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Applications of Group Theory in Quantum Mechanics
By M. I. Petrashen, E. D. Trifonov, S. Chomet, J. L. Martin Dover Publications, Inc.
Copyright © 1969 M. I. Petrashen and E. D. Trifonov
All rights reserved.
ISBN: 978-0-486-17272-9
CHAPTER 1
Introduction
In the first chapter of this monograph we shall try, in so far as it is possible at the beginning of a book, to show how one can naturally and advantageously apply the theory of groups to the solution of physical problems. We hope that this will help the reader who is mainly interested in the applications of group theory to physics to become familiar with the general ideas of abstract groups which are necessary for applications.
1.1 Symmetry properties of physical systems
It is frequently possible to establish the properties of physical systems in the form of symmetry laws. These laws are expressed by the invariance (invariant form) of the equations of motion under certain definite transformations. If, for example, the equations of motion are invariant under orthogonal transformations of Cartesian coordinates in three-dimensional space, it may be concluded that reference frames oriented in a definite way relative to each other are equivalent for the description of the motion of the physical system under consideration. Equivalent reference frames are usually defined as frames in which identical phenomena occur in the same way when identical initial conditions are set up for them. Conversely, if in a physical theory it is postulated that certain reference frames are equivalent, then the equations of motion should be invariant under the transformations relating the coordinates in these systems. For example, the postulate of the theory of relativity which demands the equivalence of all reference frames moving with uniform velocity relative to one another is expressed by the invariance of the equations of motion under the Lorentz transformation. The class of equivalent reference frames for a given problem is frequently determined from simple geometrical considerations applied to a model of the physical system. This is done, for example, in the case of symmetric molecules, crystals and so on. However, not all transformations under which the equations of motion are invariant can be interpreted as transformations to a new reference frame. The symmetry of a physical system may not have an immediate geometrical interpretation. For example, V. A. Fock has shown that the Schroedinger equation for the hydrogen atom is invariant under rotations in a four-dimensional space connected with the momentum space.
The symmetry properties of a physical system are general and very important features. Their generality usually ensures that they remain valid while our knowledge of a given physical system grows. They must not, however, be regarded as absolute properties; like any other descriptions of physical systems they are essentially approximate. The approximate nature of some symmetry properties is connected with the current state of our knowledge, while in other cases it is due to the use of simplified models of physical systems which facilitate the solution of practical problems.
Thus, by the symmetry of a system we shall not always understand the invariance of its equations of motion under a certain set of transformations. The following important property must always be remembered: if an equation is invariant under transformations A and B, it is also invariant under a transformation C which is the result of the successive application of the transformations A and B. The transformation C is usually called the product of the transformations A and B. A set of symmetry transformations for a given physical system is therefore closed with respect to the operation of multiplication which we have just defined. Such a set of transformations is called a group of symmetry transformations for the given physical system. A rigorous definition of a group is given below.
1.2 Definition of a group
A group G is defined as a set of objects or operations (elements of the group) having the following properties.
1. The set is subject to a definite 'multiplication' rule, i.e. a rule by which to any two elements A and B of the set G, taken in a definite order, there corresponds a unique element C of this set which is called the product of A and B. The product is written C = AB.
2. The product is associative, i.e. the equation (AB) D = A (BD) is satisfied by any elements A, B and D of the set. The product may not be commutative, i.e. in general AB ≠ BA. Groups for which multiplication is commutative are Abelian.
3. The set contains a unique element E (the identity or unit element) such that the equation
AE = EA = A
is satisfied by any element A in the set.
4. The set G always includes an element F (the inverse) such that for any element A
AF = E
The inverse is usually denoted by A-1.
The above four properties define a group. We see that a group is a set which is closed with respect to the given rule of multiplication. The following are consequences of the above properties.
a. The group contains only one unit element. Thus, for example, if we suppose that there are two unit elements E and E' in the group G, then in view of property 3 we have
EE' = E = E'E = E'
i.e. E = E'.
b. If F is the inverse of A, the element A will be the inverse of F, i.e. if AF = E, then FA = E. In fact, multiplying the first of these equations on the left by F, we have
FÁF = F
The element F (like any other element of the set G) has an inverse F-1. Multiplying the last equation on the right by F-1 we obtain FAFF-1 = FF-1, i.e. FA = E.
c. For each element in the set there is only one inverse element. Let us suppose that an element A in G has two inverse elements F and D, i.e. AF = E and AD = E. If this is so, then by multiplying the equation AF = AD on the left by A-1 we obtain F = D.
d. If C = AB then C-1 = B-1A-1, because of the associative property of the product of two elements in the group.
We note also that if the number of elements in a group is finite, then the group is called a finite group; if the number of elements is infinite, the group is called an infinite group. The number of elements in a finite group is the order of the group.
The following are examples of groups.
1. The set of all integers, including zero, forms an infinite group if addition is taken as group multiplication. The unit element in this group is 0, the inverse element of a number A is -A, and the group is clearly Abelian.
2. The set of all rational numbers, excluding zero, forms a group for which the multiplication rule is the same as the familiar multiplication rule used in arithmetic. The unit element is 1. This is again an infinite Abelian group. The positive rational numbers also form a group, but the negative rational numbers do not.
