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Transcript

Marshall Onellion Symmetry and Phase Transition Quantum Mechanics, Symmetry and Phase Transition Quantum mechanics: In macroscopic systems of our everyday experience, objects can have any value for physical quantities such as kinetic energy or angular momentum. Physical quantities are based on the behavior of parts of an object. For instance, the angular momentum of the Earth about its axis is due to the rotation of each small part of the Earth. In addition to an object conceivably having any value of angular momentum, all possible orientations of the angular momentum vector in space are possible. Again using the Earth as an example, the rotational angular momentum of the Earth varies around the yearly orbit, and this change of orientation is the main reason for having different seasons. Finally, measuring the value of a physical quantity, if done with sufficient care, does not alter the value that an object has of that or any other physical quantity. For atomic and sub-atomic objects, such as electrons, protons, neutrons, nuclei and atoms, none of these statements hold. The kinetic energy of an electron orbiting a nucleus can have only certain specific, discrete values, and no other values. Similarly, the orbital angular momentum can have only certain values and no others. Many atomic and sub-atomic objects, including electrons, protons, and neutrons, possess intrinsic properties, including an intrinsic angular momentum, called spin angular momentum, that is unrelated to 'parts' of the object rotating. Unlike the Earth, for instance, an electron has no 'parts'. It is a single particle, with no internal structure. Nonetheless, an electron has a spin angular momentum, and the spin angular momentum of an electron has a single magnitude. For atomic and sub-atomic objects, another new concept is that of quantization axis. Put another way, this is a special spatial direction to which all vector quantities such as angular momenta are referred. Having a special direction in space can be established by, e.g., applying an external magnetic field. In addition to having only a single magnitude to the spin angular momentum, and only certain discrete magnitudes to the orbital angular momentum, the total angular momentum vector has only a limited number of discrete orientations in space with respect to the quantization axis. This idea, of only a limited, discrete number of spatial orientations, has no counterpart in macroscopic objects. In measuring physical quantities of atomic objects, still another new idea arises, called the uncertainty principle. First, let's note what this idea is not. It is still possible to measure any one physical quantity to any accuracy that the equipment provides. There is no intrinsic limitation on accuracy of one physical quantity. However, when measuring two or more physical quantities, sometimes the act of measuring the first quantity intrinsically and unavoidably changes the value of the second quantity. A good example is measuring the spatial location ( r ) and the linear momentum ^ ( p ) of an object. If I try to know the location in the x direction and the linear ^ momentum in the y direction, there is no intrinsic effect of the first measurement on the second quantity. The reason is obvious: the two directions are perpendicular, and thus 10 Marshall Onellion Symmetry and Phase Transition ^ independent of each other. On the other hand, if I measure the location x direction and ^ the linear momentum in the x direction, then the first measurement alters the value of the second. Consequently, it is not possible to, at the same time, know both quantities to arbitrary accuracy. If I denote the uncertainty in x by (x) and the uncertainty in the xcomponent of linear momentum by (px), then the combined uncertainty is always larger than some minimum value: xp x 2h (2.1) where (h) is Planck's constant (= 6.63 x 10-34 joulesecond). A different way of saying this that provides additional insight is to denote the mathematical operator representing the measurement of the x-direction location by (x) and of the x-component of linear momentum by (p x). If we use the symbol to the right to denote what is measured first, then there is a difference between [(p x) (x)] and [(x) (p x)]. This leads us to define the commutator [A, B] of the two operators (A) and (B) as [A,B] = (AB - BA), and in the case of location and linear momentum we find that: x , px i (2.2) where i 1 . This commutator idea allows us to express the idea that some physical measurements affect other, different physical quantities. The commutator also allows us to learn the limitation of what we can know. To take a specific example, consider an electron orbiting a nucleus with an orbital angular L . For an everyday, macroscopic, object, we would expect to know all three components of the vector L . However, we find that the components Lx, Ly, and Lz of the momentum vector obey commutator relations: Li , L j ijk Lk (2.3) where ijk = +1 if (ijk) is an even permutation of (123), -1 if (ijk) is an odd permutation of (123), and 0 if (ijk) contains two or more of i, j, or k equal to