Spn Mchael Fowler /6/6 Introducton The Stern Gerlach experment for the smplest possble atom, hydrogen n ts ground state, demonstrated unambguously that the component of the magnetc moment of the atom along the z-axs could only have two values It had been well establshed by ths tme that the magnetc moment vector was along the same axs as the angular momentum Ths s obvously true for the Bohr model of hydrogen, where the crculatng electron s equvalent to a rng current, generatng a magnetc dpole The problem s, though, that a magnetc moment generated n ths way by orbtal angular momentum wll have a mnmum of three possble values of ts z-component: the lowest nonzero orbtal angular momentum s l =, wth allowed values of the z-component m, m=,, Recall, however, that n our dervaton of allowed angular momentum egenvalues from very general propertes of rotaton operators, we found that although for any system the allowed values of m form a ladder wth spacng, we could not rule out half-ntegral m values The lowest such case, l = /, would n fact have just two allowed m values: m = /, / However, ths cannot be any knd of orbtal angular momentum because the z-component of the orbtal wave functon ψ has a factor e ± ϕ, and therefore pcks up a factor - on rotatng through π, meanng ψ s not sngle-valued, whch doesn t make sense for a Schrödnger wave functon Yet the expermental result s clear Therefore, ths must be a new knd of non-orbtal angular momentum It s called spn, the smple pcture beng that just as the Earth has orbtal angular momentum n ts yearly crcle around the sun, and also spn angular momentum from ts daly turnng, the electron has an analogous spn But the analogy has obvous lmtatons: the Earth s spn s after all made up of materal orbtng around the axs through the poles, the electron s spn cannot smlarly be magned as arsng from a rotatng body, snce orbtal angular momenta always come n ntegral multples of Fortunately, ths lack of a smple quas-mechancal pcture underlyng electron spn doesn t prevent us from usng the general angular momentum machnery prevously developed, whch followed just from analyzng the effect of spatal rotaton on a quantum mechancal system Recall ths led to the spacng of the ladder of egenvalues, and to values of the matrx elements of angular momentum components J between the egenkets j, m : enough nformaton to construct matrx representatons of the rotaton operators for a system of gven angular momentum As an example, for the orbtal angular momentum j = l = state, we constructed the 3 3 matrx representaton of an arbtrary rotaton operator e n the space wth orthonormal bass,,,,, n the lm, notaton) The spn j = s = / case can be handled n exactly the same way θ J
Spnors, Spn Operators, Paul Matrces The Hlbert space of angular momentum states for spn one-half s two dmensonal Varous notatons are used: j, m becomes s, m or s, m s, or even, more graphcally,,,, Any state of the spn can be wrtten and ths two-dmensonal ket s called a spnor α + β wth α + β = β Operators on spnors are necessarly matrces We shall follow the usual practce of denotng the angular momentum components J by S for spns From our defnton of the spnor, Sz = σz, wth σz = The general formulas for rasng and lowerng operators ) ) ) ) J j, m = j j+ m m+ j, m+, J j, m = j j+ m m j, m + become for j = m= smply, S, =,, S, =, + so Sx + Sy = S+ =, Sx Sy = S = It follows mmedately that an approprate matrx representaton for spn one-half s S σ, where σ,, = = These three matrces representng the x, y, z) spn components are called the Paul spn matrces They are hermtan, traceless, and obey
j j j k 3 ) σ = I, σσ = σ σ, and σσ = σ for, j, k a cyclc permutaton of,,3 ) Ths can be wrtten σ σ = ε σ j jk k The total spn S σ 3 = 4 = 4 Any matrx can be wrtten n the form α + I ασ Exercse: prove the above statements, then use your results to show that a) n ˆ σ ) = I for any unt vector n ˆ b) σ A) σ B) = A B) I + σ A B) Relatng the Spnor to the Spn Drecton But how do α, β n α + β relate to whch way the spn s pontng? To fnd out, let s assume that t s pontng up along the unt vector n ˆ = sn θ cos ϕ, sn θ sn ϕ, cos θ), that s, n the drecton θ, ϕ ) In other words, t s n the egenstate of the operator ˆn σ havng egenvalue unty: nz nx ny nx ny n = + z β β Evaluatng, α / β n / n ) e ϕ sn θ / cos ) = = θ, usng elementary trgonometrc denttes z ϕ / e cos / = ϕ / β e sn / where we have multpled by an overall phase factor spnor s also correctly normalzed e ϕ /, to make t look ncer Note that the The physcally sgnfcant parameter for spn drecton s just the rato α / β Note that any complex number can be represented as e ϕ cot θ / ), wth θ < π, ϕ < π, so for any possble spnor, there s a drecton along whch the spn ponts up wth probablty one
