2. Postulates of Quantum Mechanics. [Last revised: Friday 7 th December, 2018, 21:27]

Transcription

1 2. Postulates of Quantum Mechancs [Last revsed: Frday 7 th December, 2018, 21:27] 24

2 States and physcal systems In the prevous chapter, wth the help of the Stern-Gerlach experment, we have shown the falure of Classcal Mechancs and the need to ntroduce a new theory able to descrbe all physcal phenomena. Notce that whatever nformaton we have about a physcal system s obtaned through expermentaton. It s useful to dvde the experment n two phases Preparaton: the expermentalst (or nature) submts the system to some condtons that defne ts state. For example, the slver atoms n the SGẑ are prepared to have well defned z-component of the magnetc moment after crossng an nhomogeneous magnetc feld appled along that drecton. By flterng those deflected upward or downward we select a value of the spn. Measurement: the expermentalst (or nature) nteracts wth the preparaton to determne the value of a partcular observable (any physcal varable that, n prncple, can be measured). For example, one can measure the observables S z or S x of the atoms prevously prepared. 25

3 States and physcal systems A preparaton: does not necessarly determne the outcome of a subsequent measurement but the probabltes of the varous possble outcomes. s ndependent of the specfc measurement that may follow t. A state s the specfcaton of a set of probabltes (or probablty dstrbutons) for the measurements of the varous observables. The concept of state n QM s very subtle and even controversal. Snce t has always been the goal of physcs to gve an objectve realstc descrpton of the world, we are tempted to nterpret the state as an element of realty descrbng the attrbutes of an ndvdual system. However such assumptons lead to contradctons and must be abandoned. The quantum state descrpton may be taken to refer to a collecton of smlarly prepared systems. 26

4 States and physcal systems For the moment we wll consder pure states, whch are those that gve maxmal (though probablstc) nformaton about the outcome of the measurements. We wll see later, n ths chapter, that n general the system s n a mxed state, specfed by a statstcal dstrbuton of pure states. For nstance, the ensemble of slver atoms comng drectly from the furnace, before gong through any SG devce; or a partally polarzed (or unpolarzed) electron beam. To construct the physcal theory t s necessary to ntroduce a few basc postulates. 27

5 States Postulate I Postulate I In QM a physcal system s assocated to a separable, complex Hlbert space and a pure state of the system at a tme t s descrbed by a unt ray a represented by a vector (ket) α or α(t) of the Hlbert space. a A unt ray s a unt vector wth arbtrary phase. Then the superposton prncple s guaranteed: f φ and ψ are states of the system then η = α φ + β ψ, wth arbtrary α, β C, s also a possble state. The Hlbert space of the system may have just two dmensons, lke n the Stern-Gerlach experment. Then we may choose an arbtrary bass of two states to represent any other state. For nstance, { S z +, S z }, { S x +, S x } and { Sy +, Sy + } are three bases, and the state S x + n the frst bass s gven by S x = 1 2 S z S z. 28

6 States Postulate I A partcularly nterestng two-dmensonal quantum mechancal system s the qubt, the quantum computer unt of nformaton. In contrast to the classcal bt that can be n just two states 0 or 1, one can prepare a qubt n any arbtrary superposton of 0 and 1. 29

7 Revew of Hlbert spaces A Hlbert space H s a vector space suppled wth an nner or scalar product that s complete respect to the norm nduced by the scalar product. It s a generalzaton of the very famlar Eucldean spaces, lke R 3, to spaces wth any fnte or nfnte number of dmensons. The vectors n a vector space are elements that can be added and multpled by a scalar. In a Hlbert space, unlke Eucldean spaces, these scalars are complex numbers: φ, ψ H, c 1, c 2 C c 1 φ + c 2 ψ H (lnear combnaton). We say that a set of vectors {φ } s lnearly ndependent f c φ = 0 c = 0. 30