each other. Notice that we cannot know two, let alone all three, components of L . However, since the magnitude of L , |L|2, commutes with all three components, we can measure |L|2 and one component, but no more than that. The result in atomic and sub-atomic systems is qualitatively different from macroscopic systems, in which we only know a vector if we know all three components 11 Marshall Onellion Symmetry and Phase Transition of the vector. In atomic and sub-atomic systems, for orbital angular momentum, by contrast, we cannot know all three components at the same time. For macroscopic objects, a given object at a particular time has only one possible value for various physical quantities such as kinetic energy, potential energy, linear momentum, angular momentum, etc. In atomic and sub-atomic systems, however, qualitatively new behavior can arise. For instance, let us suppose that there is an electron orbiting a nucleus as part of an atom, and the electron is in the first excited state. Using (n) to denote the energy level, with n= 1 the ground state, the electron is in the n= 2 state. Now, let's further suppose that there are two orbital angular momentum states having this same n=2 state energy, say the 2s and the 2p states (|L|2 = 0 for the s state and |L|2 = (2)1/2ħ for the p state). This situation would arise, for instance, if the spin-orbit interaction was negligible. Now, the total energy and the total orbital angular momentum can be known at the same time, so an electron in a n=2 energy state could be in a 2s state, or a 2p state, or in a mixed state, that is, any linear combination of 2s and 2p state. If the electron is in such a mixed state, and we measure the electron total energy, we do not change the electron state. However, what happens if we measure the total angular momentum of such a mixed state? For a macroscopic object, nothing happens, as long as we are not altering the energy, because in classical physics the act of measurement does not alter the state of the object (in this case, an electron). However, the two angular momentum states are independent of each other, so when the angular momentum is measured two things happen. First, we only get one answer, either (s) state or (p) state. Second, the act of measurement 'forces' the electron into the single angular momentum state measured. This means that if we measure and find a (s) state, then subsequent measurements of that same electron will yield a 100% chance of finding a (s) state. This change, from a mixed state to a 'pure' state, that is, to a single angular momentum value, is called the collapse of the wavefunction. In quantum mechanics, the wavefunction ((x,y,z,t)) represents everything that can be known about the object or system in question. The wavefunction has the same symmetry as the total energy, the Hamiltonian () of the physical system. Let's consider an example. Single, free electron: A 'free' electron means there is no potential energy, and no force, involved. So the Hamiltonian consists solely of the kinetic energy of the electron: () = p2/ 2m (2.4) where ( p ) is the electron linear momentum and (m) the electron rest mass. For this Hamiltonian, it does not matter where we place the original of the coordinate system. Further, if we translate the coordinate system, we still get the same linear momentum. The formal way this is described is to say the system has translational invariance. If the Hamiltonian has translational invariance, so does the wavefunction, because the wavefunction 'sees' the same energy term no matter where the coordinate system origin is placed. This is an example of the wavefunction having the symmetry of the Hamiltonian. 12 Marshall Onellion Symmetry and Phase Transition Another aspect is that for a given energy (E), a given value of the Hamiltonian, there is only one wavefunction having that energy. Mathematically this is expressed by saying: = E (2.5) You will read in physics texts terminology that () is an eigenfunction of the Hamiltonian with eigenvalue of (E). Now suppose that there is some other physical quantity represented by the mathematical operator (), and further that measuring () does not alter the energy, or vica-versa. Another way of saying this is to say that () and () commute. If this is so, if [,] = 0, then: () = () = (E ) = E () (2.6) The only way that both equations can be valid is if () is proportional to (); that is, that () is an eigenfunction of both () and (). This means that the two operators must have the same, or at least consistent, symmetry. We need to learn two sidebars before talking about group theory: complex numbers and matrices. The reason is simple necessity. Complex numbers appear constantly in describing the world, so we must learn a little about them. Matrices are often the most useful way to actually write out the elements of a group, and by doing so we learn as much as possible about the group elements. Complex numbers: A 'complex number' is a number of the form (a + ib), where (a) and (b) are real numbers and i 1 . Complex numbers have the following arithmetic: (a) Addition: (a + ib) + (c + id) = (a+c) + i (b+d); (b) Multiplication: (a + ib) (c + id) = [ac- bd] + i[ad + bc]; a ib a ib c id ac bd i bc ad . (c) Division: c id c2 d 2 c2 d 2 c2 d 2 (d) Another term we will encounter is the 'complex conjugate' of (a + ib), which is (a ib). Homework: Recall that an exponential function (ex) has properties that include: (ex) (ey) = (ex+y). Let me now mention that the exponential function can be written as an infinite series: e x n x n 1 n! (2.7) where n! is called n- factorial and n! = n(n-1)(n-2)…(3)(2)(1). Recall that 0! = 1. Let me further mention that the sine and cosine of an angle can be written in terms of infinite series as well: 13 Marshall Onellion 1 sin() Symmetry and Phase Transition n 1 n 1 n! 1 cos() n 1 n 1 (2.8) 2 n 1 2 ( n 1) (2.9) n! The problem is to prove that (ei) = cos() + i sin() (2.10) Matrices: A matrix is, for our purposes, a (N x M) array of numbers, where (N) and (M) are positive integers. We will limit our discussion to square matrices (N = M). Square matrices of the same rank ('rank' means the number of rows or columns) can be multiplied together. Let (A) and (B) be (N x N) matrices with elements (aij) and (bij), where i, j = 1, 2, …, N. Then the product of (A) (B) = (C) = (cij) is: c a b N ij k 1 ik (2.11) kj Example: 1 0 1 2 1 2 B C A 0 3 3 4 9 12 There are several properties of matrices we will find essential. (1) Trace of (A), written Tr(A), is the sum of diagonal elements. In the example above, Tr(A) = 4, Tr(B) = 5, and Tr(C) = 13. (2) Determinant of matrix (A), written Det(A), det(A), or |A|. If (A) is a (2x2) matrix, then: det(A) = (a11 a22 ) - (a12 a21) (2.12) 14 Marshall Onellion Symmetry and Phase Transition If (A) is a (3x3) matrix, then: det( A) a11 a a a a 22 23 32 33 a12 a a a a 21 23 31 33 a13 a a a a 21 22 31 32 (2.13) In the example above, det(A) = 3, det(B) = -2, and det(C) = -6. Notice that det(AB) = det(C) = [det(A)] x [det(B)]. This is always the case with matrix multiplication. (3) Diagonal matrix: This is a matrix in which only the diagonal elements are non-zero. Notice I did not say that all the diagonal elements are non-zero, merely that all the offdiagonal elements are zero. 1 0 0 Example: C 0 2 0 is a diagonal matrix. 0 0 3 (4) The inverse of a matrix (A), written as (A)-1, has the property that: (A) (A)-1 = (A)-1 (A) = (I) where (I) is the identity matrix, having diagonal terms all equal to (1) and all off-diagonal terms equal to (0). (5) The determinant of a matrix arises in many places, among them solving systems of linear equations. Example: 2x + 3y = 4 (2.14) x - 5y = 6 2 3 which can be expressed as: 1 - 5 x 4 x (A) y 6 y (2.15) Det (A) = -13. We can calculate (A)-1 and find: A 1 1 5 3 13 1 - 2 (2.16) x 4 Then : A can be solved via: y 6 A A xy A 64 131 38 8 1 1 (2.17) 15 Marshall Onellion Symmetry and Phase Transition Notice how det(A) = -13 appears in the problem. (6) If a matrix (A) = (aij), then: (i) The transpose of (A), denoted (A)T = (bij) = (aji); (ii) The adjoint of (A), denoted (A)+ = (bij) = (aji)*, where (aji)* is the complex conjugate of (aji). (iii) A 'Hermitian' matrix (H) has the property that (H)+ = (H), that is, the adjoint of the matrix is the same matrix. This will turn out to be quite important-- the matrices that represent physical observables will be Hermitian matrices. (iv) A 'unitary' matrix (U) has the property that (U)+ = (U)-1 . (7) Let (A) = (aij) be a matrix. One of the most widely encountered transformations of a matrix is a 'similarity transformation'. A similarity transformation is a matrix (C) defined as: (C) = (B)-1 (A) (B) where (B) is any matrix having an inverse. There are some properties of (A) that are invariant under a similarity transformation, including: (i) The trace of (A), Tr(A). (ii) Any matrix equation. That is, if (A) (B) = (C) and: (A)S = (D)-1 (A) (D) (B)S = (D)-1 (A) (D) (C)S = (D)-1 (A) (D) (2.18) then: (A)S (B)S = (C)S . (8) Diagonalizing a matrix. Let (A) = (aij) be a matrix. Define the secular equation of (A) as: a a 11 a12 ... 21 A I ..... ..... a 22 ... (2.19) .................................a nn Apply a similarity transformation to (A): (B) = (S)-1 (A) (S) Look at the secular equation of (B). This turns out to be: |B - I| = 0 = |A- I| (2.20) 16 Marshall Onellion Symmetry and Phase Transition That is, the solution (the 'roots') (k) of the secular equation, called the eigenvalues of the matrix (A), are invariant under a similarity transformation. In our brief discussion of quantum mechanics, we encountered equation of the form: Av E v (2.21) This is an eigenvalue equation. In the same way, matrices will be used to represent physical quantities such as (A), and the (k) will be the solutions--what we measure--of the physical quantities represented by (A). Basic calculus: Add this discussion. Points to make: * derivative of single variable * integral of single variable. Functionally, integral and derivative are linked * common derivatives: power law, exponential, trigonometric * partial derivative of multiple variable function Derivative: The basic idea of a derivative is a ratio: ratio = change of function/ change of variable Sometimes, this ratio never approaches a single value. For instance, the series of numbers (1, 0, 1, 0, 1, 0,…) never approaches (1) nor (0). However, sometimes this ratio does approach a single value as the change of variable gets small, and approaches a single value independent of exactly how the variable gets small. For such cases, the ratio is called the derivative of the function at a value of the variable. Let's look at some examples. 