Read an Excerpt
Group Theory and Quantum Mechanics
By Michael Tinkham, Gordon McKay Dover Publications, Inc.
Copyright © 1992 Michael Tinkham
All rights reserved.
ISBN: 978-0-486-13166-5
CHAPTER 1
INTRODUCTION
1-1 The Nature of the Problem
The basic task of quantum theory in the study of atoms, molecules, and solids consists in solving the time-independent Schrödinger equation
Hψn = Enψn
to determine the energy eigenvalues En and the corresponding eigenfunctions ψn. The reason for the preeminent position of energy eigenfunctions in the theory is, of course, that they form the stationary states of isolated systems in which energy must be conserved. The science of spectroscopy is devoted to discovering the eigenvalues En experimentally, so that one may work back to infer the internal structure of the system which produces them. Moreover, because of their simple time dependence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], they can be used to build up a description of the time evolution of non-stationary systems as well. Thus a major share of the theoretical work in extranuclear physics is devoted to finding eigenstates and eigenvalues of the Hamiltonian operator.
Apart from spin and other relativistic effects, the Hamiltonian operator H is obtained from the classical expression for the energy H (pi, qi) by making the usual operator replacement [gradient]. Since we shall be dealing with low-energy phenomena, we may introduce the spin effects through the standard two-component Pauli matrix treatment, without resorting to the four-component Dirac equation. In other words, there is really no longer any serious difficulty in principle in writing down a basic Hamiltonian which is accurate enough for all practical purposes in extranuclear physics. (Within the nucleus, of course, the situation is quite different.) The difficulty is simply that the eigenvalue problems to be solved are very hard and complicated.
This is illustrated by writing down our Hamiltonian.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where Hso, Hss, Hhfs, and Hext refer to spin-orbit, spin-spin, hyperfine-structure, and external-field couplings. The eigenfunctions of this operator will be functions of the space and spin coordinates of all the electrons and nuclei. Even for as simple a system as the oxygen molecule, O216, this will involve a 54-dimensional spatial function together with 16 spin functions. It is clear that exact "textbook" solutions cannot be expected except for such simple cases as harmonic oscillators, single-particle central-field problems, and noninteracting particles in a square-walled box.
One might propose a direct brute-force numerical solution by use of high-speed digital computers. The impossibility of this approach may be seen by considering the difficulty of simply stating the wavefunction obtained, which is a 3N-dimensional function (if there are N electrons and the nuclei are assumed fixed). For reasonable accuracy, we would need to divide each axis into, say, 100 units. Thus we would need to tabulate (100)3N = 106N entries. For something as simple as a four-electron system, this already exceeds the number of molecules in a mole, and our libraries could hardly contain the millions of volumes required to record the result. Needless to say, no human mind could comprehend a result hidden in such a mass of numbers. By making the sweeping, but reasonable, physical approximation of treating each particle as moving independently in an average field from the others, the problem can be reduced to large, but more manageable, dimensions. The detailed interactions could then be reintroduced as a correction, if a more refined calculation were required.
In any case, it is clear that the only practical approach to real physical problems, of any except the very simplest sort, will be an approximate one. The correct answer is approached by successively improved approximations, obtained by including successively smaller correction terms in the Hamiltonian and by increasing the accuracy with which any given approximation to the Schrödinger equation is satisfied. In many cases, this procedure can be carried far enough to yield a description adequate to explain all observed physical properties of the system under consideration, while remaining simple enough to be qualitatively helpful.
1-2 The Role of Symmetry
Being faced by the task of efficiently simplifying a problem so that it may be solved, it is clearly advantageous first to seek out any simplifications which can be made rigorously on the basis of symmetry. Only after these, have been fully exploited should one resort to approximations which reduce the generality and accuracy of the final answer. To assist us in the search for the full symmetry-based simplification of the problem, we draw upon the resources of group theory. This provides us with a systematic calculus for exploiting symmetry properties to the fullest extent. Since we require only elementary group theory for our purposes, we shall be able to first develop this machinery from the very beginning. This involves a certain amount of "capital investment" in the form of learning some new formal methods. However, once this investment has been made, we are in a position to adduce very general results in many problems in an economical way, and to gain new insight. It seems inefficient to proceed in many quantum-mechanical problems without this tool at our disposal.