8 Revew of Hlbert spaces The scalar product of any φ, ψ H s a complex number (φ, ψ) C satsfyng: () (φ, ψ) = (ψ, φ) (hermtcty). () (φ, c 1 ψ 1 + c 2 ψ 2 ) = c 1 (φ, ψ 1 ) + c 2 (φ, ψ 2 ) (lnearty of the second entry). From () and () one gets: (c 1 φ 1 + c 2 φ 2, ψ) = c1 (φ 1, ψ) + c2 (φ 2, ψ) (antlnearty of the frst entry). () (φ, φ) 0 and φ = 0 when (φ, φ) = 0. 31

9 Revew of Hlbert spaces The scalar product nduces a norm a defned by φ = (φ, φ), that generalzes the concept of length (modulus) of a vector and defnes a metrc (dstance between two vectors), gven by d(φ, ψ) = φ ψ. a The propertes of a norm are: () cφ = c φ (homogeneous). () φ + ψ φ + ψ (trangle nequalty). () φ 0 (postve defnte). The property () follows from the Schwarz nequalty: (φ, ψ) 2 (φ, φ)(ψ, ψ). 32

10 Revew of Hlbert spaces A metrc space M s complete f every Cauchy sequence n M converges n M. That s, f {ψ n } s a sequence wth d(ψ m, ψ n ) 0 when m, n then there exsts a η M such that d(ψ n, η) 0 when n. Complete normed vector spaces are called Banach spaces. A Hlbert space s a Banach space wth the norm nduced by the scalar product. One also requres that Hlbert spaces assocated to physcal sytems must be separable. Ths means that they have a countable orthonormal bass. 33

11 Revew of Hlbert spaces Let us now ntroduce lnear functonals actng on a vector space V as functons F : V C mappng vectors φ to complex numbers F(φ) satsfyng F(aφ + bψ) = af(φ) + bf(ψ), φ, ψ V, a, b C. Defnng the sum of functonals (F + G)(φ) = F(φ) + G(φ), the set of lnear over V defnes another vector space, called dual space V. These elements of the dual space are the so called covectors or one-forms. In a Hlbert space H one can defne lnear functonals F φ H from any φ H by F φ (ψ) = (φ, ψ). 34

12 Revew of Hlbert spaces Then the Resz representaton theorem apples statng that for each F H there exsts just one vector φ F H such that F(ψ) = (φ F, ψ) ψ H. Therefore, there s a bjectve mappng between V and V gven by the scalar product (H and H are somorphc; n partcular, they have the same dmenson). Ths suggests the Drac s notaton, extensvely used n Quantum Mechancs (QM): Vector ψ H ket ψ H Functonal F φ H bra φ H Acton of functonal F φ on ψ H braket φ ψ = (φ, ψ) (scalar product). In other words, every ket ψ has a correspondng bra ψ, that s unque, and the scalar product (φ, ψ) of two vectors (kets) φ and ψ s gven by the braket φ ψ = ψ φ. From now on, untl next chapter, we wll work n Hlbert spaces of fnte dmenson, although many results can be appled to nfnte dmensons. 35

13 Revew of Hlbert spaces A bass s a set of lnearly ndependent vectors { φ } ( = 1,..., d = dmh) that allows us to express any vector α H as a lnear combnaton (summatons extend from = 1 to d unless otherwse stated) α = φ α = ( ) φ 1 φ 2... where the α are the components of α n the bass { φ }. α 1 α 2., α C or α =. α 1 α 2. 36

14 Revew of Hlbert spaces An orthonormal bass { e } fulflls e k e = δ k (orthonormalty relaton). Gven a bass { φ } the Gram-Schmdt process provdes an orthornormal bass { e }: e 1 = φ 1 φ 1, φ 1 = e k+1 = φ k+1 φ k+1 k =1 e e φ k+1 φ 1 φ 1, k e e φ k+1 =1. 37

15 Revew of Hlbert spaces In an orthonormal bass the components of a vector are easy to obtan from the scalar product or braket: α = e α e k α = I = and the scalar product of two vectors reads: α β = e k e α = δ k α = α k e e (completeness or closure relaton) α e e β = e α e β = α β. In fact, the somorphsm between H and H s gven by the adjon or dagger relaton: H H { e } { e } (so called adjont bass of H ) α α = α = α e (by antlnearty of braket s left entry). 38

