8.1 The hydrogen atom  (Page 3/11)

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 ${\psi }_{nlm}$ , the magnitude of its orbital angular momentum is

where

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

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 $\varphi$ (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

where

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

where $\theta$ 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 $L_z\text{and}L$ is given in [link] .

The quantization of $L_z$ is equivalent to the quantization of $\theta$ . Substituting $\sqrt{l(l+1)}\hslash$ for L and m for $L_z$ into this equation, we find

Thus, the angle $\theta$ is quantized with the particular values

Notice that both the polar angle ( $\theta$ ) 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 $\theta$ relative to the z -axis (unless $m=0,$ in which case $\theta =90\text{°}$ and the vector points are perpendicular to 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.