which varies with respect to the radial
4:26
distance from the nucleus. While the
4:28
angular solution gives us that part of
4:31
the wave function solution which varies
4:32
as we go from north to south. And the
4:35
azimuthal solution gives us that part of
4:38
the wave function solution as you go
4:39
from west to east along the equator or
4:43
along a latitude. And the various
4:45
boundary conditions associated with
4:47
these solutions lead to three distinct
4:50
quantum numbers n, l, and m. Now, n is
4:54
related to energy of the system, so we
4:56
are not concerned with that in today's
4:57
video. L and m, however, are very much
5:01
responsible for the angular momentum of
5:03
the system. In fact, if we combine the
5:06
angular solution and the azimuthal
5:08
solution, we get what is called
5:10
spherical harmonics which are
5:12
effectively the eigen states of angular
5:15
momentum vector. So, if we apply these
5:17
operators L squared and LZ onto the
5:20
spherical harmonics, we get two very
5:23
beautiful solutions. In fact, these
5:25
equations are known as the eigenvalue
5:28
equations for angular momentum operator.
5:30
And these solutions, or the eigenvalues
5:33
corresponding to L squared and LZ, they
5:36
depend on the quantum numbers L and M.
5:39
I'll try to show you an intuitive way of
5:41
how these quantum numbers are decided.
5:43
So, first the azimuthal solutions, which
5:45
are nothing but oscillatory solutions
5:47
given by e to the power i m phi. Phi
5:51
being the angle from west to east if you
5:54
go along a latitude. Now, we can look at
5:56
the behavior of cos m phi which is
5:58
similar to that of sin m phi although
6:00
separated by a phase difference of pi by
6:02
2. So, for m is equal to 2, you end up
6:05
getting this kind of an oscillatory
6:07
solution. Now, these kinds of solutions
6:09
are easy to understand because we are
6:11
very much used to oscillatory solutions
6:13
along the x-axis. But, what if I try to
6:16
represent the same oscillation in a
6:18
polar plot because that is a much better
6:21
representation of the azimuthal nature
6:23
of the solutions. So, in this plot, the
6:26
radial distance represents the
6:27
functional value of the oscillation and
6:30
wherever the function goes to zero, the
6:32
radius becomes zero and the plot looks
6:35
something like this. I can do the same
6:37
for other values of m, m equals 0, 1, 2,
6:42
3 and we end up getting more and more
6:44
number of oscillations and as a result,
6:47
more and more lobes in the polar plot.
6:50
So, this gives you a very beautiful
6:52
visual idea of what m really represents.
6:57
It represents the oscillations of the
6:59
wave function around the azimuthal
7:01
direction and with greater and greater
7:03
value of m, you end up getting more
7:06
oscillations which by the way
7:07
corresponds to a greater value of
7:10
angular momentum because with more
7:12
oscillations, the effective wavelength
7:14
decreases and we know that wavelength
7:16
and angular momentum or momentum in
7:18
general have an inverse proportionality.
7:20
But, you may ask why integral values of
7:23
m? This is because when we undergo one
7:26
complete revolution, I want to come back
7:28
to the same point with the same value of
7:31
the function. And if I try to do that
7:33
for let's suppose m is equal to 2.5,
7:36
then that doesn't happen.
7:38
If you notice the polar plot, the wave
7:41
does not close in on itself. And these
7:44
kinds of values are therefore not
7:46
allowed. The wave function must, at the
7:48
end of the day, have the same value at
7:50
the same location, even though you took
7:53
one complete revolution and came back to
7:55
the same point. So, this boundary
7:57
condition effectively restricts the
7:59
value of the quantum number m to only
8:02
integral values. You can have 0 1 2 3 4
8:06
like that, or the negative values,
8:08
because even the negative values are
8:09
allowed. The positive and the negative
8:11
of m simply changes the direction of the
8:14
angular momentum vector.
8:17
Now, if we come to the angular
8:18
solutions, that part of the Schrödinger
8:21
equation which is responsible for theta,
8:24
then we effectively get something called
8:26
associated Legendre functions. The
8:29
associated Legendre functions gives us
8:32
how the wave function varies as we go
8:36
from north to south pole.
8:39
I know the mathematical expressions are
8:40
quite complicated here, but notice a few
8:43
essential details. The Legendre
8:45
functions are mth order derivatives of
8:48
what is known as a Legendre polynomial.
8:51
The Legendre polynomial is given by the
8:53
Rodrigues formula. Students who are
8:55
familiar with mathematical physics may
8:57
have seen these expressions before. Now,
8:59
the way to solve this kind of a
9:01
differential equation corresponding to
9:02
theta is to essentially employ what is
9:05
known as the power series method. But
9:08
the power series method does not really
9:10
give us finite solutions for all cases.
