1 The Dirac notation for vectors in Quantum Mechanics

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1 This module aims at developing the mathematical foundation of Quantum Mechanics, starting from linear vector space and covering topics such as inner product space, Hilbert space, operators in Quantum Mechanics and their matrix representation, postulates of Quantum Mechanics, Heisenberg s uncertainty relation and a particle in a quantum well having infinite potential These lectures will help the reader to understand the quantum mechanical treatment of optical near fields in a much better way In order to define a linear vector space, one has to first define a set of vectors In this lecture, the Dirac notation for vectors is dealt with The Dirac notation for vectors in Quantum Mechanics Any given vector, say V, can be completely defined if and only if all its components are specified For example, if e i are a set of unit vectors and if v i are the corresponding components of V (, then one can write V= v i e i Thus, one can efficiently implement all vector operations, namely, addition, scalar multiplication, etc, in terms of the components v i of V It can be well established that for a given set of unit vectors, the choice of the components v i of V is unique Hence there exists a one-to-one correspondence between any given vector and its tuple of components such that there exists unique -tuple of components v i for each vector V and vice versa By exhibiting this unique -tuple of components v i into a column vector, one can mathematically represent this correspondence as follows: V v v 2 v ( Again, as an adjoint form of Eq(, V can also be represented by another distinct -tuple, namely, the row vector, shown as: V ( v v 2 v (2 Bra and Ket notations in Quantum Mechanics In order to identify the aforementioned two distinct forms of the same abstract object V in Quantum Mechanics, Dirac assigned them as Ket V represented mathematically as V and Bra V represented mathematically as V Hence in Quantum Mechanics, Eq( is assigned as V ( Ket V" which is represented mathematically as V v v 2 v, (3 and Eq(2 is assigned as V ( Bra V" which is represented mathematically as V ( v, v 2,, v (4 Joint Initiative of IITs and IISc - Funded by MHRD Page 3 of 9

2 PTEL - anotechnology - anophotonics (Beyond the Diffraction Limit Thus to each ket, there exists a unique bra and vice versa More over, if α is any complex constant, then αv αv 2 α V, (5 αv and having the corresponding adjoint as α V ( α v, α v 2,, α v (6 2 Linear vector space A set of vectors, namely, { V, V 2, V 3, } can constitute a Linear vector space V if and only if they satisfy the following properties such as the set of vectors in V yield only the same vectors in V whenever the operations of addition and scalar multiplication are performed on each and every vector in V, with each vector in V obeying the following axioms given as V i + V j = V j + V i, known as commutative property of addition, ( V i + V j + V k = V i + ( V j + V k, known as associative property of addition, existence of a unique null vector / in V such that V i + / = V i = / + V i, thereby existing as an identity element of addition, existence of a unique inverse V i in addition such that V i + V i = /, α ( V i + V j = α V i +α V j, pertaining to scalar multiplication, (α+ β V i =α V i +β V i, also pertaining to scalar multiplication and α(β V i =(αβ V i, also pertaining to scalar multiplication Here, V i, V j and V k are arbitrary vectors in the linear vector space V The linear vector space can be a complex vector space or a real vector space depending on the domain of allowed values of all scalars defined over the linear vector space 2 Linear independence and linear dependence of a set of vectors in a linear vector space Out of the given set of vectors { V, V 2,,, 2,,, } constituting a linear vector space V, if there exists a linear relation of the form b j j = /, (2 j= Joint Initiative of IITs and IISc - Funded by MHRD Page 4 of 9

3 PTEL - anotechnology - anophotonics (Beyond the Diffraction Limit where / is a null vector, and if Eq(2 is satisfied only for the case for which all b j =, then such set of vectors {, 2,, } are called linear independent vectors Any other arbitrary vector V j in the linear vector space V that can be expressed as a linear combination of these linear independent vectors represented as V j = a i i, (22 is called a linear dependent vector The choice of the coefficients a i is unique for a given set of linear independent vectors 22 Dimensionality of linear vector space The dimensionality of a linear vector space or linear vector space is decided by the maximum number of linear independent vectors in that linear vector space Thus if there are at most number of linear independent vectors, the linear vector space is dimensional 3 Basis For a given set of linear independent vectors {, 2,, } in the linear vector space V, any arbitrary vector V j in the same linear vector space V can be expressed as a linear combination of these linear independent vectors given by Eq(22 Thus the above mentioned set of linearly independent vectors is called a basis that spans the linear vector space V The coefficients a i in Eq(22 are called the components of the linear dependent vector V j in this basis In Dirac notation, V j and its adjoint V j are represented as follows: Vj Vj in the given basis in the given basis a a 2 a and ( a, a 2,, a (3 4 Inner product An inner product is the scalar function of any two arbitrary vectors V i and V j taken at a time from a given set of vectors { V, V 2, V 3, } in a linear vector space V and is denoted as V i V j, which evaluates the projection of V j along V i Thus the inner product is a numerical value satisfying the following axioms: V i V i, pertaining to the positive semi definiteness property, Joint Initiative of IITs and IISc - Funded by MHRD Page 5 of 9

