Elementary quantum mechanics Going over to the continuous limit corresponds to letting oo.In this limit,the sum in(1.20)can be replaced by an integral provided that the density of states D(p)is included in the integrand and that the toas c(p)of p space.In this case,the artificial large volume y is redundant and to get rid of it,it is common to rescale cpc(p)/such that c(p)does not depend on V.With this rescaling the relations(1.21)and(r)=(r)become the expressions of the standard Fourier transforms.forward and reverse respectively p=∫de)and=∫a2 dp (1.22) Control question.Do you see how the relations(1.22)follow from(120)and(121). using the rescaling cpc(p)/V? So,what are the advantages of the Dirac formulation?First,it brings us from separate quantum particles to the notion of a quantum system.A system can be a single particle acting with each other,or an electric circuit,or a vacuun all thes examples are considered in this book.In the Dirac formulation they can be described uni formly,since all their quantum states are elements of a Hilbert space.The case of a system consisting of many particles is considered in detail.In the Schrodinger picture,the wave function of such a system becomes a function of all coordinates of all,say 1026,particles, whereas in the Dirac formulation all allowed quantum states are still simply represented by a vector.Of cou fo ra many-particle problem n more states in the Hilb rt space are rele vant.However.this does not alter the complexity of the representation:a vector remains a vector irrespective of the number of its components. Second,the Dirac formulation establishes a"democracy"in quantum mechanics.Given tdoes not matte whch as ued tor deserpton of wav fungtion.Al equal,and the choice of basis is determined by the personal taste of the descriptor rather than by any characteristic of the system Third,the formulation is practically important.It converts any quantum problem for any quantum system to a linear algebra exercise.Since wave functions are now repre- sented by vectors and operators by matrices,finding the eigenstates and eigenvalues of any observable reduces to diagonalizing the corresponding operator matrix. considering the Dirac formulation f or complex x system which is ning two separate systems.Suppose we have systems A and B whic are both described by their own wave functions.We assume that the quantum states of system A are vectors in an Ma-dimensional Hilbert space,spanned by a basis (Ina)}= ()12)...IM)).while the states of system B are vectors in an MR-dimensional space with basis (n))=(1).12) .IMg)).If we combine the two systems and create on large systemA+B.we find that the states of this combined system can be described ina Hilbert space spanned by the direct product of the two separate bases,(InA))(ng))= (11A1g).11A2g),12A18).1A38)....).which forms a (MAMg)-dimensional basis
10 Elementary quantum mechanics Going over to the continuous limit corresponds to letting V → ∞. In this limit, the sum in (1.20) can be replaced by an integral provided that the density of states D(p) is included in the integrand and that the components cp converge to a smooth function c(p) of p. This is the case when the wave functions are concentrated in a finite region of coordinate space. In this case, the artificial large volume V is redundant and to get rid of it, it is common to rescale cp → c(p)/ √ V such that c(p) does not depend on V. With this rescaling the relations (1.21) and ψ(r) = r|ψ become the expressions of the standard Fourier transforms, forward and reverse respectively, c(p) = dr e − i h¯ p·r ψ(r) and ψ(r) = dp (2πh¯)3 e i h¯ p·r c(p). (1.22) Control question. Do you see how the relations (1.22) follow from (1.20) and (1.21), using the rescaling cp → c(p)/ √ V? So, what are the advantages of the Dirac formulation? First, it brings us from separate quantum particles to the notion of a quantum system. A system can be a single particle, or 1026 particles interacting with each other, or an electric circuit, or a vacuum – all these examples are considered in this book. In the Dirac formulation they can be described uniformly, since all their quantum states are elements of a Hilbert space. The case of a system consisting of many particles is considered in detail. In the Schrödinger picture, the wave function of such a system becomes a function of all coordinates of all, say 1026, particles, whereas in the Dirac formulation all allowed quantum states are still simply represented by a vector. Of course, for a many-particle problem more states in the Hilbert space are relevant. However, this does not alter the complexity of the representation: a vector remains a vector irrespective of the number of its components. Second, the Dirac formulation establishes a “democracy” in quantum mechanics. Given a system, it does not matter which basis is used for a description of a wave function. All bases are equal, and the choice of basis is determined by the personal taste of the descriptor rather than by any characteristic of the system. Third, the formulation is practically important. It converts any quantum problem for any quantum system to a linear algebra exercise. Since wave functions are now represented by vectors and operators by matrices, finding the eigenstates and eigenvalues of any observable reduces to diagonalizing the corresponding operator matrix. Let us finish by considering the Dirac formulation for a complex system which is obtained by combining two separate systems. Suppose we have systems A and B which are both described by their own wave functions. We assume that the quantum states of system A are vectors in an MA-dimensional Hilbert space, spanned by a basis {|nA} ≡ {|1A, |2A, ... , |MA}, while the states of system B are vectors in an MB-dimensional space with basis {|nB} ≡ {|1B, |2B, ... , |MB}. If we combine the two systems and create one large system A + B, we find that the states of this combined system can be described in a Hilbert space spanned by the direct product of the two separate bases, {|nA} ⊗ {|nB} ≡ {|1A1B, |1A2B, |2A1B, |1A3B, ...}, which forms a (MAMB)-dimensional basis.