16 Revew of Hlbert spaces An operator A transforms vectors α H to other vectors A α H. Lnear operators satsfy A(a α + b β ) = aa α + ba β. Operators can be added and composed (multpled), (A + B) α = A α + B α AB α = A(B α ) and the product of operators s assocatve, A(BC) = (AB)C, but not necessarly commutatve. 39

17 Revew of Hlbert spaces To know how an operator acts on all the vectors n H t s suffcent to know how t acts on a bass of H. Gven an orthonormal bass { e }, A ej = one obtans β = A α from e A j A j = e A ej A α = A j = β = β = A j α j. j ej αj = e A j α j j e β (matrx element) 40

18 On the other hand, operators act on bras to the left: A j = e (A ej ) = ( e A) ej Revew of Hlbert spaces e A = A j ej α A = α e A = α A j ej. Notce that the vector components are bass-dependent but the sandwch α A β (and the scalar product) s bass-ndependent: α A β = j α e e A ej ej β = j α A jβ j. 41

19 Revew of Hlbert spaces The scalar product of A α and β s not α A β but α A β, that defnes the adjont operator A. Ths s because the adjont of A α s not α A but α A : A j = e A ej A j = e j A e = A j A j = A j or A = A T. 42

20 Revew of Hlbert spaces Gven φ, ψ, an useful way to defne a lnear operator s φ ψ (outer product) that actng on any η H gves a vector proportonal to φ : It s easy to check a that ( φ ψ ) η = φ ψ η. ( φ ψ ) = ψ φ. Takng a unt vector e 1 we obtan a projector, P 1 = e 1 e 1, P 2 1 = P 1, P 1 = P 1, that projects any vector α H along the vector e 1, P 1 α = e 1 e 1 α = e 1 α 1. a ( φ ψ ) η = ( η φ ψ ) = ψ η φ = ψ φ η = ( ψ φ ) η, η. 43

21 Revew of Hlbert spaces r A sum of projectors P, wth P = e e, s also a projector nto the subspace =1 spanned by the r unt vectors e =1,...,r. If { e } s an orthonormal bass of H then the P are orthogonal projectors, P 2 = P, P P j = δ j P j. d We have already seen that n fact I = e e snce =1 d α = e e α, α H. 44

22 Revew of Hlbert spaces Gven a lnear operator A, f there exst a C and α H wth α = 0 such that A φ = a φ we say that every φ s an egenvector of A wth egenvalue a. If φ =1,...,r are lnearly ndependent egenvectors of A wth the same egenvalue a (degenerate egenvalue) then obvously any lnear combnaton c φ s also an egenvector. Therefore, the egenvectors of each egenvalue form a vector subspace. And, of course, f A φ = a φ then φ A = a φ. 45

23 Revew of Hlbert spaces An operator A s self-adjont f A = A, namely, f φ A ψ = ψ A φ, φ, ψ H Actually, ths s only true n fnte dmenson, snce otherwse the domans of A and A may not concde. In the latter case, we say that A s Hermtan, but not self-adjont. In general, f A s Hermtan then all ts egenvalues are real and the egenvectors correspondng to dfferent egenvalues are orthogonal. a Furthermore, an mportant theorem states that the orthonormal set of the egenvectors of a self-adjont operator on a Hlbert space of fnte dmenson s a bass of H. a A a = a a, A a = a a, a, a R. Take a A a = a a a = a a a (a a ) a a = 0. Hence, f a = a then a a = 0. 46

24 Revew of Hlbert spaces Consder H of fnte dmenson, a self-adjont operator A and an orthonormal bass { φ } formed by the egenvectors of A. And let a be the correspondng egenvalues. We defne the othogonal projectors to the subspace of egenvalue a (perhaps degenerate) as P a = φ φ δ a a. Then, one can wrte A as follows (spectral decomposton): A = a ap a = a φ φ, a dagonal matrx n the bass of egenvectors. Ths may be used to defne a functon f of operators from the same functon of complex numbers: f (A) = f (a ) φ φ. 47

