Intrinsic Spin Angular Momentum is Quantized in Magnitude and Direction
Intrinsic Spin Angular Momentum Is Quantized in Magnitude and Direction
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
\(S=\sqrt{s(s+1)}\cfrac{h}{2\pi }\phantom{\rule{1.00em}{0ex}}\text{(}s=1/2\phantom{\rule{0.25em}{0ex}}\text{for electrons),}\)
where \(s\) is defined to be the spin quantum number. This is very similar to the quantization of \(L\) given in \(L=\sqrt{l(l+1)}\cfrac{h}{2\pi }\), 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
\({S}_{z}={m}_{s}\cfrac{h}{2\pi }\phantom{\rule{1.00em}{0ex}}({m}_{s}=-\cfrac{1}{2},+\cfrac{1}{2})\)
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 this figure.
Intrinsic Spin
In later tutorials, 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 l\((\mathrm{n, l,}\phantom{\rule{0.25em}{0ex}}{m}_{l},\phantom{\rule{0.25em}{0ex}}{m}_{s})\) —see this table. For electrons in atoms, the principal quantum number can have the values \(n=1, 2, 3, ...\). Once \(n\) is known, the values of the angular momentum quantum number are limited to \(l=1, 2, 3, ...,n-1\). For a given value of \(l\), the angular momentum projection quantum number can have only the values \({m}_{l}=-l,\phantom{\rule{0.25em}{0ex}}-l\phantom{\rule{0.25em}{0ex}}+1, ...,-1, 0, 1, ...,\phantom{\rule{0.25em}{0ex}}l-1,\phantom{\rule{0.25em}{0ex}}l\). Electron spin is independent of l\(\mathrm{n, l,}\) and \({m}_{l}\), always having \(s=1/2\). The spin projection quantum number can have two values, \({m}_{s}=1/2\phantom{\rule{0.25em}{0ex}}\text{or}\phantom{\rule{0.25em}{0ex}}-1/2\).
Atomic Quantum Numbers
| Name | Symbol | Allowed values |
|---|---|---|
| Principal quantum number | \(n\) | \(1, 2, 3, ...\) |
| Angular momentum | \(l\) | \(0, 1, 2, ...n-1\) |
| Angular momentum projection | \({m}_{l}\) | \(-l,\phantom{\rule{0.25em}{0ex}}-l\phantom{\rule{0.25em}{0ex}}+1, ...,\phantom{\rule{0.25em}{0ex}}-1, 0, 1, ...,\phantom{\rule{0.25em}{0ex}}l-1,\phantom{\rule{0.25em}{0ex}}l\phantom{\rule{0.25em}{0ex}}(\text{or}\phantom{\rule{0.25em}{0ex}}0, ±1, ±2, ...,\phantom{\rule{0.25em}{0ex}}±l)\) |
| Spin | \(s\) | \(\text{1/2}(\text{electrons})\) |
| Spin projection | \({m}_{s}\) | \(-1/2,\phantom{\rule{0.25em}{0ex}}+1/2\) |
This figure 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.
Probability clouds for the electron in the ground state and several excited states of hydrogen. The nature of these states is determined by their sets of quantum numbers, here given as \((n,\phantom{\rule{0.25em}{0ex}}l,\phantom{\rule{0.25em}{0ex}}{m}_{l})\). The ground state is (0, 0, 0); one of the possibilities for the second excited state is (3, 2, 1). The probability of finding the electron is indicated by the shade of color; the darker the coloring the greater the chance of finding the electron.
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.
This lesson is part of:
Atomic Physics