1 The Foundations Of Quantum Mechanics

1.4 COMMUTATION AND NON-COMMUTATIONj13multiplication by px and the position operator is represented by differentiation with respect to px. Explicitly:Momentum representation: x ! qhi qpxpx ! px ð1:6ÞThere are other representations. We shall normally use the position representation when the adoption of a representation is appropriate, but we shallalso see that many of the calculations in quantum mechanics can be doneindependently of a representation.1.4 Commutation and non-commutationAn important feature of operators is that in general the outcome of successiveoperations (A followed by B, which is denoted BA, or B followed by A,denoted AB) depends on the order in which the operations are carried out.That is, in general BA 6¼ AB. We say that, in general, operators do notcommute. For example, consider the operators x and px and a specifich/i)x ¼ (2 h/i)x2,function x2. In the position representation, (xpx)x2 ¼ x(2 232h/i)x . The operators x and px do not commute.whereas (pxx)x ¼ pxx ¼ (3 The quantity AB BA is called the commutator of A and B and is denoted[A, B]:½A, B ¼ AB BAð1:7ÞIt is instructive to evaluate the commutator of the position and linearmomentum operators in the two representations shown above; the procedureis illustrated in the following example.Example 1.3 The evaluation of a commutatorEvaluate the commutator [x,px] in the position representation.Method. To evaluate the commutator [A,B] we need to remember that theoperators operate on some function, which we shall write f. So, evaluate [A,B]ffor an arbitrary function f, and then cancel f at the end of the calculation.Answer. Substitution of the explicit expressions for the operators into [x,px]proceeds as follows:h qf h qðxf Þ ½x, px f ¼ ðxpx px xÞf ¼ x i qx i qxh qf hh qf f x ¼ i hf¼x i qx ii qxwhere we have used (1/i) ¼ i. This derivation is true for any function f,so in terms of the operators themselves,½x, px ¼ i hThe right-hand side should be interpreted as the operator ‘multiply by theconstant i h’.Self-test 1.3. Evaluate the same commutator in the momentum representation.[Same]

14j1 THE FOUNDATIONS OF QUANTUM MECHANICS1.5 The construction of operatorsOperators for other observables of interest can be constructed from the operators for position and momentum. For example, the kinetic energy operatorT can be constructed by noting that kinetic energy is related to linearmomentum by T ¼ p2/2m where m is the mass of the particle. It follows thatin one dimension and in the position representation p21 h d 2h d2 ¼ ð1:8ÞT¼ x ¼2m dx22m 2m i dxAlthough eqn 1.9 has explicitlyused Cartesian coordinates, therelation between the kinetic energyoperator and the laplacian is truein any coordinate system; forexample, spherical polarcoordinates.In three dimensions the operator in the position representation is()h 2 q2 q2q2h2 2 r¼ T¼ þþ2m qx2 qy2 qz22mð1:9ÞThe operator r2, which is read ‘del squared’ and called the laplacian, is thesum of the three second derivatives.The operator for potential energy of a particle in one dimension, V(x), ismultiplication by the function V(x) in the position representation. The same istrue of the potential energy operator in three dimensions. For example, in theposition representation the operator for the Coulomb potential energy of anelectron (charge e) in the field of a nucleus of atomic number Z is themultiplicative operatorV¼ Ze2 4pe0 rð1:10Þwhere r is the distance from the nucleus to the electron. It is usual to omit themultiplication sign from multiplicative operators, but it should not be forgotten that such expressions are multiplications.The operator for the total energy of a system is called the hamiltonianoperator and is denoted H:H ¼TþVð1:11ÞThe name commemorates W.R. Hamilton’s contribution to the formulationof classical mechanics in terms of what became known as a hamiltonianfunction. To write the explicit form of this operator we simply substitute theappropriate expressions for the kinetic and potential energy operators in thechosen representation. For example, the hamiltonian for a particle of mass mmoving in one dimension isH¼ 2 d2hþ VðxÞ2m dx2ð1:12Þwhere V(x) is the operator for the potential energy. Similarly, the hamiltonianoperator for an electron of mass me in a hydrogen atom isH¼ h 2 2e2r 2me4pe0 rð1:13Þ