25 Revew of Hlbert spaces Consder now A and B two self-adjont commutng operators, [A, B] = AB BA = 0, n fnte dmenson. Then there exsts a complete set of smultaneous egenvectors of A and B, that s, A and B can be dagonalzed smultaneously. If A, B, C,... are self-adjont operators commutng wth each other, then the set of ther smultaneous egenvectors a, b j, c k,..., A a, b j, c k,... = a a, b j, c k,..., B a, b j, c k,... = b j a, b j, c k,..., C a, b j, c k,... = c k a, b j, c k,..., etc. may be degenerate. But f the subspace of egenvectors for all possble sets of egenvalues has dmenson one (t s not degenerate) then A, B, C,... s a complete set of commutng (self-adjont) operators (CSCO). 48

26 Revew of Hlbert spaces As a consequence, any operator F commutng wth all the members of a CSCO s a functon of these operators and F a, b j, c k,... = f jk... a, b j, c k,..., f jk... = f (a, b j, c k,... ). 49

27 Gven two orthonormal bases { e } and { ẽ }, we may wrte ẽj = e e ẽj and defne the change of bass operator from { e } to { ẽ } as U = The operator U s untary, UU = U U = 1. ẽ e U ej = ẽj. Revew of Hlbert spaces 50

28 Revew of Hlbert spaces Notce that the bass elements and the vector components transform n an opposte way: ẽj = e e ẽj whle for any α H, α = = ẽ α = j and n fact U s untary: e e α = ẽ ẽ α = ẽj = e α ẽ α, ẽ e j ej α α = j δ k = e e k = j e ẽj ẽj ek = j e U j, U j = e U ej = e ẽj U j α j, U j = U j = ẽ ej U j U kj = j U j U jk. 51

29 Revew of Hlbert spaces On the other hand, the matrx elements of a lnear operator A transform as: Ã j = ẽ A ẽj = e U AU ej = kl = U k A klu lj. e U e k e k A e l e l U ej If A s a lnear operator and { e } s an orthonormal bass then the trace of A s Tr(A) = e A e (sum of the dagonal elements). Notce that the trace s ndependent of the bass and satsfes the propertes: () Tr(AB) = Tr(BA). () Tr(U AU) = Tr(A) f U s untary. () Tr( e ej ) = δ j. (v) Tr( φ ψ ) = ψ φ. 52

30 Observables Postulate II Postulate II Every observable of a physcal system s represented by a self-adjont lnear operator actng on the assocated Hlbert space, whose egenvalues are the only possble values of the observable. Ths justfes several ssues: The number of egenvalues of an operator actng on a space of fnte dmenson s denumerable. Hence, the values of the correspondng observable are quantzed. A self-adjont operator has real egenvalues. The values of physcal observables are always real numbers. A lnear operator respects the superposton prncple. It s not possble to measure smultaneously two observables represented by non-commutng operators because they cannot be dagonalzed n the same bass, they are ncompatble. 53

31 Observables Postulate II For example, the spn of the slver atom n the z-axs or n the x-axs are observables represented by the self-adjont operators S z and S x, respectvely. Both of them have egenvalues ± h/2. Usng ther spectral decomposton: S z = h 2 S z + S z + h 2 S. z S z = h 1 0 h σ 3, S x = h 2 S x + S x + h 2 S x S x. = h h σ 1. The matrx form of the operators has been gven n the bass { S z +, S z }, S z + =. 1, S z =. 0, S x + =. 1 1, S x = Notce that we have chosen an arbtrary phase for each of these states. The observables S x and S z are ncompatble because [S x, S z ] = 0. 54

32 Measurements Postulate III Postulate III If a physcal system s n a pure state descrbed by the normalzed vector ψ, the probablty of obtanng an egenvalue a of an observable represented by the operator A s p a = ψ P A,a ψ where P A,a s the projector nto the subspace of egenvalue a. If a s a non-degenerate egenvalue of A and a s the correspondng normalzed egenvector then P A,a = a a p a = a ψ 2. In general, let { a } be an orthonormal bass of the subspace of egenvalue a. Then P A,a = a a p a = a ψ 2. 55

