3.1OperatorsinquantummechanicsAn operator is a rule that transforms a given function into another function. E.g. d/dxsin, logAf(x) = kf(x)EigenfunctionsandEigenvaluesSuppose that the effect of operating on some function f(x) with the operator A is simply to multiply f(x) by acertain constantk.Wethen saythatf(x)is aneigenfunctionof A witheigenvaluek.EigenisaGermanwordmeaningcharacteristicOperatorsobeytheassociativelawofmultiplication:A(BC)=(AB)CA linearoperator meansA(, +2)= AW + A2Acy=cAy
3.1 Operators in quantum mechanics An operator is a rule that transforms a given function into another function. E.g. d/dx, sin, log Eigenfunctions and Eigenvalues Suppose that the effect of operating on some function f(x) with the operator  is simply to multiply f(x) by a certain constant k. We then say that f(x) is an eigenfunction of  with eigenvalue k. Af(x) kf(x) ˆ Eigen is a German word meaning characteristic. Operators obey the associative law of multiplication: C ˆ B) A ˆ ˆ C) ( B ˆ ˆ A( ˆ A linear operator means 1 2 1 Aψ2 A ˆ ψ ˆ A(ψ ψ ) ˆ A ˆ cψ cA ˆ ψ
dIsa linear operat6idxdfdgd-)[f(x)+ g(x)):)g(x)f()-dxdxdxdxdcdf-)[Cf(x)]= Cdxdxf(x)+g(x)+f(x)+/g(x)(A + B)f(x) = Af(x)+ Bf(x)(A-B)f(x)= Af(x)-Bf(x)ABf(x) = A[Bf(x)]
Is a linear operator ( )[ ( ) ( )] ( ) ( ) ( ) ( ) ( )[ ( )] d df dg d d f x g x f x g x dx dx dx dx dx d df Cf x C dx dx d dx ? f x g x f x g x ( ) ( ) ( ) ( ) ? Bf(x) ˆ Af(x) ˆ B)f(x) A ˆ ˆ ( Bf(x) ˆ Af(x) ˆ B)f(x) A ˆ ˆ ( Bf(x)] ˆ A[ ˆ Bf(x) A ˆ ˆ
3.2 HermiticityEvery operator Q has a Hermitian conjugate, conventionally denoted Q t, which has theproperty that for any 1 and 2 satisfying the boundary conditions for the problem,Jviowedt=J(otw.) ydtAn operator that is equal to its Hermitian conjugate is said to be HermitianOperators corresponding to physical observables must be Hermitian[y,x,dx=f(xy,)y,dxSo x t= x, and X is Hermitian.=--Jiv.dx=[vivL-(iv.dx= (-w) v.dxSodand d/dx is not a Hermitian operator. However id/dx is Hermitian
Every operator 𝑄 has a Hermitian conjugate, conventionally denoted 𝑄 † , which has the property that for any 𝜓1 and 𝜓2 satisfying the boundary conditions for the problem, An operator that is equal to its Hermitian conjugate is said to be Hermitian. Operators corresponding to physical observables must be Hermitian * * 1 2 1 2 ˆ ˆ Q Q d d 3.2 Hermiticity so 𝑥 ො † = 𝑥 ො, and 𝑥 ො is Hermitian. and d/dx is not a Hermitian operator. However id/dx is Hermitian * * 1 2 1 2 x x x x d d * * * * 1 2 1 2 1 2 1 2 d d d d d d x x x x x x d d d d x x So
Dirac notationWe sometimes use a notation due originally to Dirac.The idea is to reduce notational clutterandgivemoreprominencetothelabelsidentifyingthewavefunctionsIn this notation |n) is used for the wavefunction n : [n) is called a ket.(nlis abra.Thebranotation impliesthecomplexconjugate*A complete bracket expression, like (n|n) or (nQ n), implies integration over all spaceThus the notation (n|n) means the integral J Φn* Φn dt, and (m|Q|n) means J m* Qn dt.Using this notation, the expectation value integral can be written more compactly as[w'Oydt)=(g/n)(o)[y'ydt(n|n)
Dirac notation We sometimes use a notation due originally to Dirac. The idea is to reduce notational clutter and give more prominence to the labels identifying the wavefunctions. In this notation |𝑛ۧ is used for the wavefunction 𝜓𝑛 . |𝑛ۧ is called a ket. ۦ |��is a bra. The bra notation implies the complex conjugate 𝜓𝑛 * . A complete bracket expression, like 𝑛 𝑛 or 𝑛 𝑄 𝑛 , implies integration over all space. Thus the notation 𝑛 𝑛 means the integral �𝜓� ∗ 𝜓𝑛 d𝜏, and 𝑚 𝑄 𝑛 means �𝜓� ∗𝑄𝜓𝑛 d𝜏. Using this notation, the expectation value integral can be written more compactly as ˆ n Q n Q n n * *ˆ d = d Q Q
HermiticityPropertiesofHermitian operators:·Their eigenvalues are always real.· Eigenfunctions corresponding to different eigenvalues are orthogonal.In the proof of these properties we use Dirac's angle-bracket notation. First note that if Qis Hermitian, then(m[0|n) =(J ymOy,dt)=(own) v,dt)=Jw'Oymdt=(n|0|m)Note also that (m|n)* = (J m* Φn dt)* = J n*山m dt = (n|m)