4 PTEL - anotechnology - anophotonics (Beyond the Diffraction Limit V i V i =, if and only if V i = /, a null vector, When V i V j =, it means that there is no projection of V j along V i Thus V i and V j are orthogonal to each other V i V j = V j V i, pertaining to the property of skew-symmetry, V i ( α V j +β V k V i ( αv j + βv k =α Vi V j +β V i V k, pertaining to the linearity property in ket and ( αv j + βv k Vi =α V j V i +β V k V i Any vector space consisting of an inner product is called an inner product space or Hilbert space 5 orm of a vector The norm or length of a vector V is defined as follows: The unit vector i of a vector V is defined as follows: V = V V (5 i V V = V V V, (52 where the one-to-one correspondence between i and the -tuple column vector is given as follows: i in the given basis, (53 where is in the i th row of the -tuple column vector Also, i in the given basis (,,,,,,,, (54 where is in the i th column of the -tuple row vector The norm of this unit vector i is then defined as i = i i = Thus any unit vector has got an unit norm Joint Initiative of IITs and IISc - Funded by MHRD Page 6 of 9

5 PTEL - anotechnology - anophotonics (Beyond the Diffraction Limit 6 Orthonormal basis If the unit vectors of a given set of vectors in the linear vector space V are chosen as the basis vectors, they will be pairwise orthogonal to each other This is due to the fact that from the given set of unit vectors {, 2,, }, the inner product between any two arbitrary unit vectors, namely, i j takes the following form: i j δ i j = { for i= j for i j, (6 where δ i j is the usual Kronecker delta symbol Hence any given set of unit vectors chosen as the basis vectors form an othonormal basis 6 Expansion of vectors in an orthonormal basis Let X be any arbitrary vector in the linear vector space V which can be expressed as linear combination of the unit vectors, {, 2,, } which are chosen as the basis vectors, such that X = x i i, (62 where x i are the coefficients of X The equivalent matrix representation of Eq(62 is given as follows: x x 2 x = x + x 2 ++x (63 The projection of X along any one of the unit vectors j is given by the inner product j X, which is evaluated as follows: j X = = x i j i, x i δ i j, = x j, (64 which is the j th coefficient of X Thus in order to obtain the i th coefficient x i of X, one needs to obtain the inner product i X = x i Hence taking account of Eq(64, X can be expressed as follows: X = i i X (65 Joint Initiative of IITs and IISc - Funded by MHRD Page 7 of 9

6 PTEL - anotechnology - anophotonics (Beyond the Diffraction Limit Equation (65 infers that any arbitrary vector in the given Hilbert space can be expanded in an orthonormal basis Equations (62-(65 show that the best choice of basis vectors are the unit vectors Let X and Y be any two arbitrary vectors in the Hilbert space V which are expressed as linear combination of the unit vectors, {, 2,, } which are chosen as the basis vectors, such that X = Y = x i i and y j j, (66 j= where x i are the coefficients of X and y j are the coefficients of Y Then the inner product between X and Y is given as follows: X Y = { x i y j δ i j = j= x i y i for i= j for i j (67 62 Gram-Schmidt theorem for generating an orthonormal basis Let { V, V 2, V 3,, V } be a set of vectors constituting a Hilbert space V In order to generate a set of unit vectors as the orthonormal basis {, 2, 3,, } from the above mentioned Hilbert space V, the following procedure is employed: Let = V Let 2 = V 2 V 2 Then 2 = As a result, and 2 are orthogonal to each other Let 3 = V 3 V V Then 3 = 2 3 = Proceeding in this manner, the th ket can be written as = V ( i i V i = i i, which will be orthogonal to all the previous vectors Thus by following the above mentioned procedure, on finally obtains orthogonal vectors {, 2, 3,, } To obtain a set of orthonormal basis vectors, one needs to evaluate i = i i i, thereby obtaining the unit vectors {, 2,, } as the orthonormal basis vectors Joint Initiative of IITs and IISc - Funded by MHRD Page 8 of 9

7 PTEL - anotechnology - anophotonics (Beyond the Diffraction Limit 7 Additional reading and references R L Liboff, Introductory Quantum Mechanics (Addison Wesley, ew York, 98 2 R Shankar, Principles of Quantum Mechanics (Plenum Press, ew York, D Ahn and S H Park, Engineering Quantum Mechanics (IEEE Press, Singapore, 2 Joint Initiative of IITs and IISc - Funded by MHRD Page 9 of 9