1.6 INTEGRALS OVER OPERATORSj15The general prescription for constructing operators in the position representation should be clear from these examples. In short:1. Write the classical expression for the observable in terms of positioncoordinates and the linear momentum.h/i)q/qx (and likewise2. Replace x by multiplication by x, and replace px by ( for the other coordinates).1.6 Integrals over operatorsWhen we want to make contact between a calculation done using operatorsand the actual outcome of an experiment, we need to evaluate certainintegrals. These integrals all have the formZð1:14ÞI ¼ fm Ofn dtThe complex conjugate ofa complex number z ¼ a þ ibis z ¼ a ib. Complexconjugation amounts toeverywhere replacing i by i.The square modulus jzj2 is given byzz ¼ a2 þ b2 since jij2 ¼ 1.where fm is the complex conjugate of fm. In this integral dt is the volumeelement. In one dimension, dt can be identified as dx; in three dimensions it isdxdydz. The integral is taken over the entire space available to the system,which is typically from x ¼ 1 to x ¼ þ 1 (and similarly for the othercoordinates). A glance at the later pages of this book will show that manymolecular properties are expressed as combinations of integrals of this form(often in a notation which will be explained later). Certain special cases of thistype of integral have special names, and we shall introduce them here.When the operator O in eqn 1.14 is simply multiplication by 1, the integralis called an overlap integral and commonly denoted S:Zð1:15ÞS ¼ fm fn dtIt is helpful to regard S as a measure of the similarity of two functions: whenS ¼ 0, the functions are classified as orthogonal, rather like two perpendicularvectors. When S is close to 1, the two functions are almost identical. Therecognition of mutually orthogonal functions often helps to reduce theamount of calculation considerably, and rules will emerge in later sectionsand chapters.The normalization integral is the special case of eqn 1.15 for m ¼ n.A function fm is said to be normalized (strictly, normalized to 1) ifZfm fm dt ¼ 1ð1:16ÞIt is almost always easy to ensure that a function is normalized by multiplyingit by an appropriate numerical factor, which is called a normalization factor,typically denoted N and taken to be real so that N ¼ N. The procedure isillustrated in the following example.Example 1.4 How to normalize a functionA certain function f is sin(px/L) between x ¼ 0 and x ¼ L and is zero elsewhere.Find the normalized form of the function.

16j1 THE FOUNDATIONS OF QUANTUM MECHANICSMethod. We need to find the (real) factor N such that N sin(px/L) is normalized to 1. To find N we substitute this expression into eqn 1.16, evaluate theintegral, and select N to ensure normalization. Note that ‘all space’ extendsfrom x ¼ 0 to x ¼ L.Answer. The necessary integration isZZLN 2 sin2 ðpx LÞdx ¼ 12LN 2Rwhere we have usedsin2ax dx ¼ (x/2)(sin 2ax)/4a þ constant. For thisintegral to be equal to 1, we require N ¼ (2/L)1/2. The normalized function istherefore 1 22f ¼sinðpx LÞLf f dt ¼0Comment. We shall see later that this function describes the distribution of aparticle in a square well, and we shall need its normalized form there.Self-test 1.4. Normalize the function f ¼ eif, where f ranges from 0 to 2p.[N ¼ 1/(2p)1/2]A set of functions fn that are (a) normalized and (b) mutually orthogonalare said to satisfy the orthonormality condition:Zfm fn dt ¼ dmnð1:17ÞIn this expression, dmn denotes the Kronecker delta, which is 1 when m ¼ nand 0 otherwise.1.7 Dirac bracket notationWith eqn 1.14 we are on the edge of getting lost in a complicated notation. Theappearance of many quantum mechanical expressions is greatly simplified byadopting the Dirac bracket notation in which integrals are written as follows:ZhmjOjni ¼ fm Ofn dtð1:18ÞThe symbol jni is called a ket, and denotes the state described by the functionfn. Similarly, the symbol hnj is called a bra, and denotes the complex conjugateof the function, fn . When a bra and ket are strung together with an operatorbetween them, as in the bracket hmjOjni, the integral in eqn 1.18 is to beunderstood. When the operator is simply multiplication by 1, the 1 is omittedand we use the conventionZð1:19Þhmjni ¼ fm fn dtThis notation is very elegant. For example, the normalization integralbecomes hnjni ¼ 1 and the orthogonality condition becomes hmjni ¼ 0for m 6¼ n. The combined orthonormality condition (eqn 1.17) is thenhmjni ¼ dmnð1:20Þ