1. f(x) = 5. The function f(x) is a constant, independent of the variable (x). This is easy: the change of the function is zero, independent of how I change the variable, so the derivative of f(x) is zero. The derivative of f(x) with respect to (x) is written as df/dx. 2. f(x) = 2x Let x = x1 go to x = x2. f(x1) = 2x1 and f(x2) = 2x2. Then the ratio = (2x2 - 2x1)/(x2 - x1) = 2 for all changes in (x). This means that df/dx = 2. For a linear function, the derivative is identical to the slope of the line. As we will see shortly, for other functions the derivative is identical to the tangential line of the function. 17 Marshall Onellion Symmetry and Phase Transition 3. f(x) = 4 x2 . Let (x) change from (x1) to (x2). f(x1) = 4 (x1)2 and f(x2) = 4(x2)2, so: 4 x x x x ratio 4 x 4 x 4 x x x x x x 2 2 2 1 2 1 2 1 2 2 1 2 1 (2.22) 1 Now let x2 get close to x1. As x2 x1, the ratio 8x1 . As long as x2 x1, it does not matter exactly how, the ratio 8x1 . This means that df/dx = 8x, which is the slope of the tangential line of f(x). 4. In general, if f(x) = a xb, then df/dx = (a) (b) xb-1 . This is true regardless of whether (a) and (b) are integers. For instance, suppose: f ( x) 3 x 3x 1/ 2 df 3 1/ 2 (3)(1 / 2) x dx 2 x (2.23) Partial derivative: Suppose f(x,y) is a function of two variables, with the variables (x) and (y) independent of each other. What is the: change of f ratio change of y (2.24) y Treat (x) as a constant (ratio)y = 3. Similarly (ratio)x = 2. These are written as the symbols (f/y)x and (f/x)y , and are called the partial derivatives of f with respect to (y) and with respect to (x). Derivative fails: When does the idea of a derivative fail? For example, it fails for the function: 2 x 0 (2.25) f (x) 1 x 0 At or near x = 0, there is no single answer for the ratio, which means the derivative is undefined at x = 0. At all other x, the ratio is well defined, so there is a derivative. The derivative is zero, but it is well defined. As another example: 2 x x 1 f ( x) 3x 1 x 1 (2.26) Notice that this function is well defined at x= 1, f(1) = 2, but the slope of the line for x < 1, namely (2), is different from the slope (3) for x > 1. The lesson here is that f(x) must be continuous in order to have a derivative. 18 Marshall Onellion Symmetry and Phase Transition Vocabulary of functions: A function, in general, is a map from one set to another set . There is some terminology to learn. 1-to-1 function: A 1-to-1 function means exactly one element of is mapped to an element of . Examples: Suppose is all real numbers between (-1) and (+1), written Re{[-1,1]}, and that is all real numbers. (a) f(x) = x is a 1-to-1 function. (b) However, f(x) = |x| = absolute value of (x), is not 1-to-1, since both (x) and (x) are mapped to |x|. Onto: A function f: maps onto if every element (y) in has some (x) in so that y = f(x). Examples: Suppose = = Re{[0,1]}. f(x) = x maps onto . However, g(x) = (x/2) does not map onto . Isomorphism: and are isomorphic if there is a 1-to-1 mapping of onto . Homomorphism: and are homomorphic if there is a many-to-1 mapping of onto . For example: x 0 x 1 f ( x ) x - 1 1 x 2 x - 2 2 x 3 (2.27) is a homomorphism of Re{[0,3]} onto Re{[0,1]}. Group Theory For us, the purpose of group theory is to analyze the symmetry of a system and the types of phase transitions that the system can undergo. In this part of chapter 2, I closely follow Group Theory and Quantum Mechanics by Michael Tinkham, and the reader will only benefit by studying this classic and superbly written book. Let's start with the definition of a group: A set of elements = {A, B, C,…} and a mapping function (*) that maps any pair of elements into the set, with the following properties: (1) If (A) and (B) are elements of , then (A*B) is an element of . (2) Associative law holds: (A*B) * C = A * (B*C). (3) There is an unit element (I) in with the property that for any element A in : A* I = I* A = A (4) For any element A in , there is an element A-1 in with the property: A * A-1 = A-1 * A = I 19 Marshall Onellion Symmetry and Phase Transition We will use only finite groups, so the order of the group, o(), (the number of elements in the group) is well-defined. You probably noticed that commutatively is not a group property. Groups that are commutative are called Abelian groups. As an example of a group, consider all the symmetry operations of an equilateral triangle as shown in Figure One. Suppose that you operate on the triangle by two operations (A*B), so first (B), then (A). What is the resulting operation? Table I lists the results. A B C D E 1st element I I I A B C D E A A I D E B C B B E I D C A C C D E I A B D D C A B E I E E B C A I D Notice, as you can confirm by playing with the figure, that the group is not Abelian. For example: (C*A) = D (A*C) = E There are several aspects of groups in general worth mentioning. One type of group