Vincent P. Gutschick Yale University New Haven, Connecticut 06520
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I
Quantum Chemistry Easing the p a r a d o x o f the preferred axis for angular momentum
It often perplexes students of basic quantum chemistry that the angular momentum vector it has an apparent preference for one axis-call it z-particularly since the actual direction is arbitrary in the absence of a magnetic field. This preference is expressed familiarly as the existence of well-defined values (eigenvalues) simultaneously only for the operators of the square LZ and for one axial projection taken as the Cartesian component Lz.Mathematically, the development is that (1) both L Z and L, provide gwd quantum numbers, i.e., are conserved in time, in an expectation sense, or in a Heisenberg operator (1) sense
where the carets denote operators; and (2) the operators commute
[G,Ez] = 0 so that simultaneous eigenfunctionsexist (21. However. 2 is not spertal; the other two components are also conserved and du commute with r.';for example
'At leastqwo components are ill-defined. Further, the simple magnitude IL corresponds to a non-linear operator, in contrast to LZ=L.L.
Combining the (21 + 1) functions for one fixed 1, one need only adjust the coefficients to diagonalize the matrix of the ( L A enpectation values. Ladder operators or an explicit coordinate representation of L, can be used; both are given in Ref. (9):
432 / Journal of ChemicalEducatbn
If we arbitrarily choose h o have definite eigenvalues, then ex(and L,) has an uncertainty or dispersion
with bfackets denoting an expectation value. Mathematicjdly, L, cannot have eigenvalues simultaneously (2) with L, because of the non-commutation
--
[L,,L,I
=
-EL,
r0
... and cyclic
(1)
(A further consequence1 of eqn. (1) is that the vector L is essentially useless compared to t z . ) The arbitrariness of axes can be made halfway acceptable hy undoing the arbitrary choice. Any axis can be made the preferred one, using the same wave functions. The L,, L, operators still have discrete if mixed eigenvalues; and due to the L,- or m-degeneracy, linear combinations of L,-eigenfunctions can c o m p o ~ e ,say, ~ L,-eigenfunctions. Indeed, eigenfunctions can be constructed for an axis of totally arbitrary direction, via the intimidating Wigner rutation matrices (3). So, no axis is special, except for convenience. Yet a question remains, Physically, why not two axes a t once? We shall see that the difficulty lies not in mechanics, either quantum or classical, but in the properties of space itself! Briefly, we may show that two angular momentum components cannot uniquely specify the state of motion of a system. This is the same as the claim that two rotations fail to uniquely specify the motion. (Rotations are easier to comprehend, involving as they do the more concrete angular coordinates/angles, which are "canonical conjugates" to the more nebulous momenta. The operator sense of the latter can be distracting.) We must then show the equivalence of coordinates and momenta as alternative descriptions, using a piece of information lesser known to chemists. This is: In both quantum ( l b , 2b, 4a) and clasas sical (5) mechanics, the vector angular momentum operator and function, respectively, generates the (description of the) rotations of the system. This is a com-
e,
plete and generalizing restatement of the use and meaning of L. It means that, given L as a vector operator or number, we can transcribe all our detailed knowledge of the system when the latter is changed by a rotation . . . either a rotation of our viewpoint = coordinate system, or a real, physical rotation. Our detailed knowledge of all the system's motions lies, of course, in the wave function $0 (we restrict our discussion to quantum mechanics now). Let us rotate our coordinates (counterclockwise) about an axis directed along a unit vectorif, by an infinitesimal angle 68; finite mtations may be treated with somewhat more difficulty (46). The new wave function 6' a t an ohservation point 7 is the same in numerical, functional value as the old wave function II. a t the point 7 = 7 (368) x i? (The reader may convince himlherself that the preceding is a compact vector notation for the familiar Cartesian coordinate transformation usually given in terms of successive rotations 6 0 , 69 about y and z axes. One must express 60,66 in terms of n, 68.) That is
The second line here is purely mathematical, involving no added physical ideas; it is a vector form of Taylor's theorem. We may rearrange it $'