1.8 HERMITIAN OPERATORSj17A final point is that, as can readily be deduced from the definition of a Diracbracket,hmjni ¼ hnjmi 1.8 Hermitian operatorsAn operator is hermitian if it satisfies the following relation: ZZ fm Ofn dt ¼fn Ofm dtð1:21aÞfor any two functions fm and fn. An alternative version of this definition isZZð1:21bÞfm Ofn dt ¼ ðOfm Þ fn dtThis expression is obtained by taking the complex conjugate of each term onthe right-hand side of eqn 1.21a. In terms of the Dirac notation, the definitionof hermiticity ishmjOjni ¼ hnjOjmi ð1:22ÞExample 1.5 How to confirm the hermiticity of operatorsShow that the position and momentum operators in the position representation are hermitian.Method. We need to show that the operators satisfy eqn 1.21a. In some cases(the position operator, for instance), the hermiticity is obvious as soon as theintegral is written down. When a differential operator is used, it may benecessary to use integration by parts at some stage in the argument to transferthe differentiation from one function to another:ZZu dv ¼ uv v duAnswer. That the position operator is hermitian is obvious from inspection:Zfm xfndt ¼Zfn xfm dt ¼ Z fn xfm dtWe have used the facts that (f ) ¼ f and x is real. The demonstration of thehermiticity of px, a differential operator in the position representation,involves an integration by parts:ZZZ dhh fn dx ¼fm dfni dxi Zx¼1h ¼fm fn fn dfm ix¼ 1 Z 1h x¼1 d¼ fn fm dxf fn ji m x¼ 1dx 1fm px fn dx ¼fm

18j1 THE FOUNDATIONS OF QUANTUM MECHANICSThe first term on the right is zero (because when jxj is infinite, a normalizablefunction must be vanishingly small; see Section 1.12). Therefore,Z hi Zfm px fn dx ¼ ¼Zfn fnd f dxdx m dhfm dxi dx ¼ Z fn px fm dxHence, the operator is hermitian.Self-test 1.5. Show that the two operators are hermitian in the momentumrepresentation.As we shall now see, the property of hermiticity has far-reaching implications. First, we shall establish the following property:Property 1. The eigenvalues of hermitian operators are real.Proof 1.1 The reality of eigenvaluesConsider the eigenvalue equationOjoi ¼ ojoiThe ket joi denotes an eigenstate of the operator O in the sense that thecorresponding function fo is an eigenfunction of the operator O and we arelabelling the eigenstates with the eigenvalue o of the operator O. It is oftenconvenient to use the eigenvalues as labels in this way. Multiplication from theleft by hoj results in the equationhojOjoi ¼ ohojoi ¼ otaking joi to be normalized. Now take the complex conjugate of both sides:hojOjoi ¼ o However, by hermiticity, hojOjoi ¼ hojOjoi. Therefore, it follows thato ¼ o , which implies that the eigenvalue o is real.The second property we shall prove is as follows:Property 2. Eigenfunctions corresponding to different eigenvalues of anhermitian operator are orthogonal.That is, if we have two eigenfunctions of an hermitian operator O witheigenvalues o and o 0 , with o 6¼ o 0 , then hojo 0 i ¼ 0. For example, it follows atonce that all the eigenfunctions of a harmonic oscillator (Section 2.16) aremutually orthogonal, for as we shall see each one corresponds to a differentenergy (the eigenvalue of the hamiltonian, an hermitian operator).

