p. (The electron is a point particle with radius r = 0.)

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Suzan Booth
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1 - pin ½ Recall that in the H-atom olution, we howed that the fact that the wavefunction Ψ(r) i ingle-valued require that the angular momentum quantum nbr be integer: l = 0,,.. However, operator algebra allowed olution l = 0, /,, 3/, Eperiment how that the electron poee an intrinic angular momentum called pin with l = ½. By convention, we ue the letter intead of l for the pin angular momentum quantum number : = ½. The eitence of pin i not derivable from nonrelativitic QM. It i not a form of orbital angular momentum; it cannot be derived from L= r p. (The electron i a point particle with radiu r = 0.) Electron, proton, neutron, and quark all poe pin = ½. Electron and quark are elementary point particle (a far a we can tell) and have no internal tructure. However, proton and neutron are made of 3 quark each. The 3 half-pin of the quark add to produce a total pin of ½ for the compoite particle (in a ene, make a ingle ). Photon have pin, meon have pin 0, the delta-particle ha pin 3/. The graviton ha pin. (Graviton have not been detected eperimentally, o thi lat tatement i a theoretical prediction.) pin and Magnetic Moment We can detect and meaure pin eperimentally becaue the pin of a charged particle i alway aociated with a magnetic moment. Claically, µ a magnetic moment i defined a a vector µ aociated with a loop of current. The direction of µ i perpendicular to the plane of the current loop i r (right-hand-rule), and the magnitude i µ = ia = iπ r. The connection between orbital angular momentum (not pin) and magnetic moment can be een in the following claical model: Conider a particle with ma m, q i r m, charge q in circular orbit of radiu r, peed v, period T. q π r qv qv i =, v = i = µ = ia = ( π r ) = T T πr π r qvr 4/6/008 Dubon, Phy30

2 - angular momentum = L = p r = m v r, o v r = L/m, and qvr q µ= = L. m o for a claical ytem, the magnetic moment i proportional to the orbital angular momentum: ytem. µ = q L (orbital) m. The ame relation hold in a quantum In a magnetic field B, the energy of a magnetic moment i given by E = µ B = µ B (auming B = Bˆ ). In QM, L = m. Writing electron ma a m e (to avoid confuion with the magnetic quantum number m) and q = e we have e e µ = m, where m = l.. +l. The quantity µ B i called the Bohr me me magneton. The poible energie of the magnetic moment in B = Bˆ i given by E = µ B = µ Bm. orb B For pin angular momentum, it i found eperimentally that the aociated magnetic moment i twice a big a for the orbital cae: µ = q m (pin) (We ue intead of L when referring to pin angular momentum.) Thi can be written e µ = m = µ m e B m. The energy of a pin in a field i E m pin = µ B B (m = ±/) a fact which ha been verified eperimentally. The eitence of pin ( = ½) and the trange factor of in the gyromagnetic ratio (ratio of µ to ) wa firt deduced from pectrographic evidence by Goudmit and Uhlenbeck in 95. Another, even more direct way to eperimentally determine pin i with a tern-gerlach device, net page 4/6/008 Dubon, Phy30

3 -3 (Thi page from QM note of Prof. Roger Tobin, Phyic Dept, Tuft U.) tern-gerlach Eperiment (W. Gerlach & O. tern, Z. Phyik 9, (9). B B F= ( µ ib) = µ i B F = ˆ µ y Deflection of atom in -direction i proportional to -component of magnetic moment µ, which in turn i proportional to L. The fact that there are two beam i proof that l = = ½. The two beam correpond to m = +/ and m = /. If l =, then there would be three beam, correponding to m =, 0,. The eparation of the beam i a direct meaure of µ, which provide proof that µ = µ Bm The etra factor of in the epreion for the magnetic moment of the electron i often called the "g-factor" and the magnetic moment i often written a µ = gµ Bm. A mentioned before, thi cannot be deduced from non-relativitic QM; it i known from eperiment and i inerted "by hand" into the theory. However, a relativitic verion of QM due to Dirac (98, the "Dirac Equation") predict the eitence of pin ( = ½) and furthermore the theory predict the value g =. A later, better verion of relativitic QM, called Quantum Electrodynamic (QED) predict that g i a little larger than. The g- 4/6/008 Dubon, Phy30

4 -4 factor ha been carefully meaured with fantatic preciion and the latet eperiment give g = (±76 in the lat two place). Computing g in QED require computation of a infinite erie of term that involve progreively more mey integral, that can only be olved with approimate numerical method. The computed value of g i not known quite a preciely a eperiment, neverthele the agreement i good to about place. QED i one of our mot well-verified theorie. pin Math Recall that the angular momentum commutation relation [L,L ] = 0, [L i, L j] = i L k (i j k cyclic) were derived from the definition of the orbital angular momentum operator: L = r p. The pin operator doe not eit in Euclidean pace (it doen't have a poition or momentum vector aociated with it), o we cannot derive it commutation relation in a imilar way. Intead we boldly potulate that the ame commutation relation hold for pin angular momentum: [, ] 0, [ i, j] i = = k. From thee, we derive, jut a before, that 3 m = (+ ) m = m ( ince = ½ ) 4 m = m m = ± m ( ince m =,+ = /, +/ ) Notation: ince = ½ alway, we can drop thi quantum number, and pecify the eigentate of L, L by giving only the m quantum number. There are variou way to write thi: +, m = m = +,, Thee tate eit in a D ubet of the full Hilbert pace called pin pace. ince thee two tate are eigentate of a hermitian operator, they form a complete orthonormal et 4/6/008 Dubon, Phy30

