Lecture #4! Quantum Mechanics: Mathematics “If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.” -- John von Neumann • Topics – Linear vector spaces – Dirac Notation – Hilbert Space – Liouville Space • Handouts and Reading assignments – van de Ven: Appendix A. – Miller, Chapter 4. (optional) – Ernst, Chapter 2, sections 2.1.3-2.1.4, pp 17-25 (optional). – Biographies: Hamilton, Dirac, Hilbert, Liouville. 1 Review: Linear Algebra • Solving sets of linear equations Analysis involves terms ⎛ ⎞⎛ ⎞ ⎛ ⎞ x + x + x = 5 1 1 1 x1 5 1 2 3 ⎜ ⎟⎜ ⎟ ⎜ ⎟ such as inverse, transpose, 3 1 7 ⎜ x ⎟ 3 3x1 + x2 + 7x3 = 3 ⎜ ⎟ 2 = ⎜ ⎟ Ax = y determinant, row and ⎜ ⎟⎜ ⎟ ⎜ ⎟ 1 2 2 ⎜ ⎟ 2 vectors column spaces, eigen- x + 2x + 2x = 2 ⎝ ⎠ x3 ⎝ ⎠ 1 2 3 ⎝ ⎠ Matrix notation for this course vectors and values, etc. • Instead, consider an n x n matrix A as transforming a n- dimensional vector x into a new vector Ax. • Linear transformations satisfy: A(ax + by) = a(Ax) + b(Ay) a, b scalars • Matrices Linear transformations xˆ3 xˆ3 ⎛ 0 0 0 ⎞ c ⎛0⎞ ⎜ ⎟ x ⎜ ⎟ ˆ A = ⎜ 0 1 0 ⎟ Ax = ⎜b⎟ = bx2 ⎜ ⎟ ⎜ ⎟ Ax 0 0 0 0 b ⎝ ⎠ a b ⎝ ⎠ xˆ2 xˆ2 xˆ1 xˆ1 xˆ , xˆ , and xˆ are unit vectors along the three axes ˆ 1 2 3 Projection of x along x2. 2 Review: Wave Functions • For the classical concept of a trajectory, we substitute the concept of the quantum state of a particle characterized by a wave function. ! • Wavefunction ψ(r ,t) generalized variable (position, angular momentum, spin, …) - contains all info possible to obtain about the particle ∂ ! i ! - time€ evolution given by ψ(r ,t) = − Hψ(r ,t) ∂ t " - interpreted! as a probability amplitude of the particle having state r with the probability density given by: ! ! 2 ! dΡ(r ,t) = Cψ(r ,t) dr , C constant. € ! 1 ! 2 ! € ∫ dΡ(r ,t) =1 = ∫ ψ(r ,t) d 3r << ∞ C • Naturally leads us to the study of square-integrable functions. € 3 € € Functions • Important to be familiar with the concept of treating functions, e.g. ψ1(x,y,z) and ψ2(x,y,z), as elements in a vector space. • Example: consider f(n) for 1 ≤ n ≤ N, n integer. " f % f 1 3 ! $ ' f2 fN f2 f1 f f = f(n) 4 $ ' ... $ " ' n f ! # N & • We can also write f as a linear combination of vectors. 1 " 1% " 0% " 0% " 1% " % " 1% ! ! $ " ' $ 0' $ 0' $ 1' $ 0' $ 1' 1 f = f + f +#f f = f˜ + f˜ +#f˜ " 1$ "' 2$ " ' N$ "' or € 1$ "' 2$ 0' N$ ' $ ' $ ' $ ' $ ' 1 €0 0 1 1 $ " ' $ ' # & # & # & # & 0 # 0& # & What about f(n,m) or f(x) (x continuous)? € € 4 Linear Vector Spaces • The set of square-integrable functions belong to L2 ! • Each wave function ψ ( r ) is associated with an element of a linear vector space F and is denoted as ψ . ! ψ (r ) ⇔ ψ ∈ F € • Some properties: € 2 (wave functions need to F ⊂ L be sufficiently regular) € Linearity if ψ1 , ψ2 ∈ F, then € ψ = λ1 ψ1 + λ2 ψ2 ∈ F arbitrary complex numbers € Plus, other standard mathematical niceties € 5 Dirac Notation • An element of F, denoted by ψ , is called a “ket”. • A linear functional, χ , is an linear operation that associates a complex number with every ket ψ . € χ(ψ ) = number • The set of all such linear€ functionals constitute a vector space F* called the dual space of F, and an element of F* is denoted as χ and called€ a “bra”. scalar or inner product χ(ψ ) = χ ψ € “bra”+ “ket” = “bracket” € 6 Hilbert Space • The scalar product satisfies: ∗ χ ψ = ψ χ χ (ψ + ξ ) = χ ψ + χ ξ ∗ χ λψ = λ χ ψ and λχ ψ = λ χ ψ (λ scalar) € ψ ψ ≥ 0 (= 0 iff ψ = 0 ) norm: ψ = ψ ψ € • A linear vector space with a defined metric (in this case the € scalar product) is€ called a Hilbert Space. € € χ ψ Answer: Question: What is ! equivalent! r r ! ∗ ! ! to in terms of ψ ( ) and χ ( ) ? χ ψ = ψ(r )χ (r )dr ∫! r N or ∗ € = ∑ψ(n)χ (n) (discrete case) € € n=1 € 7 € Basis N • If it follows that: ∑ci f i = 0 implies ci = 0 for all i i=1 then the kets f i are called linearly independent. • If n is the maximum number of linearly independent kets in a vector€ space F, then F is call n-dimensional. € • In an n-dimensional vector space, n linearly independent kets constitute a basis, and every element ψ in F can be written as n ψ = ψ f f ∑ i i where ψi is the€ coefficient of the ket i . i=1 8 € € Orthonormal Basis • If the basis kets f i are normalized (i.e. f i f i = 1 ) and #1 if i = j f