that is Abelian is a cyclic group: Let (A) be a member of . Consider the set {A, A2, A3, …}. We know that for some power (n), An = I, so the group can be written as {A, A2, A3, …,An = I}= S. Let o(S) be the order of S. If o(S) = o(), then the cyclic group is the entire group . But what if o(S) < o()? In answering this question we learn several things. First, notice that S is itself a group. A subset S of a group that is itself a group is called a subgroup of the larger group . In trying to find a relation between o(S) and o(), we would, if we were mathematicians, eventually find a general theorem: The order of a subgroup is a divisor of the order of the group. This is important for phase transitions for several reasons. We have already mentioned that a phase transition always involves a change of symmetry. We also have discussed that changing to a state of higher order means a state of lower symmetry. If the phase transition is first order--involving a finite change of energy--there is no symmetry relation between the two phases. However, if the phase transition is second order, it means, in 20 Marshall Onellion Symmetry and Phase Transition terms of group theory, changing from a symmetry group to a subgroup. Consequently, the list of possible second order phase transitions is the same as the list of subgroups of the original group (i.e., the original symmetry state). Another result about phase transitions: What if the o() is a prime number? This means that the only divisors of o() are itself and (1). This means that there are no subgroups, and thus no second order phase transition of a system described by such a group. Indeed, we would further learn that: Any group of prime order is a cyclic Abelian group. Conjugate elements: Let (A) and (B) be elements of a group . We say that B is conjugate to A if there is some element (X) in such that B = X*A*X-1 . We also say that a class of group elements are mutually conjugate elements. As a first example, let's consider the equilateral triangle symmetry operations above. What are the classes of that group? There are six elements {I, A, B, C, D, E}. The classes are: {I}, {A, B, C} and {D, E} You should make sure you can prove this statement. Physically, elements of the same conjugate class are in some sense the 'same type' of symmetry operation. We will find that if the elements in a group are represented--written--as matrices, then all elements in the same class have the same matrix trace. New Terms: The classes of a group will be used in discussing how groups are represented--written--as matrices. Writing group elements as matrices is often the way to learn the most about group symmetry properties. There are some other terminology that will also appear. Coset: Suppose = {I, S2, S3, …, S} is a subgroup of group . o() = and o() = g. For any element (A) in , consider the set A = {I*A, S2*A, S3*A, …, S*A}. If (A) is in , then A = . If (A) is not in , we call A a right coset , and we call A = {A*I, A*S2, A*S3 , …, A*S} a left coset. Notice that a right coset or a left coset is not a group since (I) is not in either set. Theorem: A right coset A or a left coset A contains no element in common with . Theorem: Two right, or left, cosets are either identical or have no common element. Complete class: Suppose is a subgroup of a larger group . Suppose further that if (A) is in , then all elements of the form (X-1*A*X) are in , even if (X) is in but not in . Such a class containing (A) is termed a complete class. 21 Marshall Onellion Symmetry and Phase Transition Invariant subgroup: This is a subgroup of a larger group consisting entirely of complete classes. You should be able to prove that a right coset A and a left coset A of are identical when is an invariant subgroup. Class multiplication: Although it seems at first glance simply abstract mathematics, this idea is used quite often in figuring out what symmetry changes can occur. Suppose we have two classes and of a group . If I tell you that the two classes are equal, I mean that each element appears as often in as in . Theorem: Using this notation, X-1X = if and only if is any complete class of the group and (X) is any element in . Make sure you can prove both sides of this equivalence. Now, let's use some of this new terminology. Suppose we have two classes i and j . What does multiplying two classes mean? Notice that since i and j are classes: X X 1 i and i X X 1 j (2.28) j for all element (X) in . Then the product of I and j , written (i j), is: X X X X 1 i j 1 i (2.29) j for all elements (X) in . This implies that (i j) consists of complete classes. If we express the set (i j) by writing it in terms of the complete classes: c i j ijk (2.30) k k where (cijk) is an integer, denoting how often (k) appears in (i j). As an example, let's analyze the equilateral triangle. Let 1 = {I}, 2 = {A, B, C}, and 3 = {D, E}. Then: 12 = 2 13 = 3 (2.31) 22 = 31 + 33 23 = 22 Factor group: Consider a subgroup of a group . Since a coset of contains no element in common with , and that two cosets are either identical or distinct, it follows that the number of distinct cosets is [o() / o()]. Now suppose that is an invariant subgroup, meaning