5 -5 (within their part of Hilbert pace) and any, arbitrary tate in pin pace can alway be written a a χ = a + b = b (Griffith' notation i χ = aχ + + bχ ) Matri notation: 0 =, = 0. Note that = =, = 0 If we were working in the full Hilbert pace of, ay, the H-atom problem, then our bai tate would be n m m. pin i another degree of freedom, o that the full pecification of a bai tate require 4 quantum number. (More on the connection between pin and pace part of the tate later.) [Note on language: throughout thi ection I will ue the ymbol (and, etc) to refer to both the obervable ("the meaured value of i + /") and it aociated operator ("the eigenvalue of i + /").] The matri form of and in the () m bai can be worked out element by element. (Recall that for any operator A, ˆ A = m Aˆ n.) mn 3 = = 4 = +, = 0, etc., 0, etc = = Operator equation can be written in matri form, for intance, 0 = + = We are going ak what happen when we make meaurement of, a well a and y, (uing a tern-gerlach apparatu). Will need to know: What are the matrice for the operator and y? Thee are derived from the raiing and lowering operator: 4/6/008 Dubon, Phy30

6 -6 ( ) ( ) = + i = + = i = + y + y y i + To get the matri form of +,, we need a reult from the homework:,m = (+ ) m(m+ ), m + +,m = (+ ) m(m ), m For the cae = ½, the quare root factor are alway or 0. For intance, = ½, m = / give 3 ( ) ( )( ) (+ ) m(m+ ) = =. Conequently, =, = 0 and =, = 0, leading to = 0, + =, etc. and = = Notice that +, are not hermitian. + Uing = ( + ) and = ( ) yield + y i i = y = 0 i 0 Thee are hermitian, of coure. 0 0 i 0 Often written: = σ, where σ =, σ y =, σ = are 0 i 0 0 called the Pauli pin matrice. Now let' make ome meaurement on the tate a χ = a + b = b. Normaliation: χχ = a + b =. uppoe we meaure on a ytem in ome tate a χ =. Potulate ay that the b poible reult of thi meaurement are one of the eigenvalue: + / or /. 4/6/008 Dubon, Phy30

7 -7 Potulate 3 ay the probability of finding, ay /, i a Prob(find /) = χ = ( 0 ) = b. Potulate 4 ay that, a a reult b of thi meaurement, which found /, the initial tate χ collape to. But uppoe we meaure? (Which we can do by rotating the G apparatu.) What will we find? Anwer: one of the eigenvalue of, which we how below are the ame a the eigenvalue of : + / or /. (Not urpriing, ince there i nothing pecial about the -ai.) What i the probability that we find, ay, = + /? To anwer thi we need to know the eigentate of the operator. Let' call thee (o far unknown) eigentate and (Griffith call them χ and χ + ). How do we find thee? We mut olve the eigenvalue equation: χ = λ χ, where λ are the unknown eigenvalue. In matri form thi i, 0 / a a λ / a = λ which can be rewritten / 0 b b / λ b algebra, thi lat equation i called the characteritic equation. = 0. In linear Thi ytem of linear equation only ha a olution if λ / λ / Det = = 0 / λ / λ. o ( ) λ / = 0 λ = ± / A epected, the eigenvalue of are the ame a thoe of (or y ). Now we can plug in each eigenvalue and olve for the eigentate: 0 a a = 0 b b a = b ; 0 a a = a = b. 0 b b o we have and = = 4/6/008 Dubon, Phy30

8 Now back to our quetion: uppoe the ytem in the tate =, and we 0 () meaure. What i the probability that we find, ay, = + /? Potulate 3 give the recipe for the anwer: () ( ) Prob(find =+ /) = = = = / 0-8 Quetion for the tudent: uppoe the initial tate i an arbitrary tate a χ = b meaure. What are the probabilitie that we find = + / and /? and we Let' review the trangene of Quantum Mechanic. uppoe an electron i in the = + / eigentate =. If we ak: What i the value of? Then there i a definite anwer: + /. But if we ak: What i the value of, then thi i no anwer. The ytem doe not poe a value of. If we meaure, then the act of meaurement will produce a definite reult and will force the tate of the ytem to collape into an eigentate of, but that very act of meaurement will detroy the definitene of the value of. The ytem can be in an eigentate of either or, but not both. 4/6/008 Dubon, Phy30

Spin ½ (Pages 1-12 are needed)

Spin ½ (Pages 1-12 are needed) Prof. Dr. I. Nasser Phs- 55 (T-) October 3, 03 Spin3.doc Spin ½ (Pages - are needed) Recall that in the H-atom solution, we showed that the fact that the wavefunction ψ (r) is singlevalued requires that