Phys304 Quantum Physics II (2005) Quantum Mechanics Summary. 2. This kind of behaviour can be described in the mathematical language of vectors:

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1 MACQUARIE UNIVERSITY Department of Physcs Dvson of ICS Phys304 Quantum Physcs II (2005) Quantum Mechancs Summary The followng defntons and concepts set up the basc mathematcal language used n quantum mechancs, the postulates on whch quantum theory s based, and some of the fundamental deas and prncples of the theory. Physcal Motvaton 1. Quantum theory has ts orgns n the emprcal fact that physcal systems can be observed to exhbt the followng behavour: (a) Outcomes of experments repeated under dentcal condtons vary n an rreducbly random fashon. (b) Physcal systems prepared n a gven ntal state can be observed n a fnal state n a way that exhbts nterference behavour f the ntermedate state of the system between ntal preparaton and fnal observaton s not observed. Ths nterference vanshes f the ntermedate state of the system s observed. 2. Ths knd of behavour can be descrbed n the mathematcal language of vectors: (a) From part 1a, let P(φ ψ) be the (condtonal) probablty of observng, n an experment, a system to be n the state φ gven that t was prepared n a state ψ where ψ and φ are lsts of nformaton that can be used to specfy the states of the system. Then, from part 1b, the presence of nterference can be understood f ths probablty s the square of a probablty ampltude φ ψ,.e. where P(φ ψ) = φ ψ 2 φ ψ = φ n n ψ n and where the n label all the possble mutually exclusve ntermedate states of the system. (b) The probablty P(φ ψ) s then P(φ ψ) = P(φ n)p(n ψ) + 2Re ( φ n n ψ ) φ m m ψ n n,m n m where the cross terms gve rse to the nterference effects. Observaton of the ntermedate state reduces ths to the standard result of the theory of probablty: P(φ ψ) = P(φ n)p(n ψ). n

2 (c) The cancellaton of common terms then gves ψ = n n ψ n whch leads to the nterpretaton of ψ and the n s as vectors belongng to a complex nner product vector space or state space wth the probablty ampltudes n ψ as the weghtng of the ntermedate states n. (d) The probablty nterpretaton along wth the mutual exclusvty of the ntermedate states leads to the orthonormalty condton n m = δ nm (e) The states n exhaustvely cover all the possble ntermedate states and so consttute a complete orthonormal set of bass states for the state space of the system. Mathematcal Background The mathematcal background s ntated by what s effectvely a postulate, so as to motvate the mathematcal language used. The postulates themselves are consdered n detal later. 1. Every physcal state of a quantum system s represented by a symbol known as a ket wrtten... where... s a label specfyng the physcal nformaton known about the state. An arbtrary state s wrtten ψ, or φ and so on. 2. Lnear combnatons or superpostons of such states defne other possble states of the system.e. every lnear superposton of two or more kets φ 1, φ 2, φ 3,..., s also a state of the quantum system. Thus the ket ψ gven by ψ = c 1 φ 1 + c 2 φ 2 + c 3 φ also represents a physcal state of the system for all complex numbers c 1, c 2, c 3, If a state of the system s represented by a ket ψ, then the same physcal state s represented by the ket c ψ where c s any non-zero complex number. 4. The set of all kets descrbng a gven physcal system forms a complex vector space (or Hlbert space) H also known as the state space or ket space for the system. A ket s thus a vector belongng to H. A ket s also referred to as a state vector, ket vector, or sometmes just state. 5. A set of vectors φ 1, φ 2, φ 3,... s sad to be complete f every state of the quantum system can be represented as a lnear superposton of the φ s.e. for any state ψ we can wrte ψ = c φ. The set of vectors φ, = 1, 2,... are sad to span the vector space. 6. A bass for the state space H s a complete set of lnearly ndependent vectors that span all of H. 7. The number of bass states makng up a complete set of bass states for a state space H s the dmenson of the state space. Ths can be ether fnte or nfnte. In the latter case, the bass states can be denumerable (.e. dscrete) or non-denumerable (.e. contnuous). 8. In quantum mechancs, a state space of nfnte dmenson s assumed to be separable,.e. that there always exsts a denumerable set of bass states. 2

