WIXLIB

where On is a number called eigenvalue.Property 1: The eigenvalues of a hermitian operator are real.Proof: Using Definition 4, we obtain ZZ dxφn (x)Ôφn (x) dxφn (x)Ôφn (x) 0,therefore,[On On ]Zdxφn (x) φn (x) 0.Since φn (x) are square integrable functions, then,On On .Property 2: Different eigenfunctions of a hermitian operator (i.e., eigenfunctions with differenteigenvalues) are orthogonal (i.e., the scalar product of two different eigenfunctionsis equal toRzero). Mathematically, if Ôφn On φn , and Ôφm Om φm , with On 6 Om , then dxφ n φm 0.Proof: Z Zdxφ m Ôφn anddxφ n ÔφmZ[On Om ]Since On 6 Om , thenR 0,dxφ m φn 0.dxφ m φn 0.P ostulate 3 : The only possible experimental results of a measurement of an observable are theeigenvalues of the operator that corresponds to such observable.P ostulate 4 : The average value of many measurements of an observable O, when the system isdescribed by function ψ(x), is equal to the expectation value Ō, which is defined as follows,Rdxψ(x) Ôψ(x)Ō R.dxψ(x) ψ(x)P ostulate 5 :The evolution of ψ(x, t) in time is described by the following equation:i 2 ψ(x, t) Ĥψ(x, t), t2 where Ĥ 2m V̂ (x), is the operator associated with the total energy of the system, E x2p2 V (x).2m8

Expansion P ostulate : R5(15), R4(97)The eigenfunctions of a linear and hermitian operator form a complete basis set. Therefore,any function ψ(x) that is continuous, single valued, and square integrable can be expanded as alinear combination of eigenfunctions φn (x) of a linear and hermitian operator  as follows,Xψ(x) Cj φj (x),jwhere Cj are numbers(e.g., complex numbers)P called expansion coefficients.P Note that Ā j Cj Cj aj , when ψ(x) j Cj φj (x),ZÂφj (x) aj φj (x),anddxφj (x) φk (x) δjk .This is because the eigenvalues aj are the only possible experimental results of measurements of Â(according to Postulate 3), and the expectation value Ā is the Paverage value of many measurementsof  when the system is described by the expansion ψ(x) j Cj φj (x) (Postulate 4). Therefore,the product Cj Cj can be interpreted as the probability weight associated with eigenvalue aj (i.e.,the probability that the outcome of an observation of  will be aj ).Hilbert-SpaceAccording to the Expansion Postulate (together with Postulate 1), the state of a system describedby the function Ψ(x) can be expanded as a linear combination of eigenfunctions φj (x) of a linearand hermitian operator (e.g., Ψ(x) C1 φ1 (x) C2 φ2 (x) . . .). Usually, the space defined bythese eigenfunctions (i.e., functions that are continuous, single valued and square integrable) hasan infinite number of dimensions. Such space is called Hilbert-Space in honor to the mathematicianHilbert who did pioneer work in spaces of infinite dimensionality.R4(94)A representation of Ψ(x) in such space of functions corresponds to a vector-function,φ2 (x)6C2.ψ(x) .C1-φ1 (x)where C1 and C2 are the projections of Ψ(x) along φ1 (x) and φ2 (x), respectively. All othercomponents are omitted from the representation because they are orthogonal to the “plane” definedby φ1 (x) and φ2 (x).9

3Continuous RepresentationsCertain operators have a continuous spectrum of eigenvalues. For example, the coordinate operatoris one such operator since it satisfies the equation x̂ δ(x0 x) x0 δ(x0 x), where the eigenvaluesx0 define a continuum. Delta functions δ(x0 x) thus define a continuous representation (the socalled ’coordinate representation’) for whichZψ(x) dx0 Cx0 δ(x0 x),where Cx0 ψ(x0 ), sinceZZZdxδ(x β)ψ(x) dx dαCα δ(x β)δ(α x) ψ(β).When combined with postulates 3 and 4, the definition of the expansion coefficients Cx0 ψ(x0 ) implies that the probability of observing the system with coordinate eigenvalues between x0and x0 dx0 is P (x0 ) Cx0 Cx 0 dx0 ψ(x0 )ψ(x0 ) dx0 (see Note 1).In general, eigenstates φ(α, x) with a continuum spectrum of eigenvalues α define continuousrepresentations,Zψ(x) dαCα φ(α, x),Rwith Cα dxφ(α, x) ψ(x). Delta functions and the plane waves are simply two particularexamples of basis sets with continuum spectra.Note 2: According to the Expansion Postulate, a function ψ(x) is uniquely and completely definedby the coefficients Cj , associated with its expansion in a complete set of eigenfunctions φj (x).However, the coefficients of such expansion would be different if the same basis functions φjdepended on different coordinates (e.g., φj (x0 ) with x0 6 x). In order to eliminate such ambiguityin the description it is necessary to introduce the concept of vector-ket space.R4(108)4Vector SpaceVector-Ket Space ε: The vector-ket space is introduced to represent states in a convenient spaceof vectors φj , instead of working in the space of functions φj (x). The main difference is thatthe coordinate dependence does not need to be specified when working in the vector-ket space.According to such representation, function ψ(x) is the componentof vector ψ associated withPindex x (vide infra). Therefore, for any function ψ(x) j Cj φj (x), we can define a ket-vector ψ such that,X ψ Cj φj .jThe representation of ψ in space ε is,10

