Chapter 2 Introduction to Quantum Mechanics

1 Chapter 2 Introduction to Linear Vector SpacesFor our purpose the most relevantvector spaceis a finite (or countable)dimensional space ofvectorswith complex components represented in thecolumn matrix notation as| n ( )where i Cand| CnorC . definition the vector spaces comes n + n = 1+ n+ n ,( )19 Chapter 2. Introduction TO Quantum MECHANICS202. multiplication by ascalar,thatis,acomplexnumber(orc-number ),z n = z n ( )3. and azero vector, 0 ( )not to be confused with a vacuum state|0 .Spanning setin a vector space is a collection of vectors| 1 ,| 2 ..| n suchthat an arbitrary vector can be written as their liner combination,| =n'i=1ai| i.

2 ( ) Inner ProductAninner product(,)is a function which takes two vectors| and| to produce a single complex number denoted by(| ,| )with followingproperties:1.(,)is linear in the second (| ,| )=(| ,| ) 3.(| ,| ) 0with equality only iff| = example the vector spaceCnhas an inner product defined by n , n n'i=1 i i( ) Chapter 2. Introduction TO Quantum MECHANICS21 The dual vector |is a linear function from a vector space to complexnumbers defined using the inner product, |(| ) | (| ,| ).( )In matrix representation it is a raw vector (also known as a dual vector, aco-vector or a bra-vectors), | ( 1, .. n).( )when the column vector is known as a ket-vector. In a Quantum mechanicsafinitedimensionalvectorspacewi thaninnerproductiscalled a Hilbertspace.

3 A number of definitions follow Two vectors| and| areorthogonalif | =0. Normof a vector| is( | . | is aunit vectorif | =1. asetofvectors|i isorthonormalif i|i =1and i|j =0fori = an orthonormal basis an inner product can be written in amatrixrepresentation as | =)n'i=1 i|i ,n'j=1 j|j *=n'i=1n'j=1 i j i|j =n'i=1 i j=( 1, .., n) n .( ) Tensor productConsider two vector spaces ( Hilbert spaces)VandWwith basis vectors|vi s and|wi s then we can form a tensor product space (denoted byV W)with basis vectors|vi |wj (for brevity of notations often denoted by|v |w ,|v, w or even|vw ).| ='i,j ij|vi |wj For example ifVis the Hilbert space of the first q-bit andWis the Hilbertspace of the second q-bit then the tensor product space is a Hilbert space oftwo q-bits with basis vectors|00 ,|01 ,|10 and|11.

4 Chapter 2. Introduction TO Quantum MECHANICS22 The operators which act on vectors inV Ware formed by a tensorproduct of operators inVandW, (A B)'i,j ij|vi |wj 'i,j ijA|vi B|wj .Aconvenientmatrixrepresentationoftheten sorproductsisgiventheso-called Kronecker productA B= A11BA12B .. A1nBA21BA22B .. AnnB .For example+1i, + i0,= i010 orY Z=+0 ii0, +100 1,= 00 i000 0ii0000 i00 Outer ProductAnother convenient representation of linear operators is anouter productrepresentation defined by,| |(| ) | | = | | .( )For example an identity operator can be represented as a sum ofouterprod-uct of an orthonormal basis, I=n'i=1|i i|( )This is known as acompleteness relationwhich can be used to obtain anouter product representations of operators, A= I A I=n'i=1n'j=1 i| A|j |i j|.

5 ( ) Chapter 2. Introduction TO Quantum MECHANICS23 Eigenvectors|i and their respective eigenvalues iof a linear operator Aare defined by A|i = i|i .( )In a matrix representation the eigenvalues can be determinedfromacharac-teristic equationdet- A ( )Diagonal representation of an operator is given by A=n'i=1 i|i i|( )where|i are the Linear OperatorsLinear operatoron a vector space is a function which is linear in its inputs,A)n'i=1ai| i *=n'i=1aiA(| i ).( )(To emphasize the difference from c-number or complex numberstheoper-ators are sometimes called the q-numbers or Quantum numbers.) The twosimplest operators are the identity operator I,definedby I| =| for all| . zero operator 0,definedby 0| = 0for all| .Similarly to vectors, operatorsA:Cn Cncan be represented in termsof matrices.

