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 on 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 $\vphantom{0}\raisebox{1.5pt}{$-$}$$\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 $\leavevmode \kern.03em\raise.7ex\hbox{\the\scriptfont0 1}\kern-.2em /\kern-.21em\lower.56ex\hbox{\the\scriptfont0 2}\kern.05em$, which al­lows baryons to have spin $\leavevmode \kern.03em\raise.7ex\hbox{\the\scriptfont0 3}\kern-.2em /\kern-.21em\lower.56ex\hbox{\the\scriptfont0 2}\kern.05em$ or $\leavevmode \kern.03em\raise.7ex\hbox{\the\scriptfont0 1}\kern-.2em /\kern-.21em\lower.56ex\hbox{\the\scriptfont0 2}\kern.05em$. (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 ${\left\vert s\:m\right\rangle}$. For ex­am­ple, the spin-up state for an elec­tron is in­di­cated by ${\left\vert\leavevmode \kern.03em\raise.7ex\hbox{\the\scriptfont0 1}\kern-.2em ... ...n-.2em /\kern-.21em\lower.56ex\hbox{\the\scriptfont0 2}\kern.05em\right\rangle}$ and the spin-down state as ${\left\vert\leavevmode \kern.03em\raise.7ex\hbox{\the\scriptfont0 1}\kern-.2em ... ...n-.2em /\kern-.21em\lower.56ex\hbox{\the\scriptfont0 2}\kern.05em\right\rangle}$. More in­for­mally, ${\uparrow}$ and ${\downarrow}$ are of­ten used.

Key Points

$\begin{picture}(15,5.5)(0,-3) \put(2,0){\makebox(0,0){\scriptsize\bf0}} \put(12... ...\thicklines \put(3,0){\line(1,0){12}}\put(11.5,-2){\line(1,0){3}} \end{picture}$

Most par­ti­cles have in­ter­nal an­gu­lar mo­men­tum called spin.

$\begin{picture}(15,5.5)(0,-3) \put(2,0){\makebox(0,0){\scriptsize\bf0}} \put(12... ...\thicklines \put(3,0){\line(1,0){12}}\put(11.5,-2){\line(1,0){3}} \end{picture}$

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

$\begin{picture}(15,5.5)(0,-3) \put(2,0){\makebox(0,0){\scriptsize\bf0}} \put(12... ...\thicklines \put(3,0){\line(1,0){12}}\put(11.5,-2){\line(1,0){3}} \end{picture}$

Elec­trons, pro­tons and neu­trons all have spin $\leavevmode \kern.03em\raise.7ex\hbox{\the\scriptfont0 1}\kern-.2em /\kern-.21em\lower.56ex\hbox{\the\scriptfont0 2}\kern.05em$. Their spin an­gu­lar mo­men­tum in a given di­rec­tion is ei­ther $\frac12\hbar$ or $-\frac12\hbar$.

$\begin{picture}(15,5.5)(0,-3) \put(2,0){\makebox(0,0){\scriptsize\bf0}} \put(12... ...\thicklines \put(3,0){\line(1,0){12}}\put(11.5,-2){\line(1,0){3}} \end{picture}$

Pho­tons have spin one. Pos­si­ble val­ues for their an­gu­lar mo­men­tum in a given di­rec­tion are $\hbar$, zero, or $\vphantom{0}\raisebox{1.5pt}{$-$}$$\hbar$, though zero does not oc­cur in the di­rec­tion of prop­a­ga­tion.

$\begin{picture}(15,5.5)(0,-3) \put(2,0){\makebox(0,0){\scriptsize\bf0}} \put(12... ...\thicklines \put(3,0){\line(1,0){12}}\put(11.5,-2){\line(1,0){3}} \end{picture}$

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

$\begin{picture}(15,5.5)(0,-3) \put(2,0){\makebox(0,0){\scriptsize\bf0}} \put(12... ...\thicklines \put(3,0){\line(1,0){12}}\put(11.5,-2){\line(1,0){3}} \end{picture}$

The spin-up state of a spin one-half par­ti­cle like an elec­tron is usu­ally in­di­cated by ${\left\vert\leavevmode \kern.03em\raise.7ex\hbox{\the\scriptfont0 1}\kern-.2em ... ...n-.2em /\kern-.21em\lower.56ex\hbox{\the\scriptfont0 2}\kern.05em\right\rangle}$ or ${\uparrow}$. Sim­i­larly, the spin-down state is in­di­cated by ${\left\vert\leavevmode \kern.03em\raise.7ex\hbox{\the\scriptfont0 1}\kern-.2em ... ...n-.2em /\kern-.21em\lower.56ex\hbox{\the\scriptfont0 2}\kern.05em\right\rangle}$ or ${\downarrow}$.

5.4 Re­view Ques­tions

1.

Delta par­ti­cles have spin $\leavevmode \kern.03em\raise.7ex\hbox{\the\scriptfont0 3}\kern-.2em /\kern-.21em\lower.56ex\hbox{\the\scriptfont0 2}\kern.05em$. What val­ues can their spin an­gu­lar mo­men­tum in a given di­rec­tion have?

So­lu­tion spin-a

2.

Delta par­ti­cles have spin $\leavevmode \kern.03em\raise.7ex\hbox{\the\scriptfont0 3}\kern-.2em /\kern-.21em\lower.56ex\hbox{\the\scriptfont0 2}\kern.05em$. What is their square spin an­gu­lar mo­men­tum?

So­lu­tion spin-b