Jan C. A. Boeyens University of the Witwatersrand, P.O. Wits, 2050 Johannesburg, South Africa Electron spin is of obvious importance in chemistry. It is and in the same manner therefore surprising that even textbooks with "Quantum" titles often make no attempt to provide an intelligible ac- [iYi*]= LG. count thereof. It is more fashionable to introduce the con- A A cept as a mystery with its origin in either relativity or deep [L,L,] = i*iy quantum theory. The operator for the square angular momentum is con- This state of affairs is only partially explained by the fact structed from that the Schrodinger equation was formulated (1)before nnnn the recognition of electron spin (2, 3) because several pro- L~=L:+L;+L,~ posals outlining a physical basis of spin were published Now, the angular-dependent part of the central-field soon after (4, 5).The overriding consideration seems to be Schrodinger wave function, in spherical polar coordinates, that a consistent mathematical description of the two-level is an eigenfunct*n of 2'. spin system emerged for the first time in Dirac's relativis- tic wave equation (6).Thereby spin could be described as a ~~y.a(e,v)= e(e + i)t~~~(e,,~) relativistic effect without further elucidation. e=o,i, ..(n-1) S~inand Angular- Momentum In classical mechanics angular momentum is calculated as the vector ~roductof generalized coordinates and mo- A - The eigenvalues ofL, are (2t 1)-fold degenerate. States menta. + with e = 0 should have zero angular momentum, contrary e=qxp to experiment. Hydrogen s states (e = 0) show fine struc- which in component form reduces to ture in the form of doublet levels that represent an addi- tional degeneracy of 2s + 1= 2. This remarkable state of affairs implies that, whereas classical angular momentum is a con erved quantity, the observable quantity associated with 1 is not. The quan- tum-mechanical conserved quantity, implied by the rota- To convert this into a quantum-mechanical statement tional symmetry of space, is therefore associated with an- other operator (71, the momentum is represented by an operator A n n Si= -iRV J=L+S where the commutators are defined by operating on some f,for example,
A A stc., forL ,but nowS cannot be expressed in terms of$ and Operating on this with L, gives q operators. However, the following set of 2 x 2 matrices (Pauli matrices) has the required property.
Likewise
~. Using A The quantity corresponding to the eigenvalues of S,, s = *l/zh is known as the spin of the system. etc., by subtraction one has Quantum Mechanics of Spin It follows from the foregoing that quantum theory in the traditional Schrodinger formulation is not complete. On the other hand, spin appears in Dirac's equation not be- This defines the commutator cause it is relativistic, but because the wave function has the correct transformation properties. Instead of deriving a relativistic equation from the en- ergy-momentum expression,
412 Journal of Chemical Education obeys the Galilean transformation x'+Rx+ut+a by substituting a E + iti- t'+t+b at The correct Galilean invariance can he ensured by fol- p -, itiV lowing the same linearization procedure as Dirac. This means finding the linear classical form that yields the cor- to yield rect quantum formulation, that is, -ti2 -a2w = ti2c2~2v+ m2c4~ at2 (AE + B.p+ c)'~I= 0 Dirac wrote (A'(E)' + 2WEp)+ 2AC(E)+ 2BCY.p) + B'$) + c2)= 0 iti-a~ = Hv at To match eq 1this requires A2 = AB = BC = C2 = 0 and To ensure that space and time dimensions are treated in AC + CA= 2m with B2 = 1. Because B is a vector, we can the same wav. as reauired bv-. s~ecial relativity, it would not clearly formulate the required condition in terms of Pauli he acceptabcto sudstitute and other matrices as
Instead, use a linearized square mot, which formally has the form1 H = -ea.p - pmc 2 and solve for a and p. In component form one has 2 2 2 24 (a#, + a& +a& + be2)=P, +py+p2 + m c and the wave function as a column vector, This condition cannot be satisfied unless the coefficients a and p are treated as anticommuting matrices rather than complex numbers. This means that Thus. ((AC + CA)E + B~~~)= 0 The linearized Schrodinger equation in this form is The simplest solution is possible in terms of 4 x 4 matri- (AE+B.p+C)ry=O ces, that is, that is,
where 0;s are the same Pauli matrices introduced above. This requires the wave function to be formulated as a four-dimensional vector, usually written as E'P + up)^ = 0 Because o2= 1, it follows that
The two-component vectors q and x are called spinors. that is, Exactly the same procedures can be followed with the Schrodinger equation (8).In its nonrelativistic form it de- rives from the classical expression and likewise for q. 2 Both q and therefore satisfy the Schrodinger equation E-~=o x 2m with the same eigenvalue E = p2/2m. by introducing the quantum operators To describe an electron in an electromagnetic field repre- sented by a scalar potential V, and vector potential A, the quantum operators become
without demonstrating that the form
into
'This ensures that both time and space derivatives appear in first order.