f j = δij = $ i %0 if i ≠ j € Kronecker€ delta function then they constitute an orthonormal basis. € • The following€ identity (known as closure) also holds: n n ψ = ∑ f i ψ f i = ∑ fi f i ψ i=1 i=1 Simply states that a vector is equal to the sum of its projections. € € 9 Operators • An operator, O ˆ , generates one ket from another: Oˆ ψ = ξ linear operators: Oˆ ( λ ψ + µ ξ ) = λOˆ ψ + µOˆ ξ € € scalars • A vector space F is closed under O ˆ if ψ €∈ F and Oˆ ψ = ξ implies ξ ∈ F € • The adjoint of O ˆ is denoted O ˆ † as defined as € † ∗ (Oˆ ψ ) = ψ Oˆ † note : ψ Oˆ ξ = ξ Oˆ † ψ € € • O ˆ is Hermitian if Oˆ = Oˆ † € € € 10 Projection Operators ˆ • Example linear operator: Projection onto ket ψ : Pψ = ψ ψ ˆ Pψ ξ = ψ ψ ξ =λ ψ scalar € € • For the orthonormal basis kets fk € € ˆ € Pk = fk f k ˆ Eˆ ∑Pk = ∑€ fk f k = (closure) k k € identity operator • Another common operator is the transition operator. € ˆ € What’s€ that Tkl = f k fl good for? 11 € More About Operators ˆ ˆ ˆ ˆ • The product O 1 O 2 means first O 2 is applied to a ket and then O 1 is applied to the result. Oˆ Oˆ Oˆ Oˆ 1 2 ψ = 1( 2 ψ ) € ˆ ˆ ˆ ˆ • In general, O 1 O 2 ≠ O 2 O 1 , and a quantity used frequently in QM is the commutator defined as: € Oˆ ,Oˆ Oˆ Oˆ Oˆ Oˆ [ 1 2] = 1 2 − 2 1 • A function of an operator is defined via the corresponding polynomial expansion. € ˆ 1 1 e.g. eO = Eˆ + Oˆ + Oˆ O ˆ + Oˆ Oˆ O ˆ +… 2! 3! identity operator 12 € Vector and Matrix Representations ψ • When a ket, , is expressed as a linear combination! of basis kets, fi ψ , then a corresponding column vector, , can be constructed. # ψ1 & % ( N ! ψ ! € % 2 ( † ∗, ∗, , ∗ ψ = ∑ψi f i ψ = ψ ψ = ψ1 ψ2 … ψn ( ) € i 1 % " ( € = % ( $ψ n ' • Similarly, operators have matrix representations. ! ! € € € x = Oˆ y x Oy ˆ = where O is a n x n matrix with elements:Oij = f i O f j € z # x0 & # 0 & z ! % ( ! % 2 2 ( Remember: vector and 0 y x y ψ ψ = % 0 ( ψ = % 0 + 0 ( € matrix€ representations % ( % ( depends on the basis€ set! € $ z0 ' $ z0 ' € y x 0 x " 0 y basis:{ x , y , z } { x " , y " , z } 13 x y "€ € € € € € € € Change of Basis • If the kets f k constitute one orthonormal basis and the kets gk constitute another, then a change of basis (also called basis transformation) is accomplished by: € € ˆ ˆ Okl = fk O f l = ∑∑ f k gi gi O g j g j f l i j • Defining the unitary transformation matrix U as U ij = f i g j , € € † Of=U OgU € ˆ where Of and Og represent the operator matrices of O with respect to the basis f k and g k . € 14 € € Eigenkets and Eigenvalues • If Oˆ ψ = λ ψ scalar then ψ is called an eigenket of O ˆ with eigenvalue λ. € • The trace of a matrix is defined at the sum of the diagonal elements: € € Tr(O) = ∑Oii i • Since the trace of a matrix is invariant under a change of basis (see Problem Set 2), we can talk about the trace of the corresponding€ operator. Tr Oˆ = λ ˆ ( ) ∑ i where λi are the eigenvalues of O i 15 € € Liouville Space • Operators defined on an n-dimensional Hilbert space, are themselves elements of an n2-dimensional vector space known as Liouville space (sometimes called operator space or, more generally, an algebra). † • The metric in Liouville space is defined as: (Aˆ | Bˆ ) = Tr(Aˆ Bˆ ) • In contrast to Hilbert space, the product of two Liouville Space elements is defined. Aˆ , Bˆ ∈ L Aˆ B ˆ = Cˆ ∈ L a Liouville space Product is associative, but not necessarily commutative. € In general,€ Aˆ Bˆ ≠ Bˆ A ˆ For experts only. What are the remaining digits in this famous number 0.11000100000000000000000100…? 16 € Superoperators • Operators that work on elements of a Liouville space are called superoperators. ˆ ˆ ˆ AO 1 = O2 superoperator denoted by double hat • We will primarily deal with just one superoperator, namely the commutator: € ˆ Aˆ B ˆ ≡ [Aˆ , Bˆ ] = Aˆ B ˆ − Bˆ Aˆ • de Graaf text, unlike van de Ven, doesn’t use superoperators, but I think the notation and concept makes things easier. Another good reference€ for superoperators and Liouville space: R. Ernst, G. Bodenhausen, and A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions, Clarendon Press, 1987. 17 Fun with Superoperators • Cyclic Commutators: Let [Aˆ , Bˆ ] = Cˆ . Aˆ and B ˆ are said to commute cyclically if [ Aˆ, Cˆ ] = Bˆ .