1.4 and Heisenberg pictures 1.4 Schrodinger and Heisenberg pictures Suppose we are interested in the time-dependence of the expectation value of a physical observable A,so we want to know 40)=(0A). (1.23) This expression suggests the following approach.First we use the Schrodinger equation to find the time-dependent wave function lv(t)).With this solution.we can calculate expectation value of the (time-ind Tisrach is commoly called the Scrodinger pictue since it usesth the e any equation. We now note that we can write the time-evolution equation for the expectation value as =偿a+a》 (1.24) and immediately use the Schrodinger equation to bring it to the form aw-a.A: (1.25) at where the commutator [A.B]of two operators A and B is defined as [A.B]=AB-BA.This equation in fact hints at another interpretation of time-dependence in quantum mechanics. One can regad the wave function onary,and assign the time-dependenc to the operators.We see that we can reproduce the same time-dependent expectation value (A(r))if we keep the state)stationary,and let the operator A evolve obeying the(Heisenberg)equation of motion (1.26 dt Here,the subscript H indicates that we switch to an alternative picture,the Heisenberg picture,where A(0》=(1A) (1.27) any wave function.It also has a certain physical appeal:it produces quantum mechani- cal equations of motion for operators which are similar to their classical analogues.For instance,if we use the Hamiltonian in (1.5)to find the Heisenberg equations of motion for the position and momentum operator,we exactly reproduce Newton's equations(1.1) with classical variables replaced by operators.For this.we must postulate the commutatior relation between the operators p and r. [pa,iB]=-ih8aB. (1.28) o and B labeling the Cartesian components of both vectors.We note that this is consistent with the definition of the ope rators in the Schrodinger picture.where p=
11 1.4 Schrödinger and Heisenberg pictures 1.4 Schrödinger and Heisenberg pictures Suppose we are interested in the time-dependence of the expectation value of a physical observable A, so we want to know A(t)=ψ(t)|Aˆ|ψ(t). (1.23) This expression suggests the following approach. First we use the Schrödinger equation to find the time-dependent wave function |ψ(t). With this solution, we can calculate the expectation value of the (time-independent) operator Aˆ at any moment of time. This approach is commonly called the Schrödinger picture since it uses the Schrödinger equation. We now note that we can write the time-evolution equation for the expectation value as ∂A ∂t = ∂ψ ∂t |Aˆ|ψ + ψ|Aˆ| ∂ψ ∂t , (1.24) and immediately use the Schrödinger equation to bring it to the form ∂A ∂t = i h¯ ψ|[Hˆ , Aˆ]|ψ, (1.25) where the commutator [Aˆ, Bˆ] of two operators Aˆ and Bˆ is defined as [Aˆ, Bˆ] = AˆBˆ −BˆAˆ. This equation in fact hints at another interpretation of time-dependence in quantum mechanics. One can regard the wave function of a system as stationary, and assign the time-dependence to the operators. We see that we can reproduce the same time-dependent expectation value A(t) if we keep the state |ψ stationary, and let the operator Aˆ evolve obeying the (Heisenberg) equation of motion ∂Aˆ H ∂t = i h¯ [Hˆ , Aˆ H]. (1.26) Here, the subscript H indicates that we switch to an alternative picture, the Heisenberg picture, where A(t)=ψ|Aˆ H(t)|ψ. (1.27) For many practical problems the Heisenberg picture is the most convenient framework to work in. It allows us to evaluate all dynamics of a system without having to calculate any wave function. It also has a certain physical appeal: it produces quantum mechanical equations of motion for operators which are similar to their classical analogues. For instance, if we use the Hamiltonian in (1.5) to find the Heisenberg equations of motion for the position and momentum operator, we exactly reproduce Newton’s equations (1.1) with classical variables replaced by operators. For this, we must postulate the commutation relation between the operators pˆ and rˆ, [pˆα,rˆβ] = −ih¯δαβ, (1.28) α and β labeling the Cartesian components of both vectors. We note that this is consistent with the definition of the operators in the Schrödinger picture, where pˆ = −ih¯∂r