6 Inmatrix representationthe linearity condition ( ) can berewritten as, An'i=1ai| i =n'i=1ai A| i ,( )where Ais ann nmatrix and| i s are then 1column vectors. TheentriesAijof the matrix Acan be obtained for a given set of basis vectors| 1 ,| 2 ..| n from A| j ='iAij| i .( )and then one can define trace of the matrix as the sum of diagonalelementstr( A)=' 2. Introduction TO Quantum MECHANICS24An important example of operators onC2are thePauli matrices, 0 I +1001, 1 X +0110, 2 Y +0 ii0, 3 Z +100 1,.( )The Pauli matrices are related to each other through commutation rela-tions[X, Y]=2iZ[Y, Z]=2iX[Z, X]=2iYwhich can be compactly written[ j, k]=2i'l=1,2,3 jkl lwhere jkl= +1ifjklis an even premutations of 123 1ifjklis an even premutations of the Levi-Civita Adjoint OperatorsFor every linear operator Aon a Hilbert space there exist an adjoint (orHermitian conjugate) operator defined as(| , A| ) ( A | ,| ).

7 ( )It follows that(AB) =B A ( ) Chapter 2. Introduction TO Quantum MECHANICS25and- A| . = | A ,( )where| |.( )In a matrix representation an adjoint operator is defined as A - A .T( )where() is a complex conjugation and()Tis a transpose useful definitions: Ais apositive definiteoperator if(| , A| )is a positive real numberfor all| = 0. Ais aself-adjoint(or Hermitian) operator if A = operator is Hermitian. Ais anormaloperator if A A = A ,anyHermitianoperator is normal. Pis aprojectionoperator if P= , P=3ki=1|i i|,where|i is an orthonormal basis andk n. Uis aunitaryoperator if U U= | i and| i can be used to define unitary operators, U=3ni=1| i i|.Note that unitary operators preserve the inner product,( U| , U| )= | U U| = | I| = |.

8 ( ) Decompositions of operators Spectral decomposition. Any normal operatorMon vector spaceVisdiagonal with respect to some orthonormal basis|i s forV,M='i i|i i|Conversely, any diagonalizable operator is normal. For Hermitian op-erators the eigenvalues are that the spectral decomposition can be used to defined functionsof operators,f(M) 'if( i)|i i| Chapter 2. Introduction TO Quantum MECHANICS26 Polar decomposition. An arbitrary linear operator can be decomposedinto product of unitary operatorUand positive operatorsJandKsuch thatA=UJ=KUwhereJ A AK AA . Singular value decomposition. For any square matrixAthere are exitunitary matricesUandV,andadiagonalmatrixDsuch thatA= non-negative diagonal elements ofDare called the singular Quantum Classical PhysicsClassical physics is based on the two space postulate:Any closed system is associated with evendimensional space called the phase space.

9 The state is described by asingle point (or vector) in the phase state is specified byNposition (usually denoted byqi s) andNmomentum (usually denoted bypi s) coordinates:(q1,q2, .., qN,p1,p2, .., pN)( )What is the dimensionality of the phase space of a simple harmonicoscillator? What is the dimensionality of the phase space ofasingleparticle in a box? How many real numbers needed to specify a stateof the harmonic oscillator and how many needed to specify a singleparticle? postulate:Evolution of any closed system is described byfunction on the phase space. The function is called Hamiltonian andCHAPTER 2. Introduction TO Quantum MECHANICS27denoted byH(q1,q2, .., qN,p1,p2, .., pN)andtheevolutionisdescribedby the following equations pi= H qi qi= H pi.

10 ( )For a simple harmonic oscillator the Hamiltonian isH(q, p)=p22m+kq22.( )and the corresponding equations of motion are p= kq q=pm( )or equivalently q= pm= kqm.( ) Quantum PhysicsIn contrast the Quantum physics is based on three the postulates and then introduce the necessary mathematical formal-ism that goes with space postulate:Any closed system is associated with aHilbertspace. The state of the system is described by a single point(orket-vector) in the Hilbert space| ( )with unit length | | | =1( )where |is postulate:Evolution of any closed system is described byaunitaryoperator, | (t2) = U(t2 t1)| (t1) ( )where U(t2 t1) e i!(t2 t1) H.( )and His aHermitianoperator known as Hamiltonian 2. Introduction TO Quantum postulate:Ameasurementisdescribedbyacoll ec-tion of measurement operators{ Mm}with probability of an outcomemgiven byp(m)= | M m Mm| ( )where M mis theHermitianconjugate of Mmand'm M m Mm= the state after measurement Mm| (p(m).)