William H. Cropper St. Lawrence University Canton, N e w York 13617
Quantum Mechanics
with a Little Less Mystery
T o a beginning student quantum mechanics speaks in a language that borders on the fantastic. The state of a system, even one with many particles, is somehow gathered into a single function and this function is of necessity supplied with an imaginary part. Physical quantities are represented by "operators": instead of the familiar m i for a momentum component the momentum representation is -ifib/bx, an imaginary fragment of a derivative with no mass factor and no time derivative. Average values are calculated by a prescription which involves sandwiching an operator between and and integrating the entire thing over a multi-dimensional configuration space. The question of how much these peculiarities can be understood is, I think, worthy of more consideration than it usually gets. Most quantum mechanics texts make a point of emphasizing the strange things of the quantum realm, such as state functions, operator associations and average-value integrals, by stating their existence in postulates. No doubt this makes it clear how to use quantum mechanics, but to a student who has not seen the vast utility of operator methods, for example, an operator postulate makes little sense. Ultimately, of course, we must resort to postulates; no theoretical science can do otherwise. But I should like to point out that a different basis can be found which leads to the useful methods, although starting with concepts that are more acceptable to the beginning student with his naive (and often penetrating) questions. This approach has some of the "humanistic" value mentioned by Hinshelwood in the preface of his "Structure of Physical Chemistry." His treatment, he says, is "neither historical nor formally deductive, but at each stage I would try to indicate the route by which an inquiring mind might most simply and naturally proceed in its attempt to understand. . . ." A "humanistic" basis can be defined for quantum mechanics which consists largely of two concepts: the superposition principle and the correspondence principle. The former begins the argument and the latter points the direction along the way by using the known facts of classical mechanics. Several times mathematical expedience also enters; the operator method of representation seems a good idea, for example, because it does so much for the mathematical statements. This is a way of presenting quantum mechanics which has been developed especially by Dirac' and by Landau and L i f ~ h i t z . ~These authors are not known for the simplicity of their writing styles. But their basic argu-
*,
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DIRAC,P. A. M., "The Principles of Quantum Mechanics," 4th ed.. Oxford Universitv Press. Oxford. 1957. ' I.IND\C, I.. I). I V D T.IFRHITZ, t:. \I., ' ' Q ~ ~ n n ~ u.\lwhanici,'' m Addiw-Wrdcy l ' t h l i 4 1 i n ~Ct)., I k a d i n ~ ,l l : t . s . , 1958.
ments are simple and can, I think, be brought to a workable undergraduate level. There is not space here to present the full superposition-correspondence approach but some of its usefulness may be clear in the justification it offers for the mysterious -ifibpx momentum operator. This line of reasoning will be sketched with page references to the LandauLifshitz book2to fill in the gaps. The correspondence argument makes it abundantly clear that certain of the differential equations of classical mechanics, in particular the Ha,milton-Jacobi equation, are remarkably like the differential equations of quantum mechanics (p. 49). The equations in both realms utilie "state functions," that is, single functions which express the state of an entire system. The quantum state function has some peculiarities, however, the most prominent of which is the necessity that it be complex. This requirement is inescapable when the quantum state function is seen changing with time (the argument here draws from both the correspondence and the superposition aspects; pp. 25-26). It is also necessary that the state functions *A and *B from two separate noninteracting parts of a system be multiplied together to form the total state function: 9= The classical state function, on the other hand, must be additive for separate parts of the system. These various obsenrations make it reasonable to asand the classical state function S are sume that related as follows
*
*
q! = Ae'd
(1)