The symmetry which we aim to exploit via group theory is, in most cases, the symmetry of the Hamiltonian operator. To discuss this symmetry, it is convenient to introduce the concept of transformation operators which induce some particular coordinate transformation in whatever follows them. For example, the inversion operator i reverses the signs of all coordinates, taking r into -r. Other common operators are those inducing reflections, rotations, translations, or permutations of coordinates of particles. Such an operator will be a symmetry operator appropriate to a given Hamiltonian if the Hamiltonian looks the same after the coordinate transformation as it did before; in other words, if the Hamiltonian is invariant under the transformation. For example, if the potential-energy expression is an even function of coordinates about the origin, it will be invariant under inversion. Since the kinetic-energy operator involves only second derivatives, it is always even. Hence the entire Hamiltonian is invariant under inversion if the potential is. This has useful consequences, as we shall soon see.
To continue this discussion, it is useful to note that if an operator R leaves the Hamiltonian invariant, RHψ = HRψ, for any ψ. That is to say, R will commute with H if it leaves H invariant. But if R commutes with H, a general theorem tells us that there are no matrix elements of H between eigenstates of R having different eigenvalues for the operator R. The proof is simple. Namely, in the equation
RH = HR
we may expand the products in a matrix representation based on eigen-functions of R. Thus
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
But since the representation is based on eigenfunctions of R, R has a diagonal matrix and each sum reduces to a single term, namely,
RiiHik = HikRkk
or
(Rii - Rkk)Hik = 0
Thus Hik = 0 if i and k refer to different eigenvalues of R, as stated. The significance of this result to us is that, in searching for eigenfunctions that diagonalize the Hamiltonian operator, the search can be made separately within the classes of functions having different eigenvalues of a commuting symmetry operator since no off-diagonal matrix elements of H will connect functions of different symmetry. For example, considering inversion symmetry, we can find all possible energy eigenfunctions by considering only functions which are either even or odd under inversion. Roughly speaking, this cuts the detailed work in half, besides leading to some qualitative information about the solutions with practically no work at all.
If there are several mutually commuting symmetry operators, all of which commute with H, we can then choose basis functions which are simultaneous eigenfunctions of all these symmetry operators. It then follows that there are no matrix elements of H between states which differ in their classification according to any of the symmetry operators. Consequently, we must be able to find a complete set of eigenfunctions of H which are also eigenfunctions of our complete set of mutually commuting symmetry operators. Thus we may restrict our search for eigenfunctions of H to functions having a definite symmetry under this set of operators.
It is often the case that, after finding the largest group of mutually commuting symmetry operators, there may be additional symmetry operators which commute with the Hamiltonian but which do not commute with all the previous symmetry operators. Although this larger group of operators is not completely mutually commuting, some additional information can be obtained by using these symmetry operators. We shall show later that, although it is no longer possible to work with functions which are eigen-functions of all the (larger group of) operators, finite sets of functions can be found such that the effect of a symmetry operation on one function of the set produces only a linear combination of functions within the set. These linear combinations are described by matrices which replace the simple eigenvalues described above as the entity describing the symmetry properties of the functions. The dimensionality of these matrices will turn out to give the degeneracy of the quantum-mechanical eigenfunctions, and the labels characterizing the various matrices and rows within the matrices will form the "good quantum numbers" for the system. These are the generalizations of the parity, or even versus oddness, "quantum number" in the simple example of inversion symmetry treated above. These matrices are the subject of the theory of group representations, which we shall consider in Chap. 3.
The results of group theory which we shall obtain are of considerable practical value, since the method of approximate solution almost invariably consists in expanding the solution in terms of a "suitable" set of approximate functions. Our group theoretical study of the symmetry of the system will enable us to choose zero-order functions of the correct symmetry so as to eliminate most off-diagonal matrix elements of H, leaving a largely simplified problem. Since we shall also see that symmetry determines selection rules, the selection rules governing transitions between these eigenfunctions can also be determined by group-theoretical arguments, without explicit integrations to compute matrix elements. Of course, there is also a quantitative calculation left to find the actual eigenfunctions, the eigenvalues, and the transition probabilities. However, the group theory will usually provide very considerable aid in reducing the scale of this residual calculation as well as giving some rigorous qualitative results with practically no effort at all.
