Linear Vector Space and Matrix Representation

Transcription

1 Linear Vector Space and Matrix Representation Lecture Objective: In this lecture we review basic ideas and concepts in linear vector space which is the language of much of modern quantum mechanics. We shall state the axioms and give matrix representations for vectors, their products and operators defined in the basis of linearly independent vectors. he bracket notation first proposed by Dirac in quantum mechanics will be used to familiarize the student with manipulations. he concepts presented here are needed for solving quantum mechanical problems using matrix mechanics, first proposed by Werner Heisenberg, Max Born and Pascual Jordan. Lecture deliverable(s): At the end of this lecture the student will be able to write down a bra and a ket vector, be able to evaluate scalar, vector and direct products of vectors and be able to represent an operator as a matrix using basis vectors. he student will be able to represent an arbitrary vector in an n dimensional linear vector space. he student will be able to calculate inner products of vectors using a coordinate representation as an integral.. Vectors in wo Dimensions: In this lecture notes we shall study, without any mathematical rigor and without any proof, the linear algebra associated with concepts in quantum mechanics that you are already familiar. here are several texts which are important and relevant. However almost all of them start with rigorous mathematical definitions associated with linear vector spaces. I have abandoned that approach and want you to recognize the basic elements first and then lower your comfort in handling these with the introduction of formal apparatus. In the end, it is important however, that you do

2 understand mathematical rigor which is at the heart and soul of quantum mechanics and quantum chemistry. In two dimensional space we have two linearly independent vectors at the most. We can choose them as forming a basis. Any other vector can be expressed as a linear combination of the basis vectors with appropriate coefficients. Let us choose as the basis the unit vectors ˆx and ŷ on a plane perpendicular to each other and represent them by column matrices (or column vectors) as xˆ ; yˆ. he following relations must be kept in mind and can be verified (I strongly suggest that you verify all the statements made in this lecture notes). Use the definition for scalar product of two vectors by the matrix product xˆyˆ (vector notation) xˆ yˆ (matrix notation) where is the transpose. Relations: ˆx ˆx ˆx ŷ ŷ ˆx ŷ ŷ (normalization) (orthogonality) (orthogonality) (normalization) he vectors ˆx and ŷ are said to be orthonormal. Next: Any arbitrary vector in two dimensions can be expanded in terms of the two basis vectors ˆx and ŷ as A ˆx a ŷ ˆx A ; a ŷ A

3 hus a is the x -component of vector A, or better, projection of vector Ain the direction of ˆx. Likewise a is the y -component of vector A, or better, projection of vector Ain the direction of ŷ. he matrix representation of the vector A is given as follows: A a a a he scalar product of two vectors follows by the same definition of the scalar product of the basis vectors. hus if we write two vectors A and B as A ˆx a ŷ B b ˆx b ŷ hen the scalar product is given as A B a b b a b a b he normalized (unit) vector  can be given by the unit vector in the direction of A as   A A A where A A a a a a a a a. herefore We shall define the direct product of two or more vectors, also known as the tensor product of vectors, later. 3

4 . Vectors in hree Dimensions: hree linearly independent vectors are needed to define any arbitrary vector in three dimensions. In matrix notation this is given by a column matrix with three rows. he three unit vectors in the mutually perpendicular directions in the Cartesian axis system, namely xˆ, yˆ and z ˆ are chosen as the basis vectors which are linearly independent and orthogonal and are represented by column matrices as ˆ x ; yˆ ; zˆ. he scalar products which define orthonormality of this basis set are: xˆ xˆ yˆ yˆ zˆ zˆ ; In column vectors, for example, ˆ ˆ x x (normalization of the basis set) and xˆ yˆ xˆ zˆ yˆ zˆ ; In column vectors, for example, xˆ yˆ (orthogonality of the basis set) Any arbitrary vector in three dimensions can be expanded in terms of the three basis vectors ˆx, ŷ and ẑ as A ˆx a ŷ ẑ ˆx A ; a ŷ A; ẑ A hus a is the x -component of vector A, (projection of vector Ain the direction of ˆx ). Likewise for a and. he matrix representation of the vector A is given as follows: 4

5 A a a 3 a he normalized vector  can be given by the symbol of a unit vector in the direction of A as   A A A where a a a a A A a a a he scalar product of two vectors follows by the same definition of the scalar product of the basis vectors. hus if we write two vectors A and B as A ˆx a ŷ ẑ B b ˆx b ŷ b 3 ẑ hen the scalar product is given as A B a b b b 3 b a b b 3 he scalar product taken in the reverse order is B A b b b 3 a b b a b 3 5

6 Note that the two products mean the same thing if these vectors represent classical quantities. hey are equal in quantum mechanics only if the components commute, namely, a i, b j for all i and j,, 3 or a b b a for all i and j. i j j i. Quantum mechanics provides many instances when the two vectors do not commute. Examples are position and momentum vectors and position and angular momentum vectors, P x x p A position vector xˆx yŷ zẑ and B angular momentum vector L x ˆx L y ŷ L z ẑ he next step is to consider direct products and use a notation first proposed by Paul A. M. Dirac and used by quantum physicists and chemists. Represent the wave function x by the symbol (he correct notation is x, but we shall ignore that for a moment and reintroduce it later) x * (or the adjoint) by (not quite, but let us do that anyway) x he integral * x dx is written as bracket. he symbol is known as the bra state and is known as the ket state. he (the vertical bar) connects or couples the two. he bracket is bra ket C Mathematically this is a very inaccurate and incomplete description so far as I have not stated the axioms and proofs that produce the above results. But you will gain confidence by using this and slowly refining the definition one step at a time. Let us use the following symbols for what we have. hus x. 6