that if (A) is in , then B-1*A*B are in for all (B) in the larger group . Consider a new group, the factor group, consisting of all cosets of . In this factor group is the identity element. Group multiplication means: 22 Marshall Onellion Symmetry and Phase Transition BiCj = (Bi) (Cj) = Bi Cj = (BiCj) = coset of (BiCj) (2.32) Group Representations: Technically, a group representation is any group that is homomorphic to the original group. Recall that a homomorphism is a many-to-1 map from the representation group onto the original group. We will use representation groups consisting of square matrices. In representing a group by a set of matrices, we will use matrix multiplication as the group multiplication operation. So if elements (A) and (B) are in a group and (A) and (B) are the matrices representing (A) and (B), the matrix representing (A*B) is denoted (A*B) and is: (A*B) = (A) (B) (2.33) The number of rows and columns of the matrix is called the dimensionality of the representation. We went over the idea of a similarity transformation above. Note that any similarity transformation leaves the matrix equation unchanged: (2.34) ( A) (B) S S S 1 ( A)S S (B)S S ( A* B)S S ( A* B) 1 1 For this reason, group representations connected by a similarity transformation are equivalent. There is, however, one crucial difference between representations, namely reducible versus irreducible representations. Suppose (A) is in , and both (1)(A) and (2)(A) are matrix representations of (A). Notice that a representation- a matrix- of (A) written as: (1) 0 ( A) ( A) (2) 0 (B) (2.35) is also a representation of (A). (A), however, can be reduced to (1)(A) and (2)(A), so (A) is called a reducible representation. In general, a set of matrices representing a group is reducible if all the matrices can be changed to block diagonal form using the same similarity transformation. To note the structure of a reducible representation, write as consisting of certain irreducible representations. In the above example: = (1) + (2) (2.36) and in general: ai (i ) (2.37) i 23 Marshall Onellion Symmetry and Phase Transition where (ai) denotes how often (i) appears in , and Eq. (2.37) is not matrix addition. The number of irreducible representations has important physical implications. For atomic and subatomic systems, where quantum mechanics applies, the irreducible representations have the same symmetry properties as do eigenfunctions of the Hamiltonian. Different irreducible representations correspond to different eigenvalues, that is, to different energy levels. To illustrate this idea, consider an electron in a d-orbital around a nucleus. Because the Coulomb interaction between nucleus (positively charged) and electron (negatively charged) is radial, the interaction has the same symmetry as that of a sphere. All five dorbitals have the same energy. Since there is the same energy level for all d-orbitals, there is only one irreducible representation. Consider, though, what happens in an ordered solid. The highest symmetry ordered solid is a cube. We will cover crystal symmetry in more detail below, but for now the important point to note is that if there is an interaction between the electron and the nearest neighbor nuclei, the symmetry is lowered, from that of a sphere to that of a cube. Specifically, the five d-orbitals that were degenerate- degenerate means they have the same energy- now split into one group of two and a second group of three orbitals. Correspondingly, the single irreducible representation changes to two irreducible representations. Figure Three illustrates this in a simple way. With spherical symmetry, all d-orbitals have the same energy, while with cubic symmetry there is a splitting into two different energy levels. If the symmetry is lowered further by other interactions, or lower crystal symmetry, the two-fold and three-fold degeneracies are lifted. With sufficiently low symmetry, each d-orbital has a different energy. We can look at the d-orbital example from another perspective. If we measure the energy value(s) of the d-orbital electrons, the number of different energy values we measure is the number of irreducible representations. Notice that in Figure 3 we have two different energy levels for cubic symmetry but only one for spherical symmetry. That is, the symmetry controls and determines the number of energy levels. At the same time, the symmetry tells us nothing about the size, the strength (), of the interaction. Group theory and quantum mechanics: Before we continue with group theory, consider the way group theory and quantum mechanics are interrelated. Specifically, consider a group of symmetry operations that leaves the Hamiltonian invariant. The elements of the group may include rotation of the coordinates, reflections, inversions, or 24 Marshall Onellion Symmetry and Phase Transition combinations of the three. Even more specifically, consider a real orthogonal coordinate transformation (T): x T x ' ij i (2.38) j j The fact that we are consider (T) as a real, orthogonal matrix means: T-1 = T+ (2.39) so that the inverse of (T) is the transpose of (T). This means that: x T x T x 1 ij i ' ' ji j j (2.40) j j Next