9:12
It only gives us finite solutions when
9:15
the power series terminates after a
9:18
particular series number. So, the short
9:20
answer is to get a finite wave function
9:23
solution, we must terminate the power
9:26
series solution that leads to
9:29
very specific integral values of l. l
9:33
essentially represents the number of
9:35
terms present in the power series
9:37
solution. So, L therefore is now
9:39
restricted to values like 0 1 2 3 like
9:42
that. But, if you also look at the
9:43
connection between associated Legendre
9:46
function and the Rodrigues formula, the
9:47
Legendre polynomial is a polynomial of
9:50
the order of L. And if you take a
9:52
derivative of a polynomial of the order
9:54
of L, you cannot do the derivative more
9:57
than L number of times because if you do
9:59
that, you'll end up getting zero. Which
10:01
means that M is now therefore restricted
10:04
to all the values less than L. So, for
10:07
example, if L equals 0, then M can only
10:09
have a zero value. But, if L is equal to
10:12
1, M can have values of -1 0 or +1. And
10:16
then you can take it forward for L is
10:17
equal to 2 3 and further. So, given
10:20
these quantum numbers, I can write down
10:22
the mathematical expressions for each of
10:25
And I can in fact represent them in a
10:27
normal 2D plane graph. You can clearly
10:30
see the oscillatory nature of these
10:32
solutions. What's even interesting is if
10:35
I try to plot them in a polar plot with
10:38
respect to theta, then suddenly we have
10:40
these beautiful diagrams, these lobe and
10:43
petal-like shapes. In fact, if we
10:45
combine the azimuthal solutions that you
10:48
saw earlier and these associated
10:50
Legendre polynomial plots, we
10:53
effectively get an idea about the shape
10:56
of the orbitals and why orbitals have
10:58
those unique shapes of lobes and petals
11:01
etc. But, a detailed discussion on that
11:03
is probably a topic for the next video.
11:05
Today, I want to focus on angular
11:11
So, coming back to those equations, we
11:13
can now see how the various values of L
11:15
and M quantum number influences what is
11:19
going to be the angular momentum
11:20
magnitude and the angular momentum
11:24
So, for example, if we take L is equal
11:26
to 0, the S orbital, it's clear that the
11:29
magnitude of angular momentum is zero
11:32
and the Z component of angular momentum
11:35
That means the S orbital has no angular
11:43
And this is kind of the easiest example
11:45
to understand. But if we go to L is
11:47
equal to 1, what happens then? Here, the
11:51
magnitude of angular momentum is root 2
11:54
H cut. But the Z component can have
11:56
three distinct values of minus H cut,
11:59
zero, and plus H cut. How do we
12:01
represent them in a diagram, for
12:05
It can only have a fixed length, so it
12:08
can only be found on the surface of a
12:10
sphere. Any value above that or any
12:14
value below that is not allowed. So for
12:17
a P orbital, the angular momentum vector
12:20
will lie only on the surface of the
12:22
sphere. Now, what if we include the Z
12:24
component? The Z component is
12:26
effectively the component of this L
12:29
vector onto the Z axis. Now, all the
12:32
angular momentum vectors that will have
12:34
a very specific Z component lies along
12:37
the conical surface, which intersects
12:39
with the sphere and creates the circular
12:42
shape. Now, this diagram visually
12:45
demonstrates beautifully
12:47
what are the magnitudes and the
12:49
directions of the angular momentum
12:51
vector for the P orbital. As far as
12:53
magnitude is concerned, only one value
12:56
is possible, root 2 H cut. No other
12:58
value is allowed. But as far as the
13:00
direction is concerned, the angular
13:02
momentum vector can lie on any one of
13:04
these conical or circular surfaces. Now,
13:07
at this point in time, there are a few
13:09
questions that may have come up in your
13:11
mind. First of all, when we earlier
13:13
talked about the convention of L squared
13:16
and L Z, I specifically mentioned that
13:18
these are the only two quantities that
13:20
we can precisely know. But from the
13:23
diagram, you may say that wait, the
13:25
choice of coordinate axis is ours,
13:28
right? So, why don't we choose the Z
13:30
axis to be along the direction of the
13:33
angular momentum vector? Now, if you
13:34
notice, if I do make that choice, if I
13:37
choose the Z axis to be the in the
13:40
direction of the angular momentum
13:41
vector, then Ly and Lz will become
13:45
precisely equal to zero. Now, that is
13:48
not allowed in quantum mechanics. It
13:50
goes back to the uncertainty principle.