where c is a constant and A is a (slowly varying) amplitude factor. This is a state function appropriate to the "correspondence region" lying between the classical and quantum realms. The most outstanding property of the derived W is that it is in some way periodic with the phase 4 = cS. Here is an appearance in the correspondence region of that most persistent of all the quantum mysteries, the ubiquitous presence of waves. To develop the periodic behavior more fully the de Broglie and Einstein equations can be invoked as empirical statements, so as to relate the phase 4, a wave property, to particle properties such as the momentum and the energy. Among other things the constant c is found to be the universal constant l/fi (p. 20). The momentum problem is simplified considerably, in the quantum realm as well as in the classical realm, by finding means to recognize those physical quantities that perform as constants of motion, i.e., do not change with time. What is needed in quantum mechanics is a way to determine the time variation of an operator A,,: the derivative dA,,/dt must he evaluated. The relevant theorem is as follows (pp. 26-27) Volume 46, Number 12, December 1969
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dA../dt = (i/h)IHodo~l (2) . . if, as is usually the case, A , does not depend explicitly on the time variable. H , is the Hamiltonian operator and [H,, A,] stands for the "commutator" expression lHop,Aopl= H . d l o ~- A ~ O P If a tie-independent operator which commutes with the Hamiltonian operator can be found it may, according to the theorem just quoted, represent one of the physical constants of motion. The prohlem of finding these operators and establishing the necessary commutation conditions runs closely parallel to a familar classical problem. Consider a free particle from the classical viewpoint; it is free because it moves in a homogeneous, force-free space, and as a consequence of Newton's second law, its linear momentum is a constant of motion. In fact, the linear momentum of a particle can reasonably he defined as that physical entity which is conserved when the space in which the particle moves is homogeneous. This definition of the linear momentum can he used in both the classical and the quantum realms. Homogeneous space is recognized in quantum mechanics, just as it is in classical mechanics, by the condition that a system's total energy (including the potential energy terms) is independent of position. In quantum mechanics this means that the Hamiltonian operator is not changed when the system is displaced from one position to another. To express the condition of homogeneity (and also to adopt the characteristic operator language of quantum mechanics) an infinitesimal displacement operator b/bx is introduced. A system in hornogcneoiis (onedirnrr~siond)q)aw luls a II:~rnilronin~i oprrntor which is unaNccted hy this oprrator and consequently
The snace-homoeeneitv condition thus leads to two closely' parallel results "(3) and (4). The conclusion is that the momentum operator (p,),, is equivalent to d/ax multiplied perhaps by a constant c (PJOP
c=
+
Journal of Chemical Education
46.
The lines of this argument can be followed in other directions. Orbital angular momentum, for example, can he regarded as the physical entity conserved in isotropic space (all directions equivalent) and an argument parallel to that leading to the linear momentum operator is used to arrive at the angular momentum operator. Spin angular momentum presents a unique problem because no correspondence argument is possible: all spin effects vanish in the correspondence region. Nevertheless, the operator for orbital angular momentum, which can be related to a rotation operator (just as the linear momentum is represented by a displacement operator), shows how to write rotation operators for the two-dimensional space of a particle's spin state function. In effect these discussions include most of the usual postulatory scheme of quantum mechanics without identifying it as such. The intention in not pointing to the postulates is not to prove them unnecessary. They are, in fact, quite necessary for a statement of the formal working structure of quantum mechanics. But the formal structure of science is not everything. A student looking a t a list of carefully framed postulates may have the impression that they have been handed down by some divinity of the physics profession and will remain carved onto the pages of textbooks forever. we must explain that this is nonsense, that the postulates do not necessarily stand by themselves. They are the simplest and most useful mathematical forms of broader principles, and it is these broader principles-I identify them in quantum mechanics as the superposition and correspondence principles; others may have us along what Hinshelwood different ideasithat calls the route of the inquiring mind.
(4)
/
(5 )
and give the derivative aS/bx its classical identity p, (p. 39). The constant c acquires its remarkable value
lH"..a/&Tl (3) . --,. . = 0 .. A commuting operator a/bx has been found and its commutation behavior has heen &own to he a cons* quence of space homogeneity. Following the lead of classical mechanics we also say that the conserved physical quantity associated with homogeneous space is the linear momentum, that homogeneity of space implies d p d d t = 0. If p, is the linear momentum operator it follows from eqn. (2) that for each of its components, say
840
= c(a/az)
Apply this momentum operator now to the correspondence state function (1)