we consider a new group that is isomorphic to the coordinate transformation group. The elements (PT) of the new group are transformation operators that operate on functions: PTf(Tx) = f(x) PTf(x) = f(T-1x) (2.41) To show that {PT} is isomorphic to {T}, we must show that PSPT = PST : PTf(x) = f(T-1x) = g(x) PS[PTf(x)] = g(S-1x) = f(T-1(S-1x)) = f((ST)-1 x) = PST f(x) (2.42) Let's consider {PT} that commute with the Hamiltonian, that is, those that leave the Hamiltonian invariant. An example would be the Coulomb potential energy, which is radial. Because it is radial, any rotation or reflection leaves the potential energy invariant (unchanged). We'll call the symmetry operations leaving the Hamiltonian invariant the group of the Schrödinger equation. What can we learn from such operations? PT n = PT (En n) or (PT n) = En (PT n) (2.43) This means that (PT n) is also an eigenfunction of with the same energy eigenvalue. Two independent eigenfunctions with the same energy are called degenerate. A normal degeneracy is one where the symmetry of the system (PT n) yields all degenerate eigenfunctions. There are also accidental degeneracies, seemingly not explicable by symmetry. These are typically not degeneracies at all when all interactions are considered, or are due to a subtle symmetry of the Hamiltonian. 25 Marshall Onellion Symmetry and Phase Transition Suppose for an eigenvalue (En) the degeneracy is (ln). This means that there is a set of (ln) orthonormal functions having energy eigenvalue (En). These (ln) functions form the basis vectors in a (ln) dimensional vector space. Such a vector space is a subspace of the entire vector space of eigenfunctions of the Hamiltonian (). The subspace is invariant under the group of the Schrödinger equation. This means: P (n) T ln (n) (n) k (T ) (2.44) k k 1 where {k(n) } are the (ln) degenerate eigenfunctions. The ln - dimensional matrices k(n) form a (ln) dimensional representation of the group of the Schrödinger equation. To show this: PST = PS [PT ] = PS k k (T)k = k (PS k ) (T)k = k (S)k (T)k = [(S) (T)] = (ST) (2.45) which implies that (ST) = (S) (T) . Also, note that if {k(n) } are chosen to be orthnormal, then the representation {(n) } is unitary. What if we use a different set of basis function {}? l 1 k l or 1 P P l T R 1 1 k k (T ) k k , 1(T ) S (T ) (T ) 1 k ,k , k (2.46) where: S(T) = -1(T) (2.47) Thus, within a similarity transformation, there is a unique representation of the group of the Schrödinger equation for each eigenvalue of the Hamiltonian. Notice that the degeneracy is the dimensionality of the representation. By finding the dimensionality of all irreducible representations of the group of the Schrödinger equation, we find the non-accidental degeneracy of a particular Hamiltonian. For example, suppose the group of the Schrödinger equation has order = 6, such as the symmetry elements of an equilateral triangle. We know the order of any subgroup is a divisor of (6), and that: l 2 j h6 (2.48) j 26 Marshall Onellion Symmetry and Phase Transition It turns out for an order = 6 group, the dimensionalities of the subgroups are: 12 + 12 + 22 = 6 Consequently, we have two non-degenerate representations and one doubly degenerate representation. Another consequence concerns perturbations- small changes in the interactions. A perturbation lifts degeneracies when including the perturbation in the Hamiltonian reduces the symmetry group of the Hamiltonian. There are other illustrative examples. One is any Abelian group. Each element forms a class by itself, which means the group has only one dimensional representations and thus no degeneracies. One dimensional representations are merely complex numbers. Another example is a cyclic group {A, A2, …, Ah-1, I}. Suppose the number representing (A) is (r). Then: rh = 1 (2.49) which implies that r = exp(2ip/h), where p = 1, 2, 3, …, h. Cyclic groups are the symmetry groups of, for instance, a ring with (h) equal parts, or a solid with translational invariance--a straight line--having periodic boundary conditions every (h) periods. Let (A) represent a translation of one period of length (a). Then: PA s(x) = s(x+a) = (s)(A) s(x) = e2is/h s(x) (2.50) Defining L = ah, we have: s(x+a) = e2is/h s(x) = eika s(x) (2.51) where k= 2s/L. Any wavefunction satisfying (2.51) can also be written as: k(x) = uk(x) eikx (2.52) In solid state physics, this is known as Bloch's theorem and is based entirely on symmetry. Another example involves the two dimensional rotation group. Suppose we have rotations about a single axis. Such rotations form an infinite Abelian group. The group multiplication for successive rotations (1) and (2) is: (1) (2) = (1 + 2) (2.53) 27 Marshall Onellion Symmetry and Phase Transition which implies: () = eim (2.54) Now, (0) = (2) = (4) =…=1. This means that (m) is restricted to only integer values: 0, 1, 2, … Physically, this means that angular momentum along a symmetry axis is conserved and has a "good quantum number" (m). Changes of symmetry alone lead to changes in both degeneracy and energy levels. Irreducible representations lead to "good quantum numbers" in physical situations. If the symmetry group contains