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There are two more quantum numbers of immediate concern. Both were first discovered for electrons in conjunction with fine structure in atomic spectra. It is now well established that electrons and other fundamental particles have intrinsic spin , roughly analogous to a planet spinning on its axis. This spin is a fundamental characteristic of particles, and only one magnitude of intrinsic spin is allowed for a given type of particle. Intrinsic angular momentum is quantized independently of orbital angular momentum. Additionally, the direction of the spin is also quantized. It has been found that the magnitude of the intrinsic (internal) spin angular momentum , $S$ , of an electron is given by
where $s$ is defined to be the spin quantum number . This is very similar to the quantization of $L$ given in $L=\sqrt{l\left(l+1\right)}\frac{h}{\mathrm{2\pi}}$ , except that the only value allowed for $s$ for electrons is 1/2.
The direction of intrinsic spin is quantized , just as is the direction of orbital angular momentum. The direction of spin angular momentum along one direction in space, again called the $z$ -axis, can have only the values
for electrons. ${S}_{z}$ is the $z$ -component of spin angular momentum and ${m}_{s}$ is the spin projection quantum number . For electrons, $s$ can only be 1/2, and ${m}_{s}$ can be either +1/2 or –1/2. Spin projection ${m}_{s}\text{=+}1/2$ is referred to as spin up , whereas ${m}_{s}=-1/2$ is called spin down . These are illustrated in [link] .
In later chapters, we will see that intrinsic spin is a characteristic of all subatomic particles. For some particles $s$ is half-integral, whereas for others $s$ is integral—there are crucial differences between half-integral spin particles and integral spin particles. Protons and neutrons, like electrons, have $s=1/2$ , whereas photons have $s=1$ , and other particles called pions have $s=0$ , and so on.
To summarize, the state of a system, such as the precise nature of an electron in an atom, is determined by its particular quantum numbers. These are expressed in the form $\left(\mathrm{n,\; l,}\phantom{\rule{0.25em}{0ex}}{m}_{l},\phantom{\rule{0.25em}{0ex}}{m}_{s}\right)$ —see [link] For electrons in atoms , the principal quantum number can have the values $n=\mathrm{1,\; 2,\; 3,\; ...}$ . Once $n$ is known, the values of the angular momentum quantum number are limited to $l=\mathrm{1,\; 2,\; 3,\; ...},n-1$ . For a given value of $l$ , the angular momentum projection quantum number can have only the values ${m}_{l}=-l,\phantom{\rule{0.25em}{0ex}}-l\phantom{\rule{0.25em}{0ex}}+\mathrm{1,\; ...},-\mathrm{1,\; 0,\; 1,\; ...},\phantom{\rule{0.25em}{0ex}}l-\mathrm{1,}\phantom{\rule{0.25em}{0ex}}l$ . Electron spin is independent of $\mathrm{n,\; l,}$ and ${m}_{l}$ , always having $s=1/2$ . The spin projection quantum number can have two values, ${m}_{s}=1/2\phantom{\rule{0.25em}{0ex}}\text{or}\phantom{\rule{0.25em}{0ex}}-1/2$ .
Name | Symbol | Allowed values |
---|---|---|
Principal quantum number | $$n$$ | $$\mathrm{1,\; 2,\; 3,\; ...}$$ |
Angular momentum | $$l$$ | $$\mathrm{0,\; 1,\; 2,\; ...}n-1$$ |
Angular momentum projection | $${m}_{l}$$ | $$-l,\phantom{\rule{0.25em}{0ex}}-l\phantom{\rule{0.25em}{0ex}}+\mathrm{1,\; ...},\phantom{\rule{0.25em}{0ex}}-\mathrm{1,\; 0,\; 1,\; ...},\phantom{\rule{0.25em}{0ex}}l-\mathrm{1,}\phantom{\rule{0.25em}{0ex}}l\phantom{\rule{0.25em}{0ex}}(\text{or}\phantom{\rule{0.25em}{0ex}}\mathrm{0,\; \pm 1,\; \pm 2,\; ...},\phantom{\rule{0.25em}{0ex}}\pm l)$$ |
Spin The spin quantum number s is usually not stated, since it is always 1/2 for electrons | $$s$$ | $\text{1/2}(\text{electrons})$ |
Spin projection | $${m}_{s}$$ | $-\mathrm{1/2,}\phantom{\rule{0.25em}{0ex}}+\mathrm{1/2}$ |
[link] shows several hydrogen states corresponding to different sets of quantum numbers. Note that these clouds of probability are the locations of electrons as determined by making repeated measurements—each measurement finds the electron in a definite location, with a greater chance of finding the electron in some places rather than others. With repeated measurements, the pattern of probability shown in the figure emerges. The clouds of probability do not look like nor do they correspond to classical orbits. The uncertainty principle actually prevents us and nature from knowing how the electron gets from one place to another, and so an orbit really does not exist as such. Nature on a small scale is again much different from that on the large scale.
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