Hermiticity Properties of Hermitian operators: • Their eigenvalues are always real. • Eigenfunctions corresponding to different eigenvalues are orthogonal. In the proof of these properties we use Dirac’s angle-bracket notation. First note that if 𝑄 is Hermitian, then Note also that 𝑚 𝑛 �𝜓� = ∗ ∗ 𝜓𝑛 d𝜏 �𝜓� = ∗ ∗ 𝜓𝑚 d𝜏 = 𝑛 𝑚 . * * * * * * ˆ ˆ d ˆ d ˆ d ˆ m n m n n m m Q n Q Q Q n Q m
Hermiticityand orthogonalityNow consider two eigenfunctions [m)and |n).We have0[m)=qm[m)|n)=q,[n)and(nl0|m)=qm(n|m)(m|0|n)=q.(m|n)(m[0|n)=qm (m|n)wherethelastlineontheleftcomesfromtakingthecomplexconjugateSubtracting,wefind0=(q, -qm )(m|n)andfromthiswecandeduce(a) If m = n, then (m|n) = (mm) ± 0, so qm = qm * and qm is real.(b) If qm ± qn, then since both are real, qn - qm *+ 0 and (m|n) = 0Hermitian operator ensures that the eigenvalue of the operator is a real number
Hermiticity and orthogonality Now consider two eigenfunctions |𝑚ۧ and |𝑛ۧ . We have and where the last line on the left comes from taking the complex conjugate. Subtracting, we find and from this we can deduce (a) If m = n, then 𝑚 𝑛 = 𝑚 𝑚 ≠ 0, so 𝑞𝑚 = 𝑞𝑚 ∗ and 𝑞𝑚 is real. (b) If 𝑞𝑚 ≠ 𝑞𝑛 , then since both are real, 𝑞𝑛 − 𝑞𝑚 ∗ ≠ 0 and 𝑚 𝑛 = 0. * ˆ ˆ ˆ m m m Q m q m n Q m q n m m Q n q m n ˆ ˆ n n Q n q n m Q n q m n * 0 n m q q m n Hermitian operator ensures that the eigenvalue of the operator is a real number
The eigenvalue of aHermitian operator is a real numberProof:Ay = ay[y'Aydt=Jy(Ay)'dtajlyPdt=aflyP dtIyP≥0but :?0a,=a,Quantum mechanical operators have to have real eigenvalues
The eigenvalue of a Hermitian operator is a real number Proof: Quantum mechanical operators have to have real eigenvalues * * ˆ ˆ A d A d ( ) but : A a ˆ 2 * 2 a d a d | | | | 0 2 | | 0 * i i a a
The eigenfunctions of Hermitian operators are orthogonalProof:Jy, *y,dx = 8,ConsiderthesetwoeigenequationsAy,=a,nAm= ammMultiply the left of eq 1 by .* and integrate, then take the complex conjugate of eq 2multply byand integateAwd=aJJy,A'ymdx = anJynymdxSubtractingthesetwoequationsgives -JymAy,dx -Jy,A*yimdx= (a, -am)]ymy,dx=0If n = m, the integral = 1, by normalization, so a, = a. *
The eigenfunctions of Hermitian operators are orthogonal * i j ij dx Proof: Consider these two eigen equations ˆ Aψm m m a ψ ˆ Aψn n n a ψ Multiply the left of eq 1 by m* and integrate, then take the complex conjugate of eq 2, multiply by n and integrate * * n ˆ a m n m n A dx dx * * * * m ˆ a n m n m A dx dx Subtracting these two equations gives - * * * * n ˆ ˆ * (a - a ) 0 m n n m m m n A dx A dx dx ** n (a - a ) 0 m m n dx If n = m, the integral = 1, by normalization, so an = an*
If n + m, and the system is nondegenerate (i.e. different eigenfunctions do not have thesame eigenvalues, a, ± am), theny.m*y,dx = 0(a, -am)]ym*y,dx = 0Eignefunctions of B that belong to a degenerate eigenvalue can always be chosen to beorthogonal.BF=sF, BG=SGg1=F, 82=G+cFwe want Jgigzdt =0[F*(G+cF)dt =[F'Gdt+c|F*Fdt =0-[ F'GdtSchmidtOrthogonalizationF*FdtThe eigenfunctions of Hermitian operators are orthogonal Jy, *y ,dx=S
If n m, and the system is nondegenerate (i.e. different eigenfunctions do not have the same eigenvalues, an am ), then n m (a - a ) * 0 m n dx * 0 m n dx Eignefunctions of B that belong to a degenerate eigenvalue can always be chosen to be orthogonal. ˆ ˆ BF sF B G sG , 1 2 g F g G cF , 1 2 we want g g d 0 F G cF d F Gd c F Fd 0 F Gd c F Fd Schmidt Orthogonalization The eigenfunctions of Hermitian operators are orthogonal i j ij * dx
Normalized g1,g2,g3.. 1g2 > ...)(L)-' = Ig2'= g2-glg3'= g3-g1-g2( 1g2 > .)=SLis triangular matrix:SchmidtOrthogonalizationIL = LT : Lowdin Orthogonalization1
Normalized , , . 1 2 3 g g g g g g S g S g g ij i j | | . . | | | 2 1 2 1 T S LL I 1 1 g ' g ' '| ' 2 2 1 2 1 g g g g g ' '| ' '| ' 3 3 1 3 1 2 3 2 g g g g g g g g . g g g L I g L T 1 2 1 2 1 1 | | . ( ) . | | g I g g L g L T ) . | | ( . | | 2 1 1 2 1 1 L: S 1/2 L is triangular matrix: Schmidt Orthogonalization L = LT : Löwdin Orthogonalization