7.3.13. Angular Momentum

Table of Contents

Quantized Angular Momentum

In classical mechanics, angular momentum describes rotational motion. In quantum mechanics, angular momentum still plays that role, but it becomes quantized and is described by operators rather than ordinary numbers. This chapter focuses on the specifically quantum ideas of angular momentum, not on classical torque or rotational dynamics.

A quantum particle can have angular momentum even when it is not literally a tiny ball spinning in space. In quantum theory, angular momentum is a fundamental observable with special mathematical rules and measurable consequences.

Orbital Angular Momentum Operator

For a particle moving in space, the orbital angular momentum operator is defined from position and momentum, just as in classical physics:

$$
\mathbf{L} = \mathbf{r} \times \mathbf{p}
$$

In Cartesian coordinates, its components are

$$
L_x = yp_z - zp_y
$$

$$
L_y = zp_x - xp_z
$$

$$
L_z = xp_y - yp_x
$$

Since in quantum mechanics momentum is an operator,

$$
\mathbf{p} = -i\hbar \nabla
$$

the angular momentum components are also operators.

The orbital angular momentum operator is
$$
\mathbf{L} = \mathbf{r} \times \mathbf{p}
$$
with momentum operator
$$
\mathbf{p} = -i\hbar \nabla
$$
so angular momentum in quantum mechanics is an operator, not an ordinary vector.

Angular Momentum Is Not Fully Simultaneously Measurable

A major difference from classical physics is that the three components of angular momentum cannot all be known exactly at the same time. This happens because the components do not commute.

Their commutation relations are

$$
[L_x, L_y] = i\hbar L_z
$$

$$
[L_y, L_z] = i\hbar L_x
$$

$$
[L_z, L_x] = i\hbar L_y
$$

These relations mean that measuring one component precisely prevents exact knowledge of the other two components.

However, the total angular momentum squared,

$$
L^2 = L_x^2 + L_y^2 + L_z^2
$$

does commute with each component in the sense used for choosing compatible observables, especially with $L_z$:

$$
[L^2, L_z] = 0
$$

So a quantum state can have definite values of $L^2$ and one component, usually chosen to be $L_z$.

You can know exactly at the same time:
$$
L^2 \text{ and } L_z
$$
but not all three of
$$
L_x,\; L_y,\; L_z
$$
because the components satisfy nonzero commutation relations.

Quantum Numbers for Angular Momentum

The simultaneous eigenstates of $L^2$ and $L_z$ are labeled by two quantum numbers, $l$ and $m$.

They satisfy

$$
L^2 |l,m\rangle = \hbar^2 l(l+1)|l,m\rangle
$$

$$
L_z |l,m\rangle = \hbar m |l,m\rangle
$$

Here, $l$ is the angular momentum quantum number and $m$ is the magnetic quantum number.

For orbital angular momentum, the allowed values are

$$
l = 0, 1, 2, 3, \dots
$$

and for each fixed $l$,

$$
m = -l, -l+1, \dots, l-1, l
$$

So for a given $l$, there are $2l+1$ allowed values of $m$.

Meaning of the Quantum Numbers

The number $l$ determines the magnitude of angular momentum. The number $m$ determines the component along the chosen axis, usually the $z$ axis.

The magnitude of angular momentum is

$$
|\mathbf{L}| = \sqrt{l(l+1)}\,\hbar
$$

and the $z$ component is

$$
L_z = m\hbar
$$

This means the direction is not completely fixed. Only one component is sharp in a state like $|l,m\rangle$.

This is often pictured as space quantization. The angular momentum vector cannot point in just any direction if its magnitude and one component are fixed.

Rendering scientific sketch…

Space quantization of angular momentum

Allowed Values Table

The pattern of allowed values becomes clearer in a table.