1.9 STATES AND WAVEFUNCTIONSj19Proof 1.2 The orthogonality of eigenstatesSuppose we have two eigenstates joi and jo 0 i that satisfy the followingrelations:Ojoi ¼ ojoi andOjo0 i ¼ o0 jo0 iThen multiplication of the first relation by ho 0 j and the second by hoj givesho0 jOjoi ¼ oho0 joi andhojOjo0 i ¼ o0 hojo0 iNow take the complex conjugate of the second relation and subtract it fromthe first while using Property 1 (o 0 ¼ o 0 ):ho0 jOjoi hojOjo0 i ¼ oho0 joi o0 hojo0 i Because O is hermitian, the left-hand side of this expression is zero; so (notingthat o 0 is real and using hojo 0 i ¼ ho 0 joi as explained earlier) we arrive atðo o0 Þho0 joi ¼ 0However, because the two eigenvalues are different, the only way of satisfyingthis relation is for ho 0 joi ¼ 0, as was to be proved.The postulates of quantum mechanicsNow we turn to an application of the preceding material, and move into thefoundations of quantum mechanics. The postulates we use as a basis forquantum mechanics are by no means the most subtle that have been devised,but they are strong enough for what we have to do.1.9 States and wavefunctionsThe first postulate concerns the information we can know about a state:Postulate 1. The state of a system is fully described by a function C(r1,r2, . . . , t).In this statement, r1, r2, . . . are the spatial coordinates of particles 1, 2, . . .that constitute the system and t is the time. The function C (uppercase psi)plays a central role in quantum mechanics, and is called the wavefunction ofthe system (more specifically, the time-dependent wavefunction). When weare not interested in how the system changes in time we shall denote thewavefunction by a lowercase psi as c(r1, r2, . . . ) and refer to it as the timeindependent wavefunction. The state of the system may also depend on someinternal variable of the particles (their spin states); we ignore that for nowand return to it later. By ‘describe’ we mean that the wavefunctioncontains information about all the properties of the system that are open toexperimental determination.We shall see that the wavefunction of a system will be specified by a set oflabels called quantum numbers, and may then be written ca,b, . . . , wherea, b, . . . are the quantum numbers. The values of these quantum numbersspecify the wavefunction and thus allow the values of various physical

20j1 THE FOUNDATIONS OF QUANTUM MECHANICSobservables to be calculated. It is often convenient to refer to the state ofthe system without referring to the corresponding wavefunction; the state isspecified by listing the values of the quantum numbers that define it.1.10 The fundamental prescriptionThe next postulate concerns the selection of operators:Postulate 2. Observables are represented by hermitian operators chosen tosatisfy the commutation relations½q, pq0 ¼ i hdqq0½q, q0 ¼ 0 ½pq , pq0 ¼ 0where q and q 0 each denote one of the coordinates x, y, z and pq and pq 0 thecorresponding linear momenta.The requirement that the operators are hermitian ensures that the observableshave real values (see below). Each commutation relation is a basic, unprovable, and underivable postulate. Postulate 2 is the basis of the selection ofthe form of the operators in the position and momentum representations forall observables that depend on the position and the momentum.2 Thus, if wedefine the position representation as the representation in which the positionoperator is multiplication by the position coordinate, then as we saw inExample 1.3, it follows that the momentum operator must involve differentiation with respect to x, as specified earlier. Similarly, if the momentumrepresentation is defined as the representation in which the linear momentumis represented by multiplication, then the form of the position operator isfixed as a derivative with respect to the linear momentum. The coordinatesx, y, and z commute with each other as do the linear momenta px, py, and pz.1.11 The outcome of measurementsThe next postulate brings together the wavefunction and the operators andestablishes the link between formal calculations and experimental observations:Postulate 3. When a system is described by a wavefunction c, the meanvalue of the observable O in a series of measurements is equal to the expectation value of the corresponding operator.The expectation value of an operator O for an arbitrary state c is denoted hOiand defined asR c Oc dt hcjOjcihOi ¼ R ð1:23Þ¼hcjcic cdtIf the wavefunction is chosen to be normalized to 1, then the expectationvalue is simplyZð1:24ÞhOi ¼ c Oc dt ¼ hcjOjciUnless we state otherwise, from now on we shall assume that the wavefunction is normalized to 1.2. This prescription excludes intrinsic observables, such as spin (Section 4.8).