2 Elementary quantum mechanics Werner Heisenberg(1901-1976) Won the Nobel Prize in 1932 for"the creation of quantum mechanics,the ppli ation of which has nter led to the discovery of the allotropic forms of hydrogen.' In 1925 Heisenberg was only 23 years old.He was a young Privatdozent in Gottingen and was calculating the energy spectrum of hydrogen,and he wanted to describe the atom in te y.To esc ape hay fever he went to the isolated island of Heligoland,and it was there where he realized that the solution was to make the observables non-commuting.that is,matrices.Within six months he developed, together with more experienced colleagues max born and Pascual Jordan.his ideas into the first consistent quantum theory:matrix mechanics. Heisenberg stayed in Germany during the Nazi period,and was even appointed estiga at the endof the war Heisenberg was apprehended by the allies and prison for several months.In 1946.he returned to Gottingen,to become Director of the Max Planck Institute for Physics.When the Institute moved to Munich in 1958. Heisenberg moved along and held his post until his retirement in 1970. Surprisingly,the Heisenberg picture does not involve stationary states.atomic levels and mysterious wave functions ng we took for granted from the mere first introduc tory course of quantum mechanics!All these concepts are not needed here.Nevertheless. solving a problem in the Heisenberg picture produces the same results as in the Schrodinger picture,which does involve wave functions,etc.Therefore,quantum mechanics gives a clear and logical example of a physical theory where two individuals can start with very atible concepts ,and reco cile t nselves by predic detical physical obseryable One c dw her ymodem pura society where persons of opposite views can live together and collaborate successfully (unless they try to convince each other of the validity of their views). The two opposite pictures-the Heisenberg and Schrodinger pictures-are not isolated one can interp olate bet en the two.A resultinghybrid"picture is called the interaction pictre.To understand the interpolation,let us arbitrarily split the Hamiltonian into two parts. 户=1+2 (1.29) Weassiallperatratime-dependence govemned by the first part of the Hamilonian (the subscript Ihere indicates the interaction picture) (1.30)
12 Elementary quantum mechanics Werner Heisenberg (1901–1976) Won the Nobel Prize in 1932 for “the creation of quantum mechanics, the application of which has, inter alia, led to the discovery of the allotropic forms of hydrogen.” In 1925 Heisenberg was only 23 years old. He was a young Privatdozent in Göttingen and was calculating the energy spectrum of hydrogen, and he wanted to describe the atom in terms of observables only. To escape hay fever, he went to the isolated island of Heligoland, and it was there where he realized that the solution was to make the observables non-commuting, that is, matrices. Within six months he developed, together with more experienced colleagues Max Born and Pascual Jordan, his ideas into the first consistent quantum theory: matrix mechanics. Heisenberg stayed in Germany during the Nazi period, and was even appointed director of the Kaiser Wilhelm Institute in Berlin. He became one of the principal investigators involved in the “Uranverein,” the German nuclear project. This is why at the end of the war Heisenberg was apprehended by the allies and put in an English prison for several months. In 1946, he returned to Göttingen, to become Director of the Max Planck Institute for Physics. When the Institute moved to Munich in 1958, Heisenberg moved along