Ket-Space ε φ2 6C2. . ψ .C1- φ1 Note that the expansion coefficients Cj depend only on the kets ψj and not on any specificvector component. Therefore, the ambiguity mentioned above is removed.In order to learn how to operate with kets we need to introduce the bra space and the concept oflinear functional. After doing so, this section will be concluded with the description of Postulate5, and the Continuity Equation.Linear functionalsA functional χ is a mathematical operation that transforms a function ψ(x) into a number. Thisconcept is extended to the vector-ket space ε, as an operation that transforms a vector-ket into anumber as follows,χ(ψ(x)) n,orχ( ψ ) n,where n is a number. A linear functional satisfies the following equation,χ(aψ(x) bf (x)) aχ(ψ(x)) bχ(f (x)),where a and b are constants.Example: The scalar product,R4(110)Zn dxψ (x)φ(x),is an example of a linear functional, since such an operation transforms a function φ(x) into anumber n. In order to introduce the scalar product of kets, we need to introduce the bra-space.Bra Space ε : For every ket ψ we define a linear functional ψ , called bra-vector, as follows:R ψ ( φ ) dxψ (x)φ(x).Note that functional ψ is linear because the scalar product is a linear functional. Therefore, ψ (a φ b f ) a ψ ( φ ) b ψ ( f ).11

Note: For convenience, we will omit parenthesis so that the notation ψ ( φ ) will be equivalentto ψ φ . Furthermore, whenever we find two bars next to each other we can merge them intoa single one without changing the meaning of the expression. Therefore, ψ φ ψ φ .P The space of bra-vectors is called dual spaceεsimplybecausegivenaket ψ j Cj φj ,P the corresponding bra-vector is ψ j Cj φj . In analogy to the ket-space, a bra-vector ψ is represented in space ε according to the following diagram:Dual-Space ε φ2 6C2 . . ψ . C1- φ1 where Cj is the projection of ψ along φj .Projection Operator and Closure RelationGiven a ket ψ in a certain basis set φj , ψ XCj φj ,(1)jwhere φk φj δkj ,Cj φj ψ .(2)Substituting Eq. (2) into Eq.(1), we obtain ψ X φj φj ψ .jFrom Eq.(3), it is obvious thatX φj φj 1̂,Closure Relationjwhere 1̂ is the identity operator that transforms any ket, or function, into itself.12(3)

Note that P̂j φj φj is an operator that transforms any vector ψ into a vector pointingin the direction of φj with magnitude φj ψ . The operator P̂j is called the ProjectionOperator. It projects φj according to,P̂j ψ φj ψ φj .Note that P̂j2 P̂j , where P̂j2 P̂j P̂j . This is true simply because φj φj 1.4.1Exercise 1Prove thati P̂j [Ĥ, P̂j ], twhere [Ĥ, P̂j ] Ĥ P̂j P̂j Ĥ.Continuity Equation4.2Exercise 2Prove thatwhere (ψ (x, t)ψ(x, t)) j(x, t) 0, t x ψ (x, t) ψ(x, t) j(x, t) ψ(x, t)ψ (x, t).2mi x xIn general, for higher dimensional problems, the change in time of probability density, ρ(x, t) ψ (x, t)ψ(x, t), is equal to minus the divergence of the probability flux j, ρ(x, t) · j. tThis is the so-called Continuity Equation.Note: Remember that given a vector field j, e.g., j(x, y, z) j1 (x, y, z)î j2 (x, y, z)ĵ j3 (x, y, z)k̂, the divergence of j is defined as the dot product of the “del” operator ( x, y, z) and vector jas follows: j1 j2 j3 ·j . x y z5Digital Grid-Based RepresentationsThe standard formulation of quantum mechanics, presented in previous sections, relies upon thetools of calculus (e.g., derivatives, integrals, etc.) and involves equations and operations with infinitesimal quantities as well as states in Hilbert-space (the infinite dimensional space of functions13