Volume 72 Number 5 May 1995 413 An Absolute Direction This conservation law can be stated in another way, in terms of a nonobservable (111, which in this case is an ab- solute spatial direction. Whenever an absolute direction is observed, the conservation no longer holds. This could mean that in the quantum domain, space cannot be as- sumed to be rotationally symmetrical and the special di- rection corresponds to the observed spin alignment. Simple Electrodynamic Effect Alternatively it could be viewed as a simple electrody- namic effect like the instantaneous magnetic interaction where between two like charges approaching at right angles. By the right-hand rule the forces between the charges are, as V(iwA)=VxA=eurlA=B shown in the figure, equal but not opposite as required to where B is the magnetic field. conserve momentum. The missing momentum has been The equation reduces to transferred to the field, and only when the field momen- tum is added to the mechanical momentum of the charges does conservation of momentum prevail (12). Similar considerations apply to angular momentum (13). In this instance conservation requires that the field ac- which corresponds to an equation ascribed to Pauli (9)and quires angular momentum, in addition to the mechanical obtained by empirically adding an operator to the Schrod- angular momentum that a particle may have. In the case inger Hamiltonian to account for electron spin. of a quantum-mechanical particle this statement is The importance of this equation lies in the term that re- equivalent to the suggestion (5, 14) that spin should be re- flects an intrinsic magnetic moment garded as a circulating flow of momentum density in the electron field, of the same kind as carried by the fields of a fie circularly polarized electromagnetic wave. This means p=- 2mc that spin is not associated with the internal structure of the electron, but rather with the structure of its wave field. which is exactly as required to explain the fine structure of The HamiltonJacobi Equation electronic soectra in terms of a s~inaneular - momentum of 012 and a ~Lndhfactor of 2. Both of the foregoing interpretations imply that an elec- tron wave function has more than mathematical signifi- The Physical Basis of Spin cance and refers to a physically real wave as proposed by Conservation of Angular Momentum De Broglie (15) and later more fully developed by Bohm (16)and others (17)in terms of the quantum potential (16) The two-valuedness of electron spin reflected here by the and the quantum torque (17). To understand these con- matrix nature of o appears to be a strict quantum property, cepts the spinor wave function is written in terms of the but this has never been demonstrated conclusively. This Euler angles (18), question probably cannot be addressed without examining the conservation of angular momentum in more detail. It occurs as a consequence of the isotropy or spherical rota- tional symmetry of space. This is easily demonstrated (10) and substituted into the Pauli equation (see above), to pro- for rotation around an axis (z) in the x,y plane, at a dis- duct a generalized HamiltonJacobi HJ equation tance r=(.2+y2)' from the origin. The angular momentum with respect to The spin vector is the origin is fi. M=m(~-yi) S = plnE sin yr, sin 0 cosyr, cos 8) and the time rate of change follows as The HJ equation differs from the classical equation for a spinning particle by two extra terms, the usual quantum potential, Q=- fi2v2~ Because mj; and miare components of force, 2mR and a spin-dependent quantum potential,
fiZ Q. = -+(VO)~8m t sin2 8 (v~)')
Both of these terms are inversely proportional to particle mass, which is thereby identified as the parameter that distinguishes between quantum and classical systems. Thus, In terms of the spin vector,
414 Journal of Chemical Education