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