13:52
So, therefore, this is the only
13:53
reasonable explanation. And even by the
13:56
way here, as the angular momentum vector
13:58
can take any orientation on the conical
14:01
shape, if you look at its rotation at
14:04
each point along the circle, it projects
14:07
different values on the XY plane, which
14:09
is perpendicular to the Z axis. And
14:11
because it projects different values on
14:13
the XY plane, the components of Lx and
14:16
Ly is constantly changing. In fact, the
14:19
average of Lx and average of Ly comes
14:22
out to be zero because they can take
14:24
positive and negative values here. So,
14:26
the theory can only tell us what Lz and
14:30
L magnitude is. It cannot tell us what
14:33
Lx and Ly are. And one more
14:35
misconception that may arise here in
14:38
this diagram is is the angular momentum
14:40
vector precessing around the Z axis?
14:43
Even though I've shown the animation in
14:45
this manner to create a visual
14:47
representation, there is no precession
14:50
involved. If I look at all these three
14:52
distinct cases separately, what these
14:56
shapes actually mean is that the angular
14:59
momentum vector can take any direction
15:01
lying on the inverted cone, the circle,
15:04
and the cone. So, even specifying the
15:06
angular momentum vector with an arrow is
15:08
kind of misleading because it can be
15:11
anywhere in this particular shape. It is
15:14
only the magnitude and the Z component
15:16
that we are pretty much sure of. The
15:19
exact direction is still kind of smeared
15:23
along the cone or along the circular
15:27
We can do the same thing for D orbital
15:29
for quantum number L equals to two.
15:33
If I do that, we will see that the
15:34
magnitude here comes out to be root six
15:37
H cut and the possible Z components
15:39
comes out to be plus two H cut, plus H
15:42
cut, zero, minus H cut and minus two H
15:45
In a very similar manner, we can
15:47
represent them in this beautiful
15:48
diagram. The angular vector has a
15:51
magnitude which is root six H cut, which
15:53
is fixed by the radius of a sphere. And
15:57
their Z components leads to these kinds
15:59
of conical and circular surfaces where
16:02
the angular momentum vector is
16:03
effectively smeared across those
16:15
Now, till this point in time, we have
16:17
only talked about the orbital angular
16:19
momentum of an electron in the presence
16:22
of a nucleus. However, the electron also
16:25
has its own distinct spin angular
16:28
momentum. This is an intrinsic property
16:31
of the angular momentum that an electron
16:33
has and as it turns out, the eigen value
16:36
equations for the spin operator is also
16:41
The only difference is that the quantum
16:42
number S can only take values of half.
16:46
So, if we represent that visually, we
16:48
get this kind of a shape.
16:53
The electron's intrinsic angular
16:55
momentum can only have plus H cross by
16:58
two in the positive Z axis or minus H
17:01
cross by two in the negative Z [music]
17:07
Now, this is something that is verified
17:09
by what is known as the Stern-Gerlach
17:11
experiment. So, in the Stern-Gerlach
17:13
experiment, we pass a beam of electrons
17:17
through a non-uniform magnetic field.
17:20
And when we do that, because the
17:21
non-uniform magnetic field interacts
17:24
slightly differently with the spin up
17:26
and the spin down, so the beam splits
17:28
into two parts. And this result is an
17:30
actual proof that the electron has an
17:33
intrinsic spin. Now, we can perform a
17:35
similar experiment for the orbital
17:36
angular momentum case. So, for example,
17:38
if we take S orbital, L is equal to
17:41
zero, you'll end up getting a scenario
17:42
in which the beam does not split because
17:44
the S orbital has no angular momentum in
17:46
the first place. But, for L is equal to
17:48
one, there are three distinct
17:49
orientations, so the beam will split
17:51
into three spots. While for L is equal
17:55
to two, there are five distinct
17:56
orientations, so the beam will split
17:58
into five distinct spots. Now, this is
18:00
something that I've only shown for a
18:02
visual understanding perspective because
18:05
the Stern-Gerlach experiment is a little
18:06
bit hard to perform for orbital angular
18:09
momentum because usually in atoms, the
18:11
orbital angular momentum and the spin
18:13
angular momentum couple together to
18:16
create a sort of an effective angular
18:18
momentum of the system.
18:20
Nonetheless, both the spin angular
18:22
momentum and the orbital angular
18:24
momentum in an atom are quantized. They
18:27
can only have very specific magnitudes
18:30
and specific directions, which is
18:32
allowed by the quantum mechanical
18:33
theory. And this is one of the ways in
18:35
which a quantum system is so vastly
18:38
different from our classical
18:40
understanding of angular momentum. I
18:42
have made a lot of effort in creating
18:45
these visualizations to give you a
18:47
better understanding of the topic. If
18:49
this is something that you are
18:50
interested in, then please make a
18:51
comment in the video and I'll try to
18:53
create more such animated and visual
18:56
perspective of understanding various
18:58
topics in physics. I'm Divyendu Das.
19:00
This is For the Love of Physics. Thank
19:01
you so much. Take care. Bye-bye.