We now proceed to develop the group-theoretical machinery before actually considering in detail the physical problems which are our primary concern.
CHAPTER 2
ABSTRACT GROUP THEORY
2-1 Definitions and Nomenclature
By a group we mean a set of elements A, B, C, ... such that a form of group multiplication may be defined which associates a third element with any ordered pair. This multiplication must satisfy the requirements:
1. The product of any two elements is in the set; i.e., the set is closed under group multiplication.
2. The associative law holds; for example, A (BC) = (AB)C.
3. There is a unit element E such that EA = AE = A.
4. There is in the group an inverse A-1 to each element A such that AA-1 = A1A = E.
For the present we shall restrict our attention primarily to finite groups. These contain a finite number h of group elements, where h is said to be the order of the group. If group multiplication is commutative, so that AB = BA for all A and B, the group is said to be Abelian.
2-2 Illustrative Examples
An example of an Abelian group of infinite order is the set of all positive and negative integers including zero. In this case, ordinary addition serves as the group-multiplication operation, zero serves as the unit element, and -n is the inverse of n. Clearly the set is closed, and the associative law is obeyed.
An example of a non-Abelian group of infinite order is the set of all n × n matrices with nonvanishing determinants. Here the group-multiplication operation is matrix multiplication, and the unit element is the n × n unit matrix. The inverse matrix of each matrix may be constructed by the usual methods, since the matrices are required to have nonvanishing determinants.
A physically important example of a finite group is the set of covering operations of a symmetrical object. By a covering operation, we mean a rotation, reflection, or inversion which would bring the object into a form indistinguishable from the original one. For example, all rotations about the center are covering operations of a sphere. In such a group the product AB means the operation obtained by first performing B, then A. The unit operation is no operation at all, or perhaps a rotation through 2π. The inverse of each operation is physically apparent. For example, the inverse of a rotation is a rotation through the same angle in the reverse sense about the same axis.
As a complete example, which we shall often use for illustrative purposes, consider the non-Abelian group of order 6 specified by the following group-multiplication table:
[TABLE OMITTED]
The meaning of this table is that each entry is the product of the element labeling the row times the element labeling the column. For example, AB = D ≠ BA. This table results, for example, if we take our elements to be the following six matrices, and if ordinary matrix multiplication is used as the group-multiplication operation:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Verification of the table is left as a simple exercise.
The very same multiplication table could be obtained by considering the group elements A, ..., F to represent the proper covering operations of an equilateral triangle as indicated in Fig. 2-1. The elements A, B, and C are rotations by π about the axes shown. Element D is a clockwise rotation by 2π/3 in the plane of the triangle, and F is a counterclockwise rotation through the same angle. The numbering of the corners destroys the symmetry so that the position of the triangle can be followed through successive operations. If we make the convention that we consider the rotation axes to be kept fixed in space (not rotated with the object), it is easy to verify that the multiplication table given above describes this group as well.
Two groups obeying the same multiplication table are said to be isomorphic.
2-3 Rearrangement Theorem
In the multiplication table in the example above, each column or row contains each element once and only once. This rule is true in general and is called the rearrangement theorem. Stated more formally, in the sequence
EAk, A2Ak, A3Ak, ..., AhAk,
each group element Ai appears exactly once (in the form ArAk). The elements are merely rearranged by multiplying each by Ak.
PROOF: For any Ai and Ak, there exists an element Ar = AiAk-1 in the group since the group contains inverses and is closed. Since ArAk = Ai for this particular Ar, Ai must appear in the sequence at least once. But there are h elements in the group and h terms in the sequence. Hence there is no opportunity for any element to make more than a single appearance.
(Continues...)
Excerpted from Group Theory and Quantum Mechanics by Michael Tinkham, Gordon McKay. Copyright © 1992 Michael Tinkham. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.