L2 ). The equations, however, seldom can be solved analytically. Therefore, computational solutions are necessary. However, computers can not handle infinite spaces since they have only limitedmemory. In fact, all they can do is to store and manipulate discrete arrays of numbers. Therefore,the question is: how can we represent continuum states and operators in the space of memory ofdigital computers?In order to introduce the concept of a grid-representation, we consider the state,Ψ0 (x) α 1/4παe 2 (x x0 )2 ip0 (x x0 ),which can be expanded in the infinite basis set of delta functions δ(x x0 ) as follows,ZΨ0 (x) dx0 c(x0 )δ(x x0 ),(4)(5)where c(x0 ) hx0 Ψ0 i Ψ0 (x0 ). All expressions are written in atomic units, so 1.A grid-based representation of Ψ0 (x) can be obtained, in the coordinate range x (xmin , xmax ),by discretizing Eq. (5) as follows,Ψ0 (x) nXcj δ(x xj ),(6)j 1where the array of numbers cj hxj Ψ0 i represent the state Ψ0 on a grid of equally spaced coordinates xj xmin (j 1) with finite resolution (xmax xmin )/(n 1).Note that the grid-based representation, introduced by Eq. (6), can be trivially generalized toa grid-based representation in the multidimensional space of parameters (e.g., xj , pj , γj , . etc.)when expanding the target state Ψ0 (x) as a linear combination of basis functions hx xj , pj , γj i, withexpansion coefficients as cj hxj , pj , γj Ψ0 i.5.1Computational Problem 1Write a computer program to represent the wave-packet, introduced by Eq. (4) on a grid of equallyspaced coordinates xj xmin (j 1) with finite resolution (xmax xmin )/(n 1) andvisualize the output. Choose x0 0 and p0 0, in the range x (-20,20), with α ωm, wherem 1 and ω 1.Next, we consider grid-based representations in momentum space:Ψ0 (p) hp Ψ0 i.RInserting the closure relation 1̂ dx xihx in Eq. (7), we obtain thatZZ 1/2hp Ψ0 i dxhp xihx Ψ0 i (2π)dxe ipx hx Ψ0 i.14(7)(8)

is the Fourier transform of the initial state. The second equality in Eq. (8) was obtained by using:hx pi (2π) 1/2 eipx ,(9)which is the eigenstate of the momentum operator p̂ i , with eigenvalue p, since p̂hx pi phx pi.The Fourier transform can be computationally implemented in O(N log(N )) steps by usingthe Fast Fourier Transform (FFT) algorithm [see, Ch. 12 of Numerical Recipes by W.H. Press,B.P. Flannery, S.A. Teukolsky and W.T. Vetterling, Cambridge University Press, Cambridge, 1986[FFT] when hx Ψ0 i is represented on a grid with N 2n points (where n is an integer). In contrast,the implementation of the Fourier transform by quadrature integration would require O(N 2 ) steps.5.2Computational Problem 2Write a computer program to represent the initial state, introduced by Eq. (4), in the momentumspace by applying the FFT algorithm to the grid-based representation generated in Problem 1 andvisualize the output. Represent the wave-packet amplitudes and phases in the range p (-4,4) andcompare your output with the corresponding values obtained from the analytic Fourier transformobtained by using:Zpdx exp( a2 x2 a1 x a0 ) π/a2 exp(a0 a21 /(4a2 )).Next, we consider the grid-based representation of operators (e.g., x̂, p̂, V (x̂), and T̂ p̂2 /(2m))and learn how these operators act on states represented on grids in coordinate and momentumspaces. For simplicity, we assume that the potential is Harmonic:1V (x̂) mω 2 (x̂ x̄)2 .2(10)Consider first applying the potential energy operator to the initial state, as follows,V (x̂)Ψ0 (x) V (x)Ψ0 (x) Ψ̃0 (x).(11)Since Ψ̃0 (x) is just another function, Eq. (11) indicates that V (x̂) can be represented on the samegrid of coordinates as before (i.e., equally spaced coordinates xj xmin (j 1) , with finiteresolution (xmax xmin )/(n 1)). Since for each xj , Ψ̃0 (xj ) V (xj )Ψ(xj ), the operatorV (x̂) can be represented just as an array of numbers V (xj ) associated with the grid-points xj , andits operation on a state is represented on such a grid as a simple multiplication.5.3Computational Problem 3Write a computer program to compute the expectation values of the position x(0) hΨ0 x̂ Ψ0 iand the potential energy V hΨ0 V (x̂) Ψ0 i, where V (x) is defined according to Eq. (10) for the15