7 x xˆ. ˆ ˆ xx x x. yy ŷ ŷ and so on. What is xx ˆˆ? It is not a scalar product, but a direct or outer product. It is defined as ˆˆ xx. ˆˆ yy and ˆˆ zz so that xˆx yŷ zẑ 33 In bracket notation, x x y y z z (operator or a matrix here). xx ˆˆ, yy ˆˆ, and zz ˆˆ are known as orthogonal projection operators. hey form the basis for expressing operators in quantum mechanics. It is not a complete basis. You need six other quantities, 7

8 xy ˆˆ, xz ˆˆ, yx ˆˆ, yz ˆˆ, zx ˆˆ and zy ˆˆ in addition to xx ˆˆ, yy ˆˆ, and zz ˆˆ to form a complete basis for operators in quantum mechanics. 3. Operators in wo and hree Dimensions: An operator O is represented in two dimensions by the set of four elements Oxx, Oyy, Oxy and Oyx. Such that O O xx O yx O xy O yy or, equivalently. = Oxx Oxy Oyx Oyy Recall that the matrix x x likewise x y. herefore y y x y. = and O O xx x x O xy x y O yx y x O yy y y in our notation. It is usually represent-ted with O, O etc in between the ket and bra states as xx O x O xx x x O xy y y O yx x y O yy y Now you can easily verify that xy 8

9 O xx x O x, or as as simple matrix product O = xx O xy O yx O yy O xy Oxx Oxy etc. Oyx O yy O xx,o xy,o yx and O yy are called the matrix elements of the operator O in the basis set x and y. Such a representation of operators will form the basis for solving for eigenvalues of the operator. In an identical manner an operator A is represented in three dimensions using the basis vectors x, y and z 3, by the matrix A where A = 3 a a 3 3 a a 3 and A itself can be expanded in terms of the nine basis vectors combinations x x, x y, x z,..., z z. Note that x x is a direct product and gives x x 9

10 he remaining eight matrices can be identified likewise. his allows us to write A as. A a 3 3 a a where a ij i Aj, i,,3, j=,,3. Generalizing this to n dimensions one can write any operator A in n dimensions (in which there are n orthonormal basis vectors,, 3... n ) as n n A= i a ij j. i, j j Let us generalize the above concepts with a slightly more formal introduction to the linear vector spaces. 4. Introduction to Linear Vector Spaces: A linear vector space V consists of a collection of vectors (infinite in number) V, V, V 3,, V n,, such that for scalars a, b, c,, etc. the following properties are satisfied:. For all and V j, V j V j V a a (a b) a b a V j a av j V j V k V j V k (associativity). here is a null vector defined such that. 3. For every, there is a such that.

11 4. An n dimensional linear vector space contains n independent linear vectors V, V, V 3,, V n such that no one of them can be expressed as a linear combination of the remaining. In other words, the equation one trivial solution n n n n. i,n n i has only 5. Any arbitrary vector in the n dimensional vector space V can be expressed as a linear combination of the n independent linear vectors above in terms of scalars such that V c i. n i 6. he dual vector of is defined by the symbol and is used to define inner product and outer product of vectors known as operators or tensors in the space of V. It satisfies relations similar to -3 above: V j V j V a a * ; a * is the complex conjugate of a (a b) a * b * a V j a * a *V j V j V k V j V k (associativity) 7. here is a dual null vector defined such that. 8. For every, there is a such that 9. he Inner product between a vector and its dual vector is defined as where denotes the norm or length of the vector in the linear vector space. he inner product is equal to the square of the norm.

12 . he inner product of any vector with the dual of another vector can also be defined in a similar manner. he result is in general a complex number. V j c ij ; Also the inner product has the property V j V j * c ij ; or V j c ij *. For any three vectors, V j and V k, and their duals, V j and V k respectively, the inner product satisfies the following properties: av j bv k a V j b V k a bv j V k a * V k b * V j V k. wo vectors and V j are said to be orthogonal if the inner product of one of the vectors with the dual of the other is zero; V j V j * 3. A vector is called normalized or a unit vector if its norm is equal to unity, End of Lecture Additional Reading:. R. Shankar, Principles of Quantum Mechanics, Second Edition, Springer, 994. (Chapter, Sec.. to.4). homas F. Jordan, Linear Operators for Quantum Mechanics, John Wiley, 969; reprinted as Dover Publications, 6. (Chapter.) 3. J. D. Jackson, Mathematics for Quantum Mechanics, W. A. Benjamin, Inc. Reprinted, Dover Publications, 6.

13 Practice Quiz /Assignment:. A vector A with components, a and in the directions of ˆx, ŷ and ẑ, respectively, is represented in a matrix form by a. b. a c. a d. a. he scalar product of two vectors in four dimensions a a 4 and b b b 3 b 4 is given by the expression he vector 5 is orthogonal to the vector. he value of is a. 5 b. - c. d. 4. he unit vector in the direction of A is 3

14 a. / 3 / 3 / 3 b. /3 /3 /3 /9 c. /9 /9 d. / 3 / 3 / 3 5. he matrix representation for the operator x y y x is, in three dimensions, a. b. c. d. 6. he density matrix for a spin ½ system is represented by the following matrix form P where P ˆxP x ŷp y ẑp z, ˆx x ŷ y ẑ z and x, i y i and z. A compact form for is given by the matrix a. P z P x ip y P x ip y P z b. P z P x ip y P x ip y P z c. P z P y P x P z d. P z P y P x P z 7. Using the four operators 4

15 ,, 3 i i and 4, as basis operators, express the matrix a a 4 4 as c i i. Determine the i coefficients c i (i, 4). 5