5.4 Spin

At this stage, it becomes necessary to look somewhat closer at the various particles involved in quantum mechanics themselves. The analysis so far already used the fact that particles have a property called mass, a quantity that special relativity has identified as being an inter­nal amount of energy. It turns out that in addition particles have a fixed amount of “build-in” angular momentum, called “spin.” Spin reflects itself, for example, in how a charged particle such as an electron inter­acts with a magnetic field.

To keep it apart from spin, from now the angular momentum of a particle due to its motion will on be referred to as “orbital” angular momentum. As was discussed in chapter 4.2, the square orbital angular momentum of a particle is given by

$$L^2 = l(l+1)\hbar^2$$

where the azimuthal quantum number $l$ is a non­negative integer.

The square spin angular momentum of a particle is given by a similar expression:

$$S^2 = s(s+1)\hbar^2$$ (5.14)

but the “spin $s$” is a fixed number for a given type of particle. And while $l$ can only be an integer, the spin $s$ can be any multiple of one half.

Particles with half integer spin are called “fermions.” For example, electrons, protons, and neutrons all three have spin $s$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\frac12$ and are fermions.

Particles with integer spin are called “bosons.” For example, photons have spin $s$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1. The $\pi$-mesons have spin $s$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0 and gravitons, unobserved at the time of writing, should have spin $s$ $\vphantom0\raisebox{1.5pt}{$=$}$ 2.

The spin angular momentum in an arbitrarily chosen $z$-direction is

$$S_z = m\hbar$$ (5.15)

the same formula as for orbital angular momentum, and the values of $m$ range again from $-s$ to $+s$ in integer steps. For example, photons can have spin in a given direction that is $\hbar$, 0, or $\vphantom0\raisebox{1.5pt}{$-$}$$\hbar$. (The photon, a relativistic particle with zero rest mass, has only two spin states along the direction of propag­ation; the zero value does not occur in this case. But photons radiated by atoms can still come off with zero angular momentum in a direction normal to the direction of propag­ation. A derivation is in addendum {A.21.6} and {A.21.7}.)

The common particles, (electrons, protons, neutrons), can only have spin angular momentum $\frac12\hbar$ or $-\frac12\hbar$ in any given direction. The positive sign state is called “spin up”, the negative one “spin down”.

It may be noted that the proton and neutron are not elementary particles, but are baryons, consisting of three quarks. Similarly, mesons consist of a quark and an anti-quark. Quarks have spin $\leavevmode\kern.03em \raise.7ex\hbox{\the\scriptfont0 1}\kern-.2em /\kern-.2em\lower.4ex\hbox{\the\scriptfont0 2}\kern.05em$, which allows baryons to have spin $\leavevmode\kern.03em \raise.7ex\hbox{\the\scriptfont0 3}\kern-.2em /\kern-.2em\lower.4ex\hbox{\the\scriptfont0 2}\kern.05em$ or $\leavevmode\kern.03em \raise.7ex\hbox{\the\scriptfont0 1}\kern-.2em /\kern-.2em\lower.4ex\hbox{\the\scriptfont0 2}\kern.05em$. (It is not self-evident, but spin values can be additive or subtractive within the confines of their discrete allowable values; see chapter 12.) The same way, mesons can have spin 1 or 0.

Spin states are commonly shown in “ket notation” as $\big\vert s\:m\big\rangle$. For example, the spin-up state for an electron is indicated by $\big\vert\leavevmode\kern.03em \raise.7ex\hbox{\the\scriptfont0 1}\kern-.2em /\... ...\kern-.2em /\kern-.2em\lower.4ex\hbox{\the\scriptfont0 2}\kern.05em\big\rangle$ and the spin-down state as $\big\vert\leavevmode\kern.03em \raise.7ex\hbox{\the\scriptfont0 1}\kern-.2em /\... ...\kern-.2em /\kern-.2em\lower.4ex\hbox{\the\scriptfont0 2}\kern.05em\big\rangle$. More informally, ${\uparrow}$ and ${\downarrow}$ are often used.

Key Points

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Most particles have inter­nal angular momentum called spin.

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The square spin angular momentum and its quantum number $s$ are always the same for a given particle.

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Electrons, protons and neutrons all have spin $\leavevmode\kern.03em \raise.7ex\hbox{\the\scriptfont0 1}\kern-.2em /\kern-.2em\lower.4ex\hbox{\the\scriptfont0 2}\kern.05em$. Their spin angular momentum in a given direction is either $\frac12\hbar$ or $-\frac12\hbar$.

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Photons have spin one. Possible values for their angular momentum in a given direction are $\hbar$, zero, or $\vphantom0\raisebox{1.5pt}{$-$}$$\hbar$, though zero does not occur in the direction of propag­ation.

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Particles with integer spin, like photons, are called bosons. Particles with half-integer spin, like electrons, protons, and neutrons, are called fermions.

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The spin-up state of a spin one-half particle like an electron is usually indicated by $\big\vert\leavevmode\kern.03em \raise.7ex\hbox{\the\scriptfont0 1}\kern-.2em /\... ...\kern-.2em /\kern-.2em\lower.4ex\hbox{\the\scriptfont0 2}\kern.05em\big\rangle$ or ${\uparrow}$. Similarly, the spin-down state is indicated by $\big\vert\leavevmode\kern.03em \raise.7ex\hbox{\the\scriptfont0 1}\kern-.2em /\... ...\kern-.2em /\kern-.2em\lower.4ex\hbox{\the\scriptfont0 2}\kern.05em\big\rangle$ or ${\downarrow}$.

5.4 Review Questions

1. Delta particles have spin $\leavevmode\kern.03em \raise.7ex\hbox{\the\scriptfont0 3}\kern-.2em /\kern-.2em\lower.4ex\hbox{\the\scriptfont0 2}\kern.05em$. What values can their spin angular momentum in a given direction have?

   Solution spin-a

2. Delta particles have spin $\leavevmode\kern.03em \raise.7ex\hbox{\the\scriptfont0 3}\kern-.2em /\kern-.2em\lower.4ex\hbox{\the\scriptfont0 2}\kern.05em$. What is their square spin angular momentum?

   Solution spin-b