3. The set of vectors in n-dimensional linear space forms a group. The group multiplication rule is the vector addition; the unit element is the zero vector and the inverse of a vector a is — a.
4. The set of all non-singular n-th order matrices (or the corresponding linear transformations in n-dimensional space), GL(n), is an example of a non-Abelian group. It is clear that the elements of this group depend on n2 continuously varying parameters (elements of the group). Infinite groups whose elements depend on continuously varying parameters are continuous groups. The unit element of the group GL(n) is the unit matrix; the inverse elements are the corresponding inverse matrices. The operation of group multiplication is the same as the rule of multiplication of matrices, which is not commutative.
1.3 Examples of groups used in physics
Let us now list some groups which will be used in applications.
1. The three-dimensional translation group. The elements of this group are the displacements of the origin of coordinates through an arbitrary vector a:
r' = r + a
It is clear that this is a three-parameter (three components of the vector a) continuous group.
2. The rotation group O+ (3). The elements of this group are rotations of three-dimensional space, or the corresponding orthogonal matrices with a determinant equal to unity. This is also a continuous three-parameter group: the nine elements of the orthogonal transformation matrix are related by six conditions, and three angles {I, θ, ψ} can be taken as the independent rotation parameters. The polar angles I and θ define the position of the rotational axis passing through the origin, and the angle ψ defines rotation about this axis (see Exercise 1.1). Invariance with respect to the group O+ (3) expresses the isotropy of three-dimensional space, i.e. the equivalence of all directions in this space.
If we add the operations of rotation accompanied by inversion (e.g. x' = -x, y' = -y, z' = -z) to the rotation group we obtain the orthogonal group O (3).
3. Molecular symmetry groups, i.e. point groups, consist of certain orthogonal transformations of three-dimensional space. For example, the symmetry group of a molecule having the configuration of an octahedron consists of 48 elements, namely, rotations and rotations accompanied by inversion which transform the corners of a cube into one another.
4. The crystal symmetry groups, or space groups, consist of a finite number of orthogonal transformations and discrete translations, and all products of these transformations. Strictly speaking, such symmetry is exhibited only by an infinite crystal or a model of a crystal with the so-called periodic boundary conditions.
5. The permutation group which consists of all permutations of n symbols, e.g. the coordinates of n identical objects. This is a finite group of order n!.
6. The Lorentz group L+ consists of transformations relating the coordinates of two reference frames which are in uniform rectilinear relative motion. This group includes the rotation group O+(3) and depends on six parameters, namely, three angles defining the mutual orientation of the space axes, and the three components of the relative velocity. The invariance of the equations of motion under the Lorentz group is a consequence of the postulates of the theory of relativity.
The groups listed above do not, of course, exhaust all the possibilities as far as applications in physics are concerned. We shall, however, devote most of our attention to the above groups.
1.4 Invariance of equations of motion
We shall now consider the invariance of the equations of motion of a physical system with respect to transformations of its symmetry group.
In classical mechanics the motion of a system is described by Lagrange's equations. The symmetry of a physical system with respect to a given transformation group is therefore expressed through the invariance of Lagrange's equations (and additional conditions, if such exist) with respect to these transformations. Since the equations of motion written in terms of the Lagrangian L for any chosen generalized coordinates q1 are always of the same form, i.e.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
(1.1)
it follows that their invariance will be ensured if the Lagrangian itself is invariant. It is important to note, however, that the requirement that the Lagrangian should be invariant is too stringent. We know that the equations of motion remain unaltered when the Lagrangian is multiplied by a number, and a time derivative of an arbitrary function of the generalized coordinates is added to it. For example, the symmetry of the one-dimensional harmonic oscillator with respect to the interchange of coordinates and momenta (a so-called content transformation in classical mechanics) corresponds to a change of the sign of its Lagrangian
L = 1/2 p2 - 1/2 q2
In quantum mechanics the state of a physical system is described by a wave function ψ(x, t). Which is the solution of the Schroedinger equation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
(1.2)
The symmetry of a quantum-mechanical system with respect to a given group is therefore reflected in the invariance of the Schroedinger equation under the transformations in this group. If the symmetry group consists of transformations of the configuration space
x' = ux
then the invariance of the Schroedinger equation can be verified by substituting
x = u-1x', ψ'(x') = ψ(u-1x')
(1.3)
If the Schroedinger equation is invariant under the transformation u, then it should retain its form after the substitution of (1.3) in (1.2). It is clear that this will be so if the substitution does not alter the form of the Hamiltonian [??](x).
Group theory enables us to classify the states of a physical system entirely on the basis of its symmetry properties and without carrying out an explicit solution of the equations of motion. This is, in fact, the basic value of the group-theoretical method, since even an approximate solution of the equations of motion is frequently very difficult. By applying group-theoretical methods we can establish the symmetry properties of the exact solutions of these equations, and thus deduce important information about the physical system under consideration.
(Continues...)
Excerpted from Applications of Group Theory in Quantum Mechanics by M. I. Petrashen, E. D. Trifonov, S. Chomet, J. L. Martin. Copyright © 1969 M. I. Petrashen and E. D. Trifonov. Excerpted by permission of Dover Publications, Inc..
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