3 Inner Product The followng defntons and propertes defne the nner product of vectors belongng to a state space H, and ts representaton n terms of bra vectors. 1. If φ and ψ are any two vectors belongng to H, then the nner product of these two vectors s a rule that maps the par of vectors φ, ψ nto a complex number, wrtten ( φ, ψ ) wth the propertes (a) ( φ, ψ ) = c, a complex number; (b) ( φ, c 1 ψ 1 +c 2 ψ 2 ) = c 1 ( φ, ψ 1 )+c 2 ( φ, ψ 2 ) where c 1 and c 2 are complex numbers; (c) ( ψ, ψ ) 0. If ( ψ, ψ )=0 then ψ = 0, the zero vector. (d) The quantty ( ψ, ψ ) s known as the length or norm of ψ. (e) A state φ s normalzed, or normalzed to unty, f ( φ, φ ) = 1. Two states φ and ψ are orthogonal f ( φ, ψ ) = The nner product ( ψ, φ ) defnes, for all states ψ, the set of functons (or lnear functonals) ( ψ, ). The lnear functonal ( ψ, ) maps any ket vector φ nto the complex number gven by the nner product ( ψ, φ ). (a) The set of all lnear functonals ( ψ, ) forms a complex vector space H, the dual space of H. (b) The lnear functonal ( ψ, ) s wrtten ψ and s known as a bra vector. (c) To each ket vector ψ there corresponds a bra vector ψ such that f φ 1 φ 1 and φ 2 φ 2 then c 1 φ 1 + c 2 φ 2 c 1 φ 1 + c 2 φ 2. (d) In terms of the bra vector notaton, the nner product s wrtten ( ψ, φ ) = ψ φ. 3. A Hlbert space s a vector space on whch there s defned an nner product and for whch certan convergence crtera need to be satsfed, namely that every Cauchy sequence of vectors belongng to the vector space must converge to a vector that also belongs to the space, the convergence beng defned n terms of the nner product. (A condton requred n order to be able, for nstance, to defne the dervatves of state vectors). Operatons on States 1. Operators: State vectors can be transformed nto other state vectors by the acton of operators. The effect of an operator  actng on a state vector ψ s to change the state of the system to a new state φ =  ψ. An operator s fully characterzed when ts effect on every state of the state space s known. 2. Lnear Operators: An operator  s lnear f t has the property Â(c 1 ψ 1 + c 2 ψ 2 ) = c 1  ψ 1 + c 2  ψ 2 for all complex numbers c 1 and c 2 and all states ψ 1 and ψ 2. 3

4 3. Sum of Two Operators: The sum  + ˆB of two operators  and ˆB s defned by for all states ψ. ( + ˆB) ψ =  ψ + ˆB ψ 4. Product of Two Operators: The product  ˆB of two operators  and ˆB s defned by for all states ψ. ( ˆB) ψ = Â( ˆB ψ ) 5. The Zero Operator: The zero operator ˆ0 s such that ˆ0 ψ = 0, the zero vector, for all ψ. 6. The Unt Operator: The unt operator ˆ1 s such that ˆ1 ψ = ψ for all ψ. 7. Projecton Operators: An operator ˆP wth the property ˆP 2 = ˆP s known as a projecton operator. 8. Commutator: The commutator of two operators  and ˆB s defned by [Â, ˆB] =  ˆB ˆBÂ. 9. Equalty of Operators: Two operators  and ˆB are equal f  ψ = ˆB ψ for all states ψ. 10. Acton of Operator on Bra Vectors: The expresson ψ  s defned to be such that for all φ. ( ψ Â) φ = ψ(â φ ) 11. Adjont of an Operator: The adjont (or Hermtean conjugate)  of an operator  s defned such that f  ψ = φ then ψ  = φ. 12. Self Adjont or Hermtean Operators: If  =  then the operator  s self adjont or Hermtean. [Note that mathematcally speakng, these two terms do not mean qute the same thng, but n quantum mechancs, the dstncton s not usually made]. 13. Ket-Bra Notaton for Operators: An expresson of the general form wth the rules of manpulaton  = c φ ψ (a) φ (c 1 ψ 1 + c 2 ψ 2 ) = c 1 φ ψ 1 + c 2 φ ψ 2 (b)  χ = c φ ψ χ (c) χ  = c χ φ ψ s a lnear operator. Wth ths notaton t then follows that: (a) The Hermtean conjugate of the operator  s:  = ψ φ. (b) Any projecton operator ˆP can be expressed n the form ˆP = ψ ψ where ψ s normalzed to unty. 4

