S 1 Spin Recall that in the H atom solution we showed that the fact that the wavefunction 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 By convention we use the letter s instead of l for the spin angular momentum quantum number s The existence of spin is not derivable from nonrelativistic QM It is not a form of orbital angular momentum it cannot be derived from r r r L r p The electron is a point particle with radius r 0 Electrons protons neutrons and quarks all possess spin s 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 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 associated with a loop of r i current The direction of is perpendicular to the plane of the current loop right hand rule and the magnitude is m i A i p r 2 The connection between orbital angular momentum not spin and magnetic moment can be seen in the following classical model Consider a particle with mass i m q m charge q in circular orbit of radius r speed v period T i q T v 2 pr T 1 14 2019 i qv 2 pr q v 2 m i A pr 2 p r q vr 2 Dubson Phys3220 r S 2 angular momentum L p r m v r so v r L m and m q vr q L 2 2m So for a classical system the magnetic moment is proportional to the orbital angular momentum q r r m L 2m orbital The same relation holds in a quantum system r r B mz B In a magnetic field B the energy of a magnetic moment is given by E m r assuming B B z In QM L z hm Writing electron mass as me to avoid confusion with the magnetic quantum number m and q e we have mz eh m where m 2 me eh l l The quantity mB is called the Bohr magneton The possible energies of 2 me r the magnetic moment in B B z is given by E orb mz B mB B m For spin angular momentum it is found experimentally that the associated magnetic moment is twice as big as for the orbital case q r r m S m spin We use S instead of L when referring to spin angular momentum This can be written mz eh m 2 mB m The energy of a spin in a field is E spin 2mB B m m me 1 2 a fact which has been verified experimentally The existence of spin s and r r the strange factor of 2 in the gyromagnetic ratio ratio of mto S 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 1 14 2019 Dubson Phys3220 S 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 r r r r r rr F mgB m gB B z F z z z Deflection of atoms in z direction is proportional to z component of magnetic moment z which in turn is proportional to Lz The fact that there are two beams is proof that l s 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 z which provides proof that mz 2 mB m 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 mz g mB m 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 and furthermore the theory predicts the value g 2 A later better version of relativistic QM 1 14 2019 Dubson Phys3220 S 4 called Quantum Electrodynamics QED predicts that g is a little larger than 2 The gfactor 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 L2 L z 0 Li L j i hL k i j k cyclic r r r were derived from the definition of the orbital angular momentum operator L r p r 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 S2 Sz 0 Si S j i hSk From these we derive just a before that S2 s ms h2 s s 1 s ms Sz s ms hms s ms 3 2 h s ms 4 1 h s ms 2 since s since ms s s 1 2 1 2 Notation since s 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 s ms ms 12 1 2 1 14 2019 Dubson Phys3220 S 5 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 within their part of Hilbert space and any arbitrary state in spin space can always be a written as c a b Griffiths notation is c a c b c b 1 Matrix notation 0 0 Note that 1 …