initial wave-packet, introduced by Eq. (4), with various possible values of x0 and p0 , with α ωm,where m 1 and ω 1.Now consider applying the momentum operator, p̂ i , to the initial state Ψ0 (x) as follows,G(x) hx p̂ Ψ0 i i Ψ0 (x).(12)One simple way of implementing this operation, when Ψ0 (x) is represented on a grid of equallyspaced points xj xmin (j 1) , is by computing finite-increment derivatives as follows:G(xj ) iΨ0 (xj 1 ) Ψ0 (xj 1 ).2 (13)However, for a more general operator (e.g., T̂ p̂2 /(2m)) this finite increment derivative procedure becomes complicated. In order to avoid computing finite-increment derivatives, one canimplement an alternative procedure: represent the initial state in momentum-space (by Fouriertransform of the initial state); apply the operator by simple multiplication in momentum space,then transform the resulting product back to the coordinate representation (byR inverse-Fourier transform). This method can be derived by inserting the closure relation 1̂ dp pihp , in Eq. (12),ZG(x) hx p̂ Ψ0 i 1/2dphx p̂ pihp Ψ0 i (2π)Zdpeipx php Ψ0 i,(14)since hp Ψ0 i is defined, according to Eq. (8), as the Fourier transform of the initial state. Note thatthe second equality of Eq. (14) is obtained by introducing the substitutionhx pi (2π) 1/2 eixp̂ .(15)While Eq. (14) illustrates the method for the specific operator p̂, one immediately sees that anyoperator which is a function of p̂ (e.g., T̂ p̂2 /(2m)) can be analogously applied according to theFourier transform procedure.5.4Computational Problem 4Write a computer program to compute the expectation values of the initial momentum p(0) hΨ0 p̂ Ψ0 i and the kinetic energy T hΨ0 p̂2 /(2m) Ψ0 i by using the Fourier transform procedure,where Ψ0 is the initial wave-packet introduced by Eq. (4), with x0 0, p0 0, and α ωm,where m 1 and ω 1. Compute the expectation value of the energy E hΨ0 Ĥ Ψ0 i, whereĤ p̂2 /(2m) V (x̂), with V (x) defined according to Eq. (10) and compare your result with thezero-point energy E0 ω/2.6Heisenberg RepresentationMost of the problems of interest in Chemistry have equations that are too complicated to be solvedanalytically. This observation has been stated by Paul Dirac as follows: The underlying physical16

laws necessary for the mathematical theory of a large part of Physics and the whole of Chemistryare thus completed and the difficulty is only that exact application of these laws leads to the equations much too complicated to be soluble. It is, therefore, essential, to introduce numerical andapproximate methods (e.g., perturbation methods and variational methods).In this section, we describe the matrix representation, introduced by Heisenberg, which is mostuseful for numerical methods to solve the eigenvalue problem,R4(124) R3(240)Ĥ ψl i El ψl i,(16)for an arbitrary state ψl i of a system (e.g., an atom, or molecule) expanded in a basis set {φj }, asfollows:X (j) ψl i Cl φj i,(17)j(j)where Cl hφj ψl i, and hφj φk i δjk . Substituting Eq. (17) into Eq. (16) we obtain:XX(j)(j)Ĥ φj iCl El Cl φj i.jjApplying the functional hφk to both sides of this equation, we obtain:XX(j)(j)hφk Ĥ φj iCl El hφk φj iCl ,j(18)jwhere hφk φj i δkj and k 1, 2, ., n.Introducing the notation Hkj hφk Ĥ φj i we obtain, (1)(2)(3)(n)(1)(2)(n)H11 Cl H12 Cl H13 Cl . H1n Cl El Cl 0Cl . 0Cl ,(k 1) (1)(2)(3)(n)(1)(2)(n) (k 2) H21 Cl H22 Cl H23 Cl . H2n Cl 0Cl El Cl . 0Cl ,. .(k n) H C (1) H C (2) H C (3) . H C (n) 0C (1) 0C (2) . E C (n) ,n1 ln2 ln3 lnn ll lll(19)that can be conveniently written in terms of matrices and vectors as follows, (1) (1) CCH11 H12 . H1nEl 0 . 0l l(2) (2) H21 H22 . H2n Cl 0 El . 0 Cl (20) . . . . . (n)(n)Hn1 Hn2 . Hnn0 0 . ElClClThis is the Heisenberg representation of the eigenvalue problem introduced by Eq. (16). Accordingto the Heisenberg representation, also called matrix representation, the ket ψl i is represented by(j)the vector Cl , with components Cl hφj ψl i, with j 1, .,

2 The Fundamental Postulates of Quantum Mechanics Quantum Mechanics can be formulated in terms of a few postulates (i.e., theoretical principles based on experimental observations). The goal of this section is to introduce such principles, to-gether with some mathematical concepts that are necessary for that purpose. To keep the notation