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The angular momentum orbital quantum number l is associated with the orbital angular momentum of the electron in a hydrogen atom. Quantum theory tells us that when the hydrogen atom is in the state ψ n l m , the magnitude of its orbital angular momentum is

L = l ( l + 1 ) ,

where

l = 0 , 1 , 2 , , ( n 1 ) .

This result is slightly different from that found with Bohr’s theory, which quantizes angular momentum according to the rule L = n , where n = 1 , 2 , 3 , ... .

Quantum states with different values of orbital angular momentum are distinguished using spectroscopic notation ( [link] ). The designations s , p , d , and f result from early historical attempts to classify atomic spectral lines. (The letters stand for sharp, principal, diffuse, and fundamental, respectively.) After f , the letters continue alphabetically.

The ground state of hydrogen is designated as the 1 s state, where “1” indicates the energy level ( n = 1 ) and “ s ” indicates the orbital angular momentum state ( l = 0 ). When n = 2 , l can be either 0 or 1. The n = 2 , l = 0 state is designated “2 s .” The n = 2 , l = 1 state is designated “2 p .” When n = 3 , l can be 0, 1, or 2, and the states are 3 s , 3 p , and 3 d , respectively. Notation for other quantum states is given in [link] .

The angular momentum projection quantum number m is associated with the azimuthal angle ϕ (see [link] ) and is related to the z -component of orbital angular momentum of an electron in a hydrogen atom. This component is given by

L z = m ,

where

m = l , l + 1 , , 0 , , + l 1 , l .

The z -component of angular momentum is related to the magnitude of angular momentum by

L z = L c o s θ ,

where θ is the angle between the angular momentum vector and the z -axis. Note that the direction of the z -axis is determined by experiment—that is, along any direction, the experimenter decides to measure the angular momentum. For example, the z -direction might correspond to the direction of an external magnetic field. The relationship between L z and L is given in [link] .

An x y z coordinate system is shown. The vector L is at an angle theta to the positive z axis and has positive z component L sub z equal to m times h bar. The x and y components are positive but not specified.
The z -component of angular momentum is quantized with its own quantum number m .
Spectroscopic notation and orbital angular momentum
Orbital Quantum Number l Angular Momentum State Spectroscopic Name
0 0 s Sharp
1 2 h p Principal
2 6 h d Diffuse
3 12 h f Fundamental
4 20 h g
5 30 h h
Spectroscopic description of quantum states
l = 0 l = 1 l = 2 l = 3 l = 4 l = 5
n = 1 1 s
n = 2 2 s 2 p
n = 3 3 s 3 p 3 d
n = 4 4 s 4 p 4 d 4 f
n = 5 5 s 5 p 5 d 5 f 5 g
n = 6 6 s 6 p 6 d 6 f 6 g 6 h

The quantization of L z is equivalent to the quantization of θ . Substituting l ( l + 1 ) for L and m for L z into this equation, we find

m = l ( l + 1 ) cos θ .

Thus, the angle θ is quantized with the particular values

θ = cos −1 ( m l ( l + 1 ) ) .

Notice that both the polar angle ( θ ) and the projection of the angular momentum vector onto an arbitrary z -axis ( L z ) are quantized.

The quantization of the polar angle for the l = 3 state is shown in [link] . The orbital angular momentum vector lies somewhere on the surface of a cone with an opening angle θ relative to the z -axis (unless m = 0 , in which case θ = 90 ° and the vector points are perpendicular to the z -axis).

Seven vector, all of the same length L, are drawn at 7 different angles to the z axis. The z components of the vectors are indicated both by horizontal lines from the tip of the vector to the z axis and by labels on the z axis. For four of the vectors, the angle between the z axis and the vector is also labeled. The z component values are 3 h bar at angle theta sub three, 2 h bar at angle theta sub two, h bar at angle theta sub one, zero at angle theta sub zero, minus h bar, minus 2 h bar, and minus 3 h bar.
The quantization of orbital angular momentum. Each vector lies on the surface of a cone with axis along the z -axis.

A detailed study of angular momentum reveals that we cannot know all three components simultaneously. In the previous section, the z -component of orbital angular momentum has definite values that depend on the quantum number m . This implies that we cannot know both x- and y -components of angular momentum, L x and L y , with certainty. As a result, the precise direction of the orbital angular momentum vector is unknown.

Practice Key Terms 5

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Source:  OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4
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