Understanding Electron Spin 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 therefore surprising that even textbooks with "Quantum" titles often make no attempt to provide an intelligible ac- count thereof. It is more fashionable to introduce the con- cept as a mystery with its origin in either relativity or deep quantum theory.

This state of affairs is only partially explained by the fact that the Schrodinger equation was formulated (1) before the recognition of electron spin (2, 3) because several pro- posals outlining a physical basis of spin were published soon after (4, 5). The overriding consideration seems to be that a consistent mathematical description of the two-level spin system emerged for the first time in Dirac's relativis- tic wave equation (6). Thereby spin could be described as a relativistic effect without further elucidation.

S ~ i n and Angular Momentum - In classical mechanics angular momentum is calculated

as the vector ~roduct of generalized coordinates and mo- - menta.

e = q x p

which in component form reduces to

To convert this into a quantum-mechanical statement the momentum is represented by an operator

Si = -iRV

operating on some f, for example,

Operating on this with L, gives

Likewise

Using

etc., by subtraction one has

This defines the commutator

and in the same manner

[iYi*] = LG. A A

[L,L,] = i*iy The operator for the square angular momentum is con-

structed from n n n n L ~ = L : + L ; + L , ~

Now, the angular-dependent part of the central-field Schrodinger wave function, in spherical polar coordinates, is an eigenfunct*n of 2'.

~ ~ y . a ( e , v ) = e(e + i ) t ~ ~ ~ ( e , , ~ )

e=o, i , . . (n-1)

A

The eigenvalues ofL, are ( 2 t + 1)-fold degenerate. States with e = 0 should have zero angular momentum, contrary to experiment. Hydrogen s states ( e = 0) show fine struc- 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- tional symmetry of space, is therefore associated with an- other operator (71,

A n n J = L + S

where the commutators are defined by

A A

stc., forL , but nowS cannot be expressed in terms of$ and q operators. However, the following set of 2 x 2 matrices (Pauli matrices) has the required property.

~. A

The quantity corresponding to the eigenvalues of S,, s = *l/zh is known as the spin of the system.

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- 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

by substituting a

E + iti- at

p -, itiV

to yield a2w -ti2 - = t i2c2~2v + m 2 c 4 ~ at2

Dirac wrote a~ iti- = H v at

To ensure that space and time dimensions are treated in the same wav. as reauired bv s~ecial relativity, it would not - . he acceptabcto sudstitute

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 2 4 (a#, + a& +a& + b e 2 ) =P, + p y +p2 + m c

This condition cannot be satisfied unless the coefficients a and p are treated as anticommuting matrices rather than complex numbers. This means that

The simplest solution is possible in terms of 4 x 4 matri- ces, 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

The two-component vectors q and x are called spinors. Exactly the same procedures can be followed with the

Schrodinger equation (8). In its nonrelativistic form it de- rives from the classical expression

2 E - ~ = o

2m by introducing the quantum operators

without demonstrating that the form

'This ensures that both time and space derivatives appear in first order.

obeys the Galilean transformation

x ' + R x + u t + a

t ' + t + b

The correct Galilean invariance can he ensured by fol- lowing the same linearization procedure as Dirac. This means finding the linear classical form that yields the cor- rect quantum formulation, that is,

(AE + B.p + c)'~I = 0

(A'(E)' + 2 W E p ) + 2AC(E) + 2BCY.p) + B'$) + c2) = 0

To match eq 1 this requires A2 = AB = B C = C2 = 0 and AC + C A = 2m with B 2 = 1. Because B is a vector, we can clearly formulate the required condition in terms of Pauli and other matrices as

and the wave function as a column vector,

Thus. ((AC + CA)E + B ~ ~ ~ ) = 0

The linearized Schrodinger equation in this form is

(AE+B.p+C)ry=O

that is,

E'P + up)^ = 0

Because o2 = 1, it follows that

that is,

and likewise for q. Both q and x therefore satisfy the Schrodinger equation

with the same eigenvalue E = p2/2m. To describe an electron in an electromagnetic field repre-

sented by a scalar potential V, and vector potential A , the quantum operators become

into

Volume 72 Number 5 May 1995 413

An Absolute Direction

where

V ( i w A ) = V x A = e u r l A = B

where B is the magnetic field. The equation reduces to

which corresponds to an equation ascribed to Pauli (9) and obtained by empirically adding an operator to the Schrod- inger Hamiltonian to account for electron spin.

The importance of this equation lies in the term that re- flects an intrinsic magnetic moment

fie p=-

2mc

which is exactly as required to explain the fine structure of electronic soectra in terms of a s ~ i n aneular momentum of - 012 and a ~Lndh factor of 2.

The Physical Basis of Spin Conservation of Angular Momentum

The two-valuedness of electron spin reflected here by the matrix nature of o appears to be a strict quantum property, but this has never been demonstrated conclusively. This 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 ( 1 0 ) for rotation around an axis (z) in the x,y plane, at a dis- tance

r=(.2+y2)'

from the origin. The angular momentum with respect to the origin is

M = m ( ~ - y i )

and the time rate of change follows as

Because mj; and m i are components of force,

Thus,

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 between two like charges approaching at right angles. By the right-hand rule the forces between the charges are, as shown in the figure, equal but not opposite as required to conserve momentum. The missing momentum has been 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- quires angular momentum, in addition to the mechanical angular momentum that a particle may have. In the case of a quantum-mechanical particle this statement is equivalent to the suggestion (5, 14) that spin should be re- garded as a circulating flow of momentum density in the electron field, of the same kind as carried by the fields of a circularly polarized electromagnetic wave. This means that spin is not associated with the internal structure of the electron, but rather with the structure of its wave field.

The HamiltonJacobi Equation

Both of the foregoing interpretations imply that an elec- tron wave function has more than mathematical signifi- cance and refers to a physically real wave as proposed by De Broglie ( 1 5 ) and later more fully developed by Bohm ( 1 6 ) and others ( 1 7 ) in terms of the quantum potential ( 1 6 ) and the quantum torque (17) . To understand these con- cepts the spinor wave function is written in terms of the Euler angles (18) ,

and substituted into the Pauli equation (see above), to pro- duct a generalized HamiltonJacobi HJ equation

The spin vector is

f i . S = p l n E sin yr, sin 0 cosyr, cos 8)

The HJ equation differs from the classical equation for a spinning particle by two extra terms, the usual quantum potential,

f i2v2~ Q = - 2mR

and a spin-dependent quantum potential,

f i Z 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.

In terms of the spin vector,

414 Journal of Chemical Education