$l$	Allowed $m$ values	Number of states $2l+1$
0	0	1
1	$-1, 0, 1$	3
2	$-2, -1, 0, 1, 2$	5
3	$-3, -2, -1, 0, 1, 2, 3$	7

For orbital angular momentum,
$$
l = 0,1,2,\dots
$$
and
$$
m = -l,-l+1,\dots,l
$$
The total number of possible $m$ values for a given $l$ is
$$
2l+1
$$

Angular Momentum in Wave Functions

In position space, orbital angular momentum is closely connected with the angular dependence of the wave function. For central potentials, such as the hydrogen atom, solutions naturally separate into radial and angular parts. The angular part is described by spherical harmonics, written as

$$
Y_l^m(\theta,\phi)
$$

These functions are eigenfunctions of both $L^2$ and $L_z$:

$$
L^2 Y_l^m = \hbar^2 l(l+1) Y_l^m
$$

$$
L_z Y_l^m = \hbar m Y_l^m
$$

So the quantum numbers $l$ and $m$ appear directly in the spatial form of the wave function.

Ladder Operators

Angular momentum has a very useful pair of operators called raising and lowering operators:

$$
L_\pm = L_x \pm iL_y
$$

These operators change the value of $m$ without changing $l$:

$$
L_\pm |l,m\rangle \propto |l,m\pm1\rangle
$$

More precisely,

$$
L_\pm |l,m\rangle = \hbar \sqrt{l(l+1)-m(m\pm1)}\,|l,m\pm1\rangle
$$

This shows that starting from one state, we can move step by step through the allowed $m$ values. The process must stop at the top and bottom values, which is why $m$ only runs from $-l$ to $l$.

The ladder operators
$$
L_\pm = L_x \pm iL_y
$$
change only the magnetic quantum number:
$$
m \to m \pm 1
$$
They do not change $l$.

Why the Magnitude Is $\sqrt{l(l+1)}\hbar$

A beginner might expect the magnitude to be simply $l\hbar$, but quantum mechanics gives

$$
|\mathbf{L}| = \sqrt{l(l+1)}\hbar
$$

This result comes from the operator structure and commutation relations. It is one of the clear signs that quantum angular momentum is not just a classical vector with restricted values. The extra $+1$ is a purely quantum feature.

Orbital Angular Momentum and Atomic States

Orbital angular momentum is especially important in atoms. In atomic notation, different values of $l$ are given letter names.

$l$	Letter
0	s
1	p
2	d
3	f

These labels are widely used in atomic physics. For example, an electron in a p state has $l=1$, and so it has three possible $m$ values.

Distinguishing Orbital and Other Angular Momenta

This chapter is about quantum angular momentum in general, with orbital angular momentum as the main example. In quantum mechanics there are also other forms of angular momentum, especially spin, which is treated separately. Spin obeys similar mathematical rules, but it is not produced by spatial motion in the same way as orbital angular momentum.

So it is important not to assume that all angular momentum comes from a particle moving around in space.

Visual Pattern of States

For each value of $l$, there is a set of allowed $m$ values arranged symmetrically around zero.

Rendering scientific sketch…

Allowed magnetic quantum numbers for several l values

Summary

Quantum angular momentum is described by operators with special commutation relations. For orbital angular momentum, the key observables are $L^2$ and one component such as $L_z$. Their eigenvalues are labeled by the quantum numbers $l$ and $m$:

$$
L^2 = \hbar^2 l(l+1), \qquad L_z = m\hbar
$$

For orbital angular momentum, $l$ is a nonnegative integer and $m$ runs from $-l$ to $l$ in integer steps. This gives $2l+1$ allowed states for each $l$. These rules are central to the quantum description of atoms and many other microscopic systems.

The most important results are
$$
L^2 |l,m\rangle = \hbar^2 l(l+1)|l,m\rangle
$$
$$
L_z |l,m\rangle = \hbar m |l,m\rangle
$$
with
$$
l = 0,1,2,\dots
$$
and
$$
m = -l,-l+1,\dots,l
$$
These are the basic quantization rules for orbital angular momentum.

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