non-commuting elements, there will be representations having dimensionality greater than (1) that may lead to degeneracy. We need more information on representations. There are several results worth noting. (1) Any representation by matrices having non-zero determinants is, through a similarity transformation, equivalent to a representation by unitary matrices. (2) Any matrix that commutes with all the matrices of an irreducible representation is a multiple of the identity matrix. (3) Suppose we consider two irreducible representations (1)(A) and (2)(A) of the same group. These representations have dimensionality (l1) and (l2). Suppose further that there is a rectangular matrix (M) such that: M A A M (1) (2) i (2.55) i Then: (a) If l1 l2, M = 0. (b) If l1 = l2, and if det(M) 0, then the two representations are equivalent. (4) Orthogonality theorem: Let (i) denote all the inequivalent, irreducible, and unitary representations of a group , (h) equal the order of the group , and (li) the dimensionality of (i) . Then if I sum over the elements (R) of , I find: (i)R ( j)R l * h (2.56) ij R If we view i (i)R as a "vector" in the h-dimensional "space" of the group elements, then the vectors are orthogonal in the group element space. From this we also find: (a) For a (li)- dimensional matrix, there are (li)2 entries. Summing over all the 2 irreducible representations (i), the number of orthogonal vectors is l i . In a hi dimensional space, the maximum number of orthogonal vectors is (h). In fact: 28 Marshall Onellion Symmetry and Phase Transition l h 2 (2.57) i i Characters: Now, two equivalent matrices, related by a unitary similarity transformation, are not identical. What do they have in common? They have the same trace. This leads to the character of a representation, which is the set of (h) numbers { (j)(Ai), i= 1, 2, …, h} where: ( j) lj (j) ( R) Tr ( R) ( j ) 1 R (2.58) Further, all elements in the same class have the same character. This means we can state the character of a representation by listing the character of one element from each class. Writing the kth class of a group as Єk, we denote the character by (j) (Єk). We can rewrite the orthogonality theorem in terms of characters: R (i) R * (j) R h ij (i)k N h * (j) k k (2.59) ij k where Nk is the number of group elements in class consequence of (2.59) is: k . Another importance Number of irreducible representations = Number of classes Now consider two square matrices Q1 and Q2. (2) (1) 1 * N1 1 * N1 ..... h h (1) (1) ...... 1 2 (2) (1) (2) * N2 2 * N2 (2) 2 .... Q1 1 2 ..... Q2 h h ............ .............. ..... .......................... ....................... .......... Multiply these together: 29 (2.60) Marshall Onellion Symmetry and Phase Transition N Q1Q2 k h k * ( j) (i) k ij ij (2.61) k This means that Q2 = (Q1)-1 so that: h k N (i ) (i) l i k (2.62) kl Character Table: Using these results, we display the characters of representations in a character table. The columns are the classes and list the number (Nk) of group elements in each class ( k). The rows are the irreducible representations. The entries are the characters (i ) ( k ). The rows of the character table are orthogonal as per Eq. (2). The columns make orthogonal vectors. There is a 'recipe' for constructing a character table. (1) Find the number of classes of the group, which is also the number of irreducible representations. (2) Use Eq. (2.57) to find the dimensionalities of the irreducible representations. The identity element is represented by a unit matrix, so the trace of a matrix of the identity class is (li), which determines the first column of the character table. Further, we always have the totally symmetric representation in which every group element is represented by (1), so the first row is: 1 . (1) k (3) Use the fact that the rows of the character table are orthogonal, normalized to (h = order of group), and have weighting factors (Nk) as per Eq. (2.59). (4) Use the fact that the columns of the character table are orthogonal, normalized to (h/Nk) as per Eq. (2.62). (5) Finally, if needed, the elements within a given row, say the ith row, are related: N N l c N (i ) j (i ) j k (i ) k i jkl l l (2.63) l where the (cjkl) coefficients are the constants obtained through class multiplication by using Eq. (2.30). Let's apply the recipe to finding the character table of our equilateral triangle group. Items (1) and (2) tell us: 30 Marshall Onellion (2) (3) (1) Symmetry and Phase Transition 1 3 2 1 a c 1 1 2 2 3 1 B D The first and second columns are orthogonal: 1 + a + 2c = 0 The first and third columns are orthogonal: 1 + b + 2d = 0 The first and second rows are orthogonal: 1 + 3a + 2b = 0 (2.66) The first and third rows are orthogonal: 2 + 3c + 2d = 0 (2.67) (2.64) (2.65) Use Eq. (2.59) for the second column: 12 + a2 + c2 = (6/3) = 2 (2.68) This means that either (i) a = 0, c = 1 or (ii) a = 1, c = 0. Compare the result of Eq. (2.68) with Eq. (2.64) a = -1, c= 0. Use Eq. (2.59) for the third column: 12 + b2 + d2 = (6/2) = 3 (2.69) This means that b = 1, d = 1. Compare the result of Eq. (2.69) with Eq. (2.65) b = 1, d = -1. The result is: (1) (2) (3) 1 1 1 2 3 2 1 -1 0 31 2 3 1 1 -1