and held his post until his retirement in 1970. Surprisingly, the Heisenberg picture does not involve stationary states, atomic levels and mysterious wave functions – everything we took for granted from the mere first introductory course of quantum mechanics! All these concepts are not needed here. Nevertheless, solving a problem in the Heisenberg picture produces the same results as in the Schrödinger picture, which does involve wave functions, etc. Therefore, quantum mechanics gives a clear and logical example of a physical theory where two individuals can start with very different and seemingly incompatible concepts, and reconcile themselves by predicting identical physical observables. One can draw here an analogy with a modern pluralistic society where persons of opposite views can live together and collaborate successfully (unless they try to convince each other of the validity of their views). The two opposite pictures – the Heisenberg and Schrödinger pictures – are not isolated, one can interpolate between the two. A resulting “hybrid” picture is called the interaction picture. To understand the interpolation, let us arbitrarily split the Hamiltonian into two parts, Hˆ = Hˆ 1 + Hˆ 2. (1.29) We assign to all operators a time-dependence governed by the first part of the Hamiltonian (the subscript I here indicates the interaction picture), ∂AˆI ∂t = i h¯ [Hˆ 1, AˆI], (1.30)
13 1.5 Perturbation theory and the dynamics of the wave functions are determined by the second part h恤= (1.31) Note that the Hamiltonian A used in (1.31).being an operator,has also acquired the time-dependence governed by(1.30). Control question.Which choice of H and A2 reproduces the Heisenberg picture? And which reproduces the Schrodinger picture? This interaction picture is useful for time-dependent perturbation theory outlined in Section 1.6. 1.5 Perturbation theory equations are known for very few turbation theory being an indispensable tool for this.Let us consider a"composite" Hamiltonian =0+ (1.32 anry leumhe puram he ma e see later the precise meaning of this.This smallness ensures that the energy levels and to those of ony by mceonand the To do so,let us consider an auxiliary Hamiltonian A=Ho+aA' (1.33) We expand its eigenstates in the convenient basis of (1.34 where the coefficients cm should satisfy the normalization condition dnm=∑cncm (1.35 Control and(1.l7)2 The Schrodinger equation in these notations becomes {Ea(a)-E9om=a∑CnpMmp (1.36) where Mm are the matrix elements of the perturbation.Mm=(nAm)
13 1.5 Perturbation theory and the dynamics of the wave functions are determined by the second part, ih¯ ∂|ψI ∂t = Hˆ 2,I|ψI. (1.31) Note that the Hamiltonian Hˆ 2,I used in (1.31), being an operator, has also acquired the time-dependence governed by (1.30). Control question. Which choice of Hˆ 1 and Hˆ 2 reproduces the Heisenberg picture? And which reproduces the Schrödinger picture? This interaction picture is useful for time-dependent perturbation theory outlined in Section 1.6. 1.5 Perturbation theory The precise analytical solutions to Schrödinger equations are known for very few Hamiltonians. This makes it important to efficiently find approximate solutions, perturbation theory being an indispensable tool for this. Let us consider a “composite” Hamiltonian Hˆ = Hˆ 0 + Hˆ , (1.32) which consists of a Hamiltonian Hˆ 0 with known eigenstates |n(0) and corresponding energy levels E(0) n , and a perturbation Hˆ . The perturbation is assumed to be small, we see later the precise meaning of this. This smallness ensures that the energy levels and eigenstates of Hˆ differ from those of Hˆ 0 only by small corrections, and the goal is thus to find these corrections. To do so, let us consider an auxiliary Hamiltonian Hˆ = Hˆ 0 + αHˆ . (1.33) We expand its eigenstates |n in the convenient basis of |n(0), |n = m cnm|m (0), (1.34) where the coefficients cnm should satisfy the normalization condition δmn = p c∗ mpcnp. (1.35) Control question. Do you see the equivalence between this normalization condition and (1.17)? The Schrödinger equation in these notations becomes En(α) − E(0) m cnm = α p cnpMmp, (1.36) where Mnm are the matrix elements of the perturbation, Mnm = n|Hˆ |m
