S-1 Spin ½ Recall that in the H-atom solution, we showed that the fact that the wavefunction Ψ(r) is single-valued requires that the angular momentum quantum nbr be integer: l = 0, 1, 2.. However, operator algebra allowed solutions l = 0, 1/2, 1, 3/2, 2… Experiment shows that the electron possesses an intrinsic angular momentum called spin with l = ½. By convention, we use the letter s instead of l for the spin angular momentum quantum number : s = ½. The existence of spin is not derivable from non-relativistic QM. It is not a form of orbital angular momentum; it cannot be derived from . (The electron is a point particle with radius r = 0.) Lrp=×Electrons, protons, neutrons, and quarks all possess spin s = ½. Electrons and quarks are elementary point particles (as far as we can tell) and have no internal structure. However, protons and neutrons are made of 3 quarks each. The 3 half-spins of the quarks add to produce a total spin of ½ for the composite particle (in a sense, ↑↑↓ makes a single ↑). Photons have spin 1, mesons have spin 0, the delta-particle has spin 3/2. The graviton has spin 2. (Gravitons have not been detected experimentally, so this last statement is a theoretical prediction.) Spin and Magnetic Moment We can detect and measure spin experimentally because the spin of a charged particle is always associated with a magnetic moment. Classically, a magnetic moment is defined as a vector µ associated with a loop of current. The direction of µ is perpendicular to the plane of the current loop (right-hand-rule), and the magnitude is 2iA i rµ==π. The connection between orbital angular momentum (not spin) and magnetic moment can be seen in the following classical model: Consider a particle with mass m, charge q in circular orbit of radius r, speed v, period T. µ ()2q2r qv qvi,v i iA rTT 2r 2r⎛⎞π==⇒=µ== π=⎜⎟ππ⎝⎠qvr2 r i i r m, q 4/26/2008 Dubson, Phys3220S-2 | angular momentum | = L = p r = m v r , so v r = L/m , and qvr qL22mµ= =. So for a classical system, the magnetic moment is proportional to the orbital angular momentum: qL (orbital)2mµ=. The same relation holds in a quantum system. In a magnetic field B, the energy of a magnetic moment is given by (assuming zEB=−µ⋅ = −µBˆBBz=). In QM, zLm=. Writing electron mass as me (to avoid confusion with the magnetic quantum number m) and q = –e we have zeem2mµ=−, where m = − l .. +l. The quantity Bee2mµ≡is called the Bohr magneton. The possible energies of the magnetic moment in ˆBBz= is given by orb z BEB=− Bmµ=−µ. For spin angular momentum, it is found experimentally that the associated magnetic moment is twice as big as for the orbital case: qS(spin)mµ= (We use S instead of L when referring to spin angular momentum.) This can be written zeem2mµ=− =−µBmm. The energy of a spin in a field is spin BE2B=−µ (m = ±1/2) a fact which has been verified experimentally. The existence of spin (s = ½) and the strange factor of 2 in the gyromagnetic ratio (ratio of to Sµ) was first deduced from spectrographic evidence by Goudsmit and Uhlenbeck in 1925. Another, even more direct way to experimentally determine spin is with a Stern-Gerlach device, next page 4/26/2008 Dubson, Phys3220S-3 (This page from QM notes of Prof. Roger Tobin, Physics Dept, Tufts U.) Stern-Gerlach Experiment (W. Gerlach & O. Stern, Z. Physik 9, 349-252 (1922). y x z B ()FBB=−∇ µ =−µ∇ ii ⎟⎠⎞⎜⎝⎛∂∂=zBzFzzµˆ Deflection of atoms in z-direction is proportional to z-component of magnetic moment µz, which in turn is proportional to Lz. The fact that there are two beams is proof that l = s = ½. The two beams correspond to m = +1/2 and m = –1/2. If l = 1, then there would be three beams, corresponding to m = –1, 0, 1. The separation of the beams is a direct measure of µz, which provides proof that zB2mµ=−µ The extra factor of 2 in the expression for the magnetic moment of the electron is often called the "g-factor" and the magnetic moment is often written as zBgmµ=−µ. As mentioned before, this cannot be deduced from non-relativistic QM; it is known from experiment and is inserted "by hand" into the theory. However, a relativistic version of QM due to Dirac (1928, the "Dirac Equation") predicts the existence of spin (s = ½) and furthermore the theory predicts the value g = 2. A later, better version of relativistic QM, called Quantum Electrodynamics (QED) predicts that g is a little larger than 2. The g-4/26/2008 Dubson, Phys3220S-4 factor has been carefully measured with fantastic precision and the latest experiments give g = 2.0023193043718(±76 in the last two places). Computing g in QED requires computation of a infinite series of terms that involve progressively more messy integrals, that can only be solved with approximate numerical methods. The computed value of g is not known quite as precisely as experiment, nevertheless the agreement is good to about 12 places. QED is one of our most well-verified theories. Spin Math Recall that the angular momentum commutation relations 2zijk[L ,L ] 0 , [L , L ] i L (i j k cyclic)== were derived from the definition of the orbital angular momentum operator: Lr. p=×The spin operator S does not exist in Euclidean space (it doesn't have a position or momentum vector associated with it), so we cannot derive its commutation relations in a similar way. Instead we boldly postulate that the same commutation relations hold for spin angular momentum: 2zij[S ,S ] 0 , [S , S ] i S= k=. From these, we derive, just a before, that 22 2ss3S sm s(s 1) sm sm4=+ =s ( since s = ½ ) zs ss s1Ssm msm sm2==± ( since ms = −s ,+s = −1/2, +1/2 ) Notation: since s = ½ always, we can drop this quantum number, and specify the eigenstates of L2 , Lz by giving only the ms quantum number. There are various ways to write this: 1122ss,sm m ,,+−==+−↑↓ These states exist in a 2D subset of the full Hilbert Space called spin space. Since these two states are eigenstates of a hermitian operator, they form a complete orthonormal set 4/26/2008 Dubson, Phys3220S-5 (within their part of Hilbert space) and any, arbitrary state in spin space can always be written as aabb⎛⎞χ= ↑+ ↓ =⎜⎟⎝⎠ (Griffiths' notation is ab+−χ=χ+χ)