4 The Spn Rotaton Operator The rotaton operator for rotaton through an angleθ about an axs n the drecton of the unt vector ˆ,, ) n = nx ny nz s, usng J = S = σ, e θ nˆ J = e /) ˆ θ nσ ) Warnng: we re followng standard notaton here, but don t confuse ths θ --angle turned through wth the θ n wrtng ˆn n terms of Expandng the exponental, and usng n σ ) θ, ϕ )!) 3 θ ) ˆ / σ ) θ ˆ σ) θ ˆ σ) θ ˆ σ ) n 3 e = I + n + n + n +! 3! ˆ = I, 4 ) ˆ θ / n σ) θ θ e = I + + +! 4! 3 θ ˆ θ + n σ) + nˆ σ) + 3! ˆ σ ) θ θ = Icos n sn Wrtng ths n the same D-notaton we used for orbtal angular momentum earler the superscrpt refers to the j-value) Explctly, t s ) ) θ nˆ J /) nˆ ) σ) / θ σ ˆ θ cos ˆ θ D R n = e = e = I n sn The rotaton operator ) /) ˆ ) D R θn s a matrx operatng on the ket space, α + β = β
5 nz nx + ny) nx + ny) sn θ / ) cos θ / ) + nzsn θ / ) /) cos / sn / sn / D R θnˆ )) = Notce that ths matrx has the form wth a b * * b a a + b = The nverse of ths rotaton operator s clearly gven by replacng θ wth θ, that s, * a b a b = b a b a * * * These matrces have determnant a + b =, and so are untary They clearly form a group, snce they represent operatons of rotaton on a spn Ths group s called SU), the refers to the dmensonalty, the U to ther beng untary, and the S sgnfyng determnant + Note that for rotaton about the z-axs, n = ) rotaton operator becomes ˆ,,, t s more natural to replace ) ϕ / ˆ e D R z = ϕ / e /) ϕ ) θ wth ϕ and the In partcular, the wave functon s multpled by - for a rotaton of π Snce ths s true for any ntal wave functon, t s clearly also true for rotaton through π about any axs δθ nˆ J Exercse: wrte down the nfntesmal verson of the rotaton operator e for spn ½, and ˆ δθ n J δθ nˆ J prove that e σ e = σ + δθ nˆ σ, that s, σ s rotated n the same way as an ordnary three-vector note partcularly that the change depends on the angle rotated through, as opposed to the half-angle, so, reassurngly, there s no - for a complete rotaton as there cannot be the drecton of the spn s a physcal observable, and cannot be changed on rotatng the measurng frame through π )
6 Spn Precesson n a Magnetc Feld As a warm up exercse, consder a magnetzed classcal object spnnng about ts center of mass, wth angular momentum L and parallel magnetc moment μ, μ = γl The constant γ s called the gyromagnetc rato Now add a magnetc feld B, say n the z-drecton Ths wll exert a torque T = μ B = γl B = dl / dt, easly solved to fnd the angular momentum vector L precessng about the magnetc feld drecton wth angular velocty of precesson ω =γ B Bt Proof: from dl / dt = γ L B, take L Lx Ly, dl / dt BL, L L+ e γ + = + + = γ + + = Of course, dl z /dt =, snce dl / dt = γ L B s perpendcular to B, whch s n the z-drecton) The exact same result comes from the quantum mechancs of an electron spn n a magnetc feld The electron has magnetc dpole moment μ = γ S γ = g e/mc and g known, where ) as the Landé g-factor) s very close to Ths g-factor termnology s used more wdely: the magnetc moment of an atom s wrtten μ = gμb, where μb = e / mcs the Bohr magneton, and g depends on the total orbtal angular momentum and total spn of the partcular atom) The Hamltonan for the nteracton of the electron s dpole moment wth the magnetc feld s H =μ B =γs B, hence the tme development s wth the propagator ψ t) = U t) ψ ) Ht / Bt / ) = = U t e e γσ but ths s exactly the rotaton operator as shown earler) through an angle γ Bt about B! For an arbtrary ntal spn orentaton ϕ / e cos / = ϕ /, β e sn / the propagator for a magnetc feld n the z-drecton so the tme-dependent spnor s ) U t β ωt/ Bt/ e ωt/ γσ = e =, e ) t ) t ϕ ω t) + / e cos / = ϕ+ ω t) / e sn /
7 The angle θ between the spn and the feld stays constant, the azmuthal angle around the feld ncreases as ϕ = ϕ ω t, + exactly as n the classcal case Exercse: for a spn ntally pontng along the x-axs, prove that x ) = ) ω S t / cos t Paramagnetc Resonance We have shown that the spn precesson frequency s ndependent of the angle of the spn to the feld Consder how all ths looks n a frame of reference whch s tself rotatng wth angular velocty ω about the z-axs Let s call the magnetc feld B = Bˆ z, because we ll soon be addng another one In the rotatng frame, the observed precesson frequency s ωr = γ B + ω/ γ), so there s a dfferent effectve feld B + ω / γ n the rotatng frame Obvously, f the frame rotates exactly at the precesson frequency, ω = ω =γb, spns pontng n any drecton wll reman at rest n that frame there s no effectve feld at all Suppose now we add a small rotatng magnetc feld wth angular frequency ω n the x,y plane, so the total magnetc feld B = Bz ˆ + B x cos t y sn t ˆ ω ˆ ω ) The effectve magnetc feld n the frame rotatng wth the same frequency ω as the small added feld s / ) ˆ Br = B + ω γ z +Bx ˆ Now, f we tune the angular frequency of the small rotatng feld so that t exactly matches the precesson frequency n the orgnal statc magnetc feld, ω = ω =γb, all the magnetc moment wll see n the rotatng frame s the small feld n the x-drecton! It wll therefore precess about the x-drecton at the slow angular speed γ B Ths matchng of the small feld rotaton frequency wth the large feld spn precesson frequency s the resonance If the spns are lned up preferentally n the z-drecton by the statc feld, and the small resonant oscllatng feld s swtched on for a tme such that γ Bt = π /, the spns wll be preferentally n the y-drecton n the rotatng frame, so n the lab they wll be rotatng n the x,y plane, and a col wll pck up an ac sgnal from the nduced emf