1.11 THE OUTCOME OF MEASUREMENTSj21The meaning of Postulate 3 can be unravelled as follows. First, supposethat c is an eigenfunction of O with eigenvalue o; thenZZZ ð1:25ÞhOi ¼ c Oc dt ¼ c oc dt ¼ o c c dt ¼ oThat is, a series of experiments on identical systems to determine O will givethe average value o (a real quantity, because O is hermitian). Now supposethat although the system is in an eigenstate of the hamiltonian it is not in aneigenstate of O. In this case the wavefunction can be expressed as a linearcombination of eigenfunctions of O:Xcn cn where Ocn ¼ on cnc¼nIn this case, the expectation value is! !ZZ XXXhOi ¼cm cm Ocn cn dt ¼c m cn c m Ocn dtmnm, nZX¼c m cn on c m cn dtm, nBecause the eigenfunctions form an orthonormal set, the integral in the lastexpression is zero if n 6¼ m, is 1 if n ¼ m, and the double sum reduces to asingle sum:ZXX 2X cn cn on c n cn dt ¼c n cn on ¼jcn j onð1:26ÞhOi ¼nnnThat is, the expectation value is a weighted sum of the eigenvalues of O,the contribution of a particular eigenvalue to the sum being determined by thesquare modulus of the corresponding coefficient in the expansion of thewavefunction.We can now interpret the difference between eqns 1.25 and 1.26 in theform of a subsidiary postulate:Postulate 30 . When c is an eigenfunction of the operator O, the determination of the property O always yields one result, namely the correspondingeigenvalue o. The expectation value will simply be the eigenvalue o. When cis not an eigenfunction of O, a single measurement of the property yieldsa single outcome which is one of the eigenvalues of O, and the probability thata particular eigenvalue on is measured is equal to jcnj2, where cn is thecoefficient of the eigenfunction cn in the expansion of the wavefunction.One measurement can give only one result: a pointer can indicate only onevalue on a dial at any instant. A series of determinations can lead to a series ofresults with some mean value. The subsidiary postulate asserts that a measurement of the observable O always results in the pointer indicating one ofthe eigenvalues of the corresponding operator. If the function that describesthe state of the system is an eigenfunction of O, then every pointer reading isprecisely o and the mean value is also o. If the system has been prepared in astate that is not an eigenfunction of O, then different measurements givedifferent values, but every individual measurement is one of the eigenvalues of

22j1 THE FOUNDATIONS OF QUANTUM MECHANICSO, and the probability that a particular outcome on is obtained is determinedby the value of jcnj2. In this case, the mean value of all the observations is theweighted average of the

The foundations of quantum mechanics Operators in quantum mechanics 1.1 Linear operators 1.2 Eigenfunctions and eigenvalues 1.3 Representations 1.4 Commutation and non-commutation 1.5 The construction of operators 1.6 Integrals over operators 1.7 Dirac bracket notation 1.8 Hermitian operators The postulates of quantum mechanics 1.9 States and .