4 Elementary quantum mechanics Control question.Can you derive(1.36)from the Schrodinger equation? To proceed,we seek for En and cm in the form of a Taylor expansion in En(a)=Et)+aEd)+a2)+.... (1.37) cmm =crom+acm+a.cm+.. The perturbation theory formulated in (1.35)and (1.36)can be solved by subsequent orrections of any order N can be expressed in terms of the cor- Tomake pracieal ne rerictonnd t I at the end of the calculation. The resulting corrections are simple only for the first and second order,and those are widely used.Let us give their explicit form. Ed Man (1.38) 出=E西a≠m=0 (1.39) (1.40) 品-乃 MmpMpn Mmn Mmn -0”-0-” forn≠m (1.41 鼎=∑ (E0-P with the same energy.In this case the above expressions blow up and one has to use a degenerate perturbation theory.This observation also sets a limit on the relative magni- tude of Ho and A'.All elements of A'must be much smaller than any energy difference E)-Ebetween eigenstates of Ao. 1.6 Time-dependent perturbation theory Let us consider again the standard framework of perturbation theory,where the Hamiltonian car be split into uncomplicate time-independent term,and a smal perturbation which we now allow to be time-dependent. 户=Ao+0). (1.42) In this case,we cannot apply the perturbation theory of Section 1.5 since the HamiltonianA has no stationary eigenstates and we cannot write a related stationary eigenvalue problem. remove the ambiguity.we require ()=0
14 Elementary quantum mechanics Control question. Can you derive (1.36) from the Schrödinger equation? To proceed, we seek for En and cnm in the form of a Taylor expansion in α, En(α) = E(0) n + αE(1) n + α2E(2) n + ... , cnm = c(0) nm + αc(1) nm + α2c(2) nm + ... (1.37) The perturbation theory formulated in (1.35) and (1.36) can be solved by subsequent approximations:1 the corrections of any order N can be expressed in terms of the corrections of lower orders. To make it practical, one restricts to a certain order, and sets α to 1 at the end of the calculation. The resulting corrections are simple only for the first and second order, and those are widely used. Let us give their explicit form, E(1) n = Mnn, (1.38) c(1) nm = Mmn E(0) n − E(0) m for n = m, c(1) nn = 0, (1.39) E(2) n = p=n MnpMpn E(0) n − E(0) p , (1.40) c(2) nm = p=n MmpMpn (E(0) n − E(0) m )(E(0) n − E(0) p ) − MmnMnn (E(0) n − E(0) m )2 for n = m, c(2) nn = −1 2 p=n MnpMpn (E(0) n − E(0) p )2 . (1.41) From these expressions we see that the theory breaks down if there exist degenerate eigenstates of Hˆ 0, i.e. E(0) n = E(0) m for some m = n, or in words, there exist multiple eigenstates with the same energy. In this case the above expressions blow up and one has to use a degenerate perturbation theory. This observation also sets a limit on the relative magnitude of Hˆ 0 and Hˆ . All elements of Hˆ must be much smaller than any energy difference E(0) n − E(0) m between eigenstates of Hˆ 0. 1.6 Time-dependent perturbation theory Let us consider again the standard framework of perturbation theory, where the Hamiltonian can be split into a large, uncomplicated, time-independent term, and a small perturbation which we now allow to be time-dependent, Hˆ = Hˆ 0 + Hˆ (t). (1.42) In this case, we cannot apply the perturbation theory of Section 1.5 since the Hamiltonian Hˆ has no stationary eigenstates and we cannot write a related stationary eigenvalue problem. 1 Equations (1.35) and (1.36) do not determine cnm unambiguously. The point is that any quantum state is determined up to a phase factor: the state |ψ = eiφ|ψ is equivalent to the state |ψ for any phase φ. To remove the ambiguity, we require ( ∂ ∂α n|)|n = 0