5 14. Egenvalues and Egenvectors: A vector s an egenvector, egenket, or egenstate of an operator  f  ψ = a ψ where a s n general a complex number known as the egenvalue belongng to the egenvector ψ. The usual notaton s to dentfy the egenvector by ts egenvalue.e. n ths case ψ becomes a. 15. Dscrete and Contnuous Egenvalues: An operator  may have dscrete egenvalues a 1, a 2, a 3,..., or a contnuous set of egenvectors, e. g. α 1 < a < α 2 or both. The set of egenvalues s known as the spectrum of the operator. 16. Degeneracy: If two or more dstnct egenvectors have the same egenvalue, the egenvalue s sad to be degenerate. 17. Propertes of Hermtean operators: If  s Hermtean then: (a) All egenvalues of  are real, and egenvectors belongng to dfferent egenvalues are orthogonal. (b) For dscrete egenvalues, the assocated egenvectors of an Hermtean operator,  say, can be normalzed to unty so that a a j = δ j. For contnuous egenvalues, the assocated egenvectors can be delta functon normalzed. e. a a = δ(a a ). In ether case the egenvectors are sad to be orthonormal. (c) The egenstates of a Hermtean operator form a complete set. (d) The operator  can be expanded (spectral decomposton) as:  = a a a + a a a da. 18. Egenvectors of Commutng Operators: If two Hermtean operators  and ˆB commute, then they have n common a complete set of egenvectors ab such that  ab = a ab and ˆB ab = b ab. 19. The Inverse of an Operator: An operator  has an nverse f there exsts an operator ˆB such that  ˆB = ˆB = ˆ1. Usually wrte ˆB =  Untary Operator: An operator  s untary f  =  1. A State Space for Combned Systems 1. If a system s made up of two parts S 1 and S 2, each of whch can be consdered as a system on ts own, and the state spaces H 1 and H 2 of the respectve systems are spanned by the complete sets of states ψ 1, ψ 2, ψ 2,..., and φ 1, φ 2, φ 2,..., then every state of the combned system can be expressed n the form Ψ = c 1 ψ 1 φ 1 + c 2 ψ 2 φ where the tensor product state ψ φ represents the state of the combned system n whch the system S 1 s n state ψ 1 and system S 2 n state φ 2. In other words, the set of states ψ φ, = 1, 2,... span the tensor product Hlbert space H 1 H 2, the state space of the combned system. The notaton ψ φ s usually replaced by the smpler notaton ψ φ. 2. The product states ψ φ have the propertes (a) ψ (c 1 φ 1 + c 2 φ 2 ) = c 1 ψ φ 1 + c 2 ψ φ 2. 5

6 (b) (c 1 ψ 1 + c 2 ψ 2 ) φ = c 1 ψ 1 φ + c 2 ψ 2 φ. (c) The bra vector correspondng to the ket vector ψ φ s φ ψ [note the reversal of order of the factors]. (d) ( φ 1 ψ 1 )( ψ 2 φ 2 ) = φ 1 φ 2 ψ 1 ψ If  1 s an operator actng on states of S 1 and  2 an operator actng on states of S 2 then an operator actng on states of the combned system, wrtten  1  2, can be defned wth the propertes (a) ( 1  2 ) ψ φ =  1 ψ  2 φ. (b)  1 +  2  1 ˆ1 2 + ˆ1 1  2. (c) Wth the smpler notaton  1  2  1  2, t then also follows that [ 1,A 2 ] = 0. The above defntons can be readly generalzed for systems made up of more than two subsystems. Basc Postulates and Concepts of Quantum Mechancs 1. Every state of a physcal system s represented by a vector, and conversely, each lnear superposton of vectors s representatve of a possble state of the system. The set of all state vectors descrbng a gven physcal system forms a complex vector space (or Hlbert space) also known as the state space for the system. If a state of the system s represented by a vector ψ, then the same state s represented by the vector c ψ where c s any non-zero complex number. 2. To each physcally measurable property (called an observable ) of a physcal system there corresponds a Hermtean operator (also called an observable ) whose egenvalues are the possble results of measurements of the correspondng observable. Conversely, any Hermtean operator s understood as representng a possble measureable property of the system. 3. The egenvectors of each observable assocated wth a physcal system s assumed to form a complete set so that any state of the system can be expressed as a lnear superposton of the egenvectors of any observable of the system. 4. The orthonormal egenvectors a (dscrete egenvalues) and a (contnuous egenvalues) of any observable are assumed to satsfy the closure or completeness relaton a a + a a da = ˆ1. 5. Probablty Interpretaton: A (a) Only those states that have fnte norm (and hence can be normalzed to unty) can represent possble physcal states of a system. (However, certan states of nfnte norm can be sad to represent a possble physcal state of a system under certan crcumstances.) (b) If the two states φ and ψ are both normalzed to unty,.e. φ φ = ψ ψ = 1, then: () The nner product φ ψ s the probablty ampltude of observng the system to be n the state φ gven that t s n the state ψ. 6

