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# APPENDIX A: Further Development of Quantum Mechanics, Angular Momentum, and Spin of the Electron

Before embarking on an analysis of angular momentum and spin of the electron, we collect together here our main findings of Quantum Theory. In section 1.2 and 1.3 we learned that the energies of the electrons are given by the eigenvalues of the hamiltonian H and the latter is constructed by substituting for the momentum the operator —iftV in the classical expression for the energy. The eigenfunctions Ч'(г) of the hamiltonean Я, are then used to deduce the average position and momentum of an electron in a potential energy field. This concept was further generalized in section 1.4. Just as the expression for the classical energy of an electron was turned into an operator, the hamiltonian operator Я, the expression for any classical quantity Q is turned into an operator Q in Quantum Mechanics by exactly the same rules as for the hamiltonean Я, i.e. 1) p —> —itiV and 2) leaving the position vector r as r. Then the possible measured values of that physical quantity are the eigenvalues q„ of that operator. Mathematically we should write

where qn) are the eigenvectors of Q. The average value of repeated measurements is then Note that this is a generalization of equations 1.11 and 1.12.

The physical thinking behind equation A1 is that the act of measurement and the result of measurement are not the same thing in Quantum Mechanics. The measurement of variable A and then of variable В will not necessarily give the same result if it is performed in the opposite order, first measurement of B, then A. To account for this, we must in some way have, (not always),

This can only be achieved with operators. The act of measurement in Quantum Mechanics is represented by Hermitian operators which have the property of having real eigenvalues and for which Amj, = (Amjl) (where the star denotes complex conjugate). When A3 is indeed different from zero, we say that A and В do not commute and [ А,й] is called the commutator of A,B. However, in some cases, if AB = BA, this has a significant physical consequence: it means that the two physical variables-observables can be measured simultaneously. In this case, the two operators have common eigenfunctions. An example of this pair is the momentum and the energy in a zero potential field (where the energy is just the kinetic one only). The angular momentum L is defined classically by

The components of L are classically and then quantum mechanically as follows:

It is easier to work in the spherical coordinates (r,0,cp) given by Elementary calculus manipulations give

It is easy to see that

However we find that Lz commutes with the magnitude of L which is given by

This can easily be deduced by inspection of equation A9. Therefore L2 and Lz can be measured simultaneously and hence they must have common eigenfunctions. These eigenfunctions are the spherical harmonics Y/,m(0,(p) that we have encountered in the theory of the Hydrogen atom. A rather lengthy purely mathematical argument shows that the eigenvalue equations for L2 and Lz take the form

and

Therefore the eigenvalues of L2 are Ti2l(l+1) and the eigenvalues of Lz are bmt. The integers l and mi are identical to the quantum numbers we have encountered in the electron states of the Hydrogen atom. We conclude that the hydrogenic electrons with a given l state are in a state of constant (magnitude of) angular momentum.

The above theory is not complete. The famous Stern-Gerlach experiment, which we describe immediately below, has shown that the electron possesses in addition to the orbital angular momentum (due to its motion around the nucleus) an inherent “spin” angular momentum as if it were spinning around itself. The experiment briefly is as follows: A beam of Silver (Ag) atoms is produced in an oven and passed through a magnetic

FIGURE A1 Apparatus of the Stern-Gerlach experiment.

field pointing in the z direction, see figure Al. The Ag atoms have an atomic number Z = 47 with 46 electrons in a closed inner shell with no angular momentum (the sum of the orbital momenta along the z direction cancels out) and the 47th electron is in a 5s1 orbital which also has zero “orbital” angular momentum (because / = 0 in this state).

Classically a circulating charge (the electron) possesses a magnetic moment perpendicular to its plane of circulation and of magnitude jX = 1A where I is the current produced by the charge and A the area enclosed by the circulating charge. The magnetic field of the magnet used in the experiment is non-uniform so that a force Fz will be exerted on the outer electrons of Ag given by

where

Since the magnetic moment p is proportional to the angular momentum that creates it and the z-component of the latter is quantized (according to Al 1) we should see as many spots, after the beam has passed through the magnet, as the number of possible values of Lz.

Two spots of Silver atoms are observed on the screen. This is however in contradiction to equations A12 and A13. If / = 0 there should have been one spot and if / > 1 there should have been three or more. The conclusion of the Stern-Gerlach experiment is that the electron possesses an angular momentum additional to the orbital angular momentum called spin angular momentum which is given the symbol S because it is a different quantity from L. For the eigenvalues Sz we should have in analogy with equations All and A13

Therefore ms plays the role of a fourth (fractional) quantum number in addition to n, /, and mt that we have encountered in chapter 1. Each state defined by {n,l,m,) can be occupied by two electrons, one with Sz=ft/2 and another with Sz=—fi / 2. We usually refer to them as spin up and spin down electrons. Note that the Pauli principle is not violated because no two electrons have the same (four) quantum numbers.

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