# 30.8 Quantum numbers and rules  (Page 4/11)

 Page 4 / 11

We will see that the quantum numbers discussed in this section are valid for a broad range of particles and other systems, such as nuclei. Some quantum numbers, such as intrinsic spin, are related to fundamental classifications of subatomic particles, and they obey laws that will give us further insight into the substructure of matter and its interactions.

## Phet explorations: stern-gerlach experiment

The classic Stern-Gerlach Experiment shows that atoms have a property called spin. Spin is a kind of intrinsic angular momentum, which has no classical counterpart. When the z-component of the spin is measured, one always gets one of two values: spin up or spin down.

## Section summary

• Quantum numbers are used to express the allowed values of quantized entities. The principal quantum number $n$ labels the basic states of a system and is given by
$n=1,\phantom{\rule{0.25em}{0ex}}2,\phantom{\rule{0.25em}{0ex}}3,\text{.}\text{.}\text{.}.$
• The magnitude of angular momentum is given by
$L=\sqrt{l\left(l+1\right)}\frac{h}{2\pi }\phantom{\rule{1.00em}{0ex}}\left(l=0, 1, 2, ...,\phantom{\rule{0.25em}{0ex}}n-1\right),$
where $l$ is the angular momentum quantum number. The direction of angular momentum is quantized, in that its component along an axis defined by a magnetic field, called the $z$ -axis is given by
${L}_{z}={m}_{l}\frac{h}{2\pi }\phantom{\rule{1.00em}{0ex}}\left({m}_{l}=-l,-l+1, ...,\phantom{\rule{0.25em}{0ex}}-1, 0, 1, ...\phantom{\rule{0.25em}{0ex}}l-1,\phantom{\rule{0.25em}{0ex}}l\right)\text{,}$
where ${L}_{z}$ is the $z$ -component of the angular momentum and ${m}_{l}$ is the angular momentum projection quantum number. Similarly, the electron’s intrinsic spin angular momentum $S$ is given by
$S=\sqrt{s\left(s+1\right)}\frac{h}{2\pi }\phantom{\rule{1.00em}{0ex}}\text{(}s=1/2\phantom{\rule{0.25em}{0ex}}\text{for electrons),}$
$s$ is defined to be the spin quantum number. Finally, the direction of the electron’s spin along the $z$ -axis is given by
${S}_{z}={m}_{s}\frac{h}{2\pi }\phantom{\rule{1.00em}{0ex}}\left({m}_{s}=-\frac{1}{2},+\frac{1}{2}\right)\text{,}$
where ${S}_{z}$ is the $z$ -component of spin angular momentum and ${m}_{s}$ is the spin projection quantum number. Spin projection ${m}_{s}\text{=+}1/2$ is referred to as spin up, whereas ${m}_{s}=-1/2$ is called spin down. [link] summarizes the atomic quantum numbers and their allowed values.

## Conceptual questions

Define the quantum numbers $\mathrm{n, l,}\phantom{\rule{0.25em}{0ex}}{m}_{l}\mathrm{, s}$ , and ${m}_{s}$ .

For a given value of $n$ , what are the allowed values of $l$ ?

For a given value of $l$ , what are the allowed values of ${m}_{l}$ ? What are the allowed values of ${m}_{l}$ for a given value of $n$ ? Give an example in each case.

List all the possible values of $s$ and ${m}_{s}$ for an electron. Are there particles for which these values are different? The same?

## Problem exercises

If an atom has an electron in the $n=5$ state with ${m}_{l}=3$ , what are the possible values of $l$ ?

$l=4, 3$ are possible since $l and $\mid {m}_{l}\mid \text{≤}l$ .

An atom has an electron with ${m}_{l}=2$ . What is the smallest value of $n$ for this electron?

What are the possible values of ${m}_{l}$ for an electron in the $n=4$ state?

$n=4⇒l=3, 2, 1, 0⇒{m}_{l}=±3,\phantom{\rule{0.25em}{0ex}}±\text{2,}\phantom{\rule{0.25em}{0ex}}±\text{1, 0}$ are possible.

What, if any, constraints does a value of ${m}_{l}=1$ place on the other quantum numbers for an electron in an atom?

(a) Calculate the magnitude of the angular momentum for an $l=1$ electron. (b) Compare your answer to the value Bohr proposed for the $n=1$ state.

(a) $1\text{.}\text{49}×{\text{10}}^{-\text{34}}\phantom{\rule{0.25em}{0ex}}\text{J}\cdot s$

(b) $1\text{.}\text{06}×{\text{10}}^{-\text{34}}\phantom{\rule{0.25em}{0ex}}\text{J}\cdot s$

(a) What is the magnitude of the angular momentum for an $l=1$ electron? (b) Calculate the magnitude of the electron’s spin angular momentum. (c) What is the ratio of these angular momenta?

Repeat [link] for $l=3$ .

(a) $3\text{.}\text{66}×{\text{10}}^{-\text{34}}\phantom{\rule{0.25em}{0ex}}\text{J}\cdot s$

(b) $s=9\text{.}\text{13}×{\text{10}}^{-\text{35}}\phantom{\rule{0.25em}{0ex}}\text{J}\cdot s$

(c) $\frac{L}{S}=\frac{\sqrt{\text{12}}}{\sqrt{3/4}}=4$

(a) How many angles can $L$ make with the $z$ -axis for an $l=2$ electron? (b) Calculate the value of the smallest angle.

What angles can the spin $S$ of an electron make with the $z$ -axis?

$\theta =\text{54.7º, 125.3º}$

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