7 () The probablty of observng the system to be n the state φ gven that t s n the state ψ s φ ψ 2. (c) In the lnear superposton of a state ψ n terms of the egenvectors of the observable Â: ψ = a a ψ + a a ψ da the quantty a ψ 2 s the probablty of obtanng the (dscrete) result a n a measurement of  and a ψ 2 da s the probablty of obtanng a result n the range a to a + da for the contnuous set of egenvalues, provded ψ ψ = Expectaton Value of Observable: The expectaton or mean or average value of the results of observng  for a large number of dentcal systems all n the state ψ s ψ  ψ f ψ ψ = Uncertanty: The uncertanty or varance A n the results of measurements of an observable  on a large number of dentcal systems all n the state ψ s gven by: provded ψ ψ = 1. ( A) 2 = ψ (  ) 2 ψ =  2  2 8. Change of State by Measurement: If as a result of a measurement of an observable Â, the result a s obtaned, then mmedately after the measurement, the system s n an egenstate of  wth egenvalue a. (Ths s the smplest possble assumpton concernng the change of state by measurement). 9. If two observables  and ˆB do not commute, then measurements of  and ˆB nterfere n the sense that alternate measurements of  and ˆB wll yeld randomly varyng results. Observables  and ˆB cannot smultaneously have precsely defned values, and the observables are sad to be ncompatble. 10. If two observables  and ˆB do commute, then there exsts states of the system for whch these observables smultaneously have precsely defned values. These are the common egenstates ab. The observables are then sad to be compatble. 11. The Hesenberg Uncertanty Relaton: The uncertantes A and B n two observables for a system n a state ψ are related by the nequalty A A B 1 2 [Â, ˆB]. 12. Complete Set of Commutng Observables: A complete set of commutng observables Â, ˆB, Ĉ,... are a set of observables that represent the maxmum number of physcal observables of a system that smultaneously can have precsely defned values. Ther common egenvectors abcd... are completely and unquely characterzed by ther egenvalues a, b, c,...,.e. no two egenvectors have dentcal sets of egenvalues. 13. If  s an observable for a quantum system wth dscrete egenvalues then states and operators of the system can be represented n terms of ther components wth respect to the egenstates of  as bass vectors n the followng manner: (a) The ket vector ψ s represented by a column vector of components a 1 ψ a ψ 2 ψ a 3 ψ a 1 ψ 7

8 (b) The bra vector s represented by the row vector of components ψ a ψ ( ψ a 1 ψ a 2 ψ a 3... ) (c) The operator ˆB s represented by a matrx wth components a ˆB a j a 1 ˆB a 1 a 1 ˆB a 2 a 1 ˆB a 3... a 2 ˆB a 1 a 2 ˆB a 2 a 2 ˆB a 3... ˆB a 3 ˆB a 1 a 3 ˆB a 2 a 3 ˆB a Note the use of the symbol rather than = to ndcate that the vectors (or operators) are represented by the correspondng column or row vectors, or matrx. Relatons between abstract vectors and operators can then be represented as equatons n terms of row and column vectors and square matrces. 14. If  s an observable wth contnuous egenvalues, then a representaton n terms of column or row vectors and matrces s no longer explcty possble. Nevertheless, vectors and operators can stll be represented n terms of ther components. If  s an observable for a quantum system wth contnuous egenvalues then: (a) The components of the ket vector ψ wth respect to the bass states a are a ψ = ψ(a). The quantty ψ(a) s known as the wave functon of the state ψ n the Â- representaton. (b) The components of the bra vector ψ are ψ a = ψ (a). (c) The components of the operator ˆB are gven by a ˆB a. Note that, n general, these components wll be Drac delta functons or ts dervatves. 15. Infntesmal tme evoluton of a quantum system: If a quantum system s n the state ψ(t) at tme t, the state of the system an nfntesmal tme nterval later s assumed to be gven by ψ(t + dt) = (ˆ1 ˆΩ(t)dt) ψ(t) where ˆΩ s a (lnear) Hermtean operator whch may also be tme dependent. 16. The Hamltonan: Typcally, ˆΩ s wrtten as Ĥ/ where Ĥ s known as the Hamltonan of the system, and s Planck s constant. 17. The tme evoluton operator: (a) Over the fnte tme nterval (t 0, t) ψ(t) = Û(t, t 0 ) ψ(t 0 ) where Û(t, t 0 ) s a untary operator known as the tme evoluton, or tme development operator. (b) If the Hamltonan s tme ndependent, then Û(t, t 0 ) = e Ĥ(t t 0 )/ If the Hamltonan s tme dependent, there s n general no closed form expresson for the tme evoluton operator. 8

9 18. The Schrödnger equaton: In general, the tme dependent state vector ψ(t) satsfes the equaton Ĥ(t) ψ(t) = d ψ(t). dt 19. Statonary states: If the Hamltonan Ĥ s ndependent of tme, then the tme dependence of the egenstates E of Ĥ, where Ĥ E = E E takes the form E(t) = e Et/ E. so that the probablty for the outcome of the measurement of any observable s ndependent of tme. a E(t) 2 = a E Conserved quanttes: The observable Ô s sad to be conserved f ts expectaton value Ô(t) = ψ(t) Ô ψ(t), or for all ntal states ψ(t 0 ). d Ô(t) dt = [Ĥ, Ô] = The nfntesmal dsplacement operator: In one dmenson, f a quantum system s n the egenstate x of the poston operator ˆx, then the state of the system after undergong an nfntesmal dsplacement dx s where ˆK s a (lnear) Hermtean operator. x + dx = (ˆ1 + ˆKdx) x 22. The momentum operator: Typcally, ˆK s wrtten as ˆp/ where ˆp s known as the momentum operator of the system. 23. The dsplacement operator: If a quantum system n the state ψ undergoes a fnte dsplacement a, the state of the system after the dsplacement s ˆD(a) ψ where s a untary operator. ˆD(a) = e ˆpa/ 24. Symmetres: An operaton performed on a system (as represented by a untary operator Ŝ say) s a symmetry operaton f the evoluton of the system s unaffected by the operaton. (a) Ths s to be understood n the sense that Ŝ 1 Û(t, t 0 )Ŝ ψ = Û(t, t 0 ) ψ.e. performng the operaton (Ŝ ), allowng the system to evolve (Û(t, t 0 )), and then reversng the operaton (Ŝ 1 ) produces the same outcome as not performng the operaton at all, for all states ψ of the system. (b) As a consequence, the operaton represented by the untary operator Ŝ s a symmetry operaton f [Ŝ, Ĥ] = 0 where Ĥ s the Hamltonan of the system. 9

10 25. Symmetres and Conservaton Laws: Assocated wth each symmetry of a quantum system s a conserved observable. In partcular, f the space dsplacement operator Ŝ = ˆD(a) s a symmetry operaton, then the conserved quantty s the momentum ˆp. 26. Canoncal commutaton relatons: From the defnton of the space dsplacement operator t follows that [ ˆx, ˆp] = or, more generally [ ˆx, f ( ˆp)] = d f ( ˆp) d ˆp. 27. Canoncal quantzaton: The quantum descrpton of a classcal physcal system can be constructed by demandng that (a) All symmetry propertes of the classcal system also be symmetres of the quantum system. (b) The conserved quanttes of the quantum system be dentfed, up to a constant of proportonalty, as the Hermtean operators (observables) of the correspondng classcally conserved quanttes. whch results n the rules: (a) Replace, n the classcal Hamltonan, all generalzed postons and momenta p and q, = 1, 2,..., by Hermtean operators ˆp and ˆq. (b) The classcal Hamltonan H(q, p) becomes the quantum Hamltonan Ĥ(ˆq, ˆp). (c) The quantum operators ˆp and ˆq are requred to satsfy the commutaton rules [ˆq, ˆp j ] = δ j [ˆq, ˆq j ] = [ ˆp, ˆp j ] = The poston representaton: Ths representaton s defned n terms of the egenstates of the poston operator ˆx (n one dmenson). Ths operator has contnuous egenvalues, so that: (a) The wave functon of the ket vector n the poston representaton s x ψ = ψ(x). (b) The poston operator n the poston representaton s x ˆx x = δ(x x ) (c) The momentum operator n the poston representaton s x ˆp x = δ(x x ) x or, n a form often more convenent x ˆp ψ = dψ(x) dx 29. Schrödnger equaton n the poston representaton: The Schrödnger equaton for a partcle wth Hamltonan Ĥ = ˆp2 + V( ˆx, t) 2m s, n the poston representaton 2 2 Ψ(x, t) Ψ(x, t) 2m x 2 + V(x, t)ψ(x, t) = t otherwse known as the Schrödnger wave equaton. Updated: 1 st August2005 at4:26pm. 10