33 Measurements Postulate III Notce that: If the state of the system was already n the subspace of egenvalue a, ψ H a p a = ψ P A,a ψ = ψ ψ = 1. If ψ H a then p a = ψ P A,a ψ = 0. The probablty s p a (0, 1) otherwse. The sum of probabltes to obtan any possble value s one, as t should be, snce the egenvectors form a complete set, I = a P A,a a p a = a ψ P A,a ψ = ψ ψ = 1. And what s the state after the measurement? 56

34 Measurements Postulate IV Postulate IV If a physcal system s n a pure state descrbed by the normalzed vector ψ and one measures A obtanng a, the system s left n the state ψ = P A,a ψ P A,a ψ. In other words, after the measurement, the state of the system s projected nto a partcular state of the subspace wth egenvalue a. It s often sad that the state ψ collapses nto the egenstate state ψ of A. But one can also vew t n a dfferent way: There s no measurement wthout nteracton wth the measurng nstrument (another system). Hence, we must always consder our system as a part of a composte system. As we wll see later, the states of the Hlbert space of ths composte system are vectors of the tensor product of the Hlbert spaces of ts subsystems. 57

35 Measurements Postulate IV Some of these states are entangled,.e. they cannot be wrtten as the product of a vector of each space, they are a non separable combnaton. For nstance, Now, assume a that the nteracton entangles the measurng nstrument wth the system we wsh to study. Let us take that, after crossng SGẑ, + s the state for the atoms devated upward wth S z = + h/2 and the opposte for. The entangled state above s none of them but a superposton. b a Why? How? Ths vew s not a soluton but another way to formulate the measurement problem. b If you replace the states, by unbroken or broken posson flask and +, by cat alve or dead, ths descrbes the famous Schrödnger s cat states: 58

36 Measurements Postulate IV The fact s we do not really know whether the atom s n state + or, snce we just measure that t leaves the SG as or after experencng countless (uncontrolled) nteractons wth the magnetc feld. Ths partal knowledge causes the decoherence. Thus the nteracton: allows for the creaton of superpostons of macroscopc objects, and at the same tme breaks the coherence of ts subsystems. 59

37 Measurements Postulate IV (a) Let s apply these postulates to our sequence of Stern-Gerlach experments: ψ = S z + p Sz,+ = S z + S z + 2 = 1, p Sz, = S z + S z 2 = 0 ψ = S z + ; (never happens). 60

38 Measurements Postulate IV (b) Let s apply these postulates to our sequence of Stern-Gerlach experments: ψ = S z + p Sx,+ = S z + S x + 2 = 1 2, ψ = S x + ; p Sx, = S z + S x 2 = 1 2, ψ = S x. 61

39 Measurements Postulate IV (c) Let s apply these postulates to our sequence of Stern-Gerlach experments: ψ = S x + (after flterng one half of the atoms n (b)) p Sz,+ = S x + S z + 2 = 1 2, ψ = S z + ; p Sz, = S x + S z 2 = 1 2, ψ = S z. 62

40 Measurements Expectaton value and uncertanty relatons Consder a macroscopc object, lke a bar, whose length L we want to measure. The procedure conssts of takng several measurements and then averagng. Suppose that, wthn the precson of the ruler, we obtan L 1 (n 1 tmes), L 2 (n 2 tmes), etc. If the total number of measurements s n then the mean value of the bar length s L = L n n where n /n s the relatve frequency of every result. We expect that L approaches the actual value of L for large n. 63

41 Measurements Expectaton value and uncertanty relatons If you want to measure an observable A n a quantum state ψ of a physcal system you must prepare many replcas of the system n the same state and then measure A. Accordng to the postulates, the result of every measurement s an egenvalue a of A and the mean value of all measurements, A ψ = a ap a = a a ψ P A,a ψ = ψ a ap A,a ψ = ψ A ψ. Ths s called the expectaton value of the observable A n the pure state ψ. We can also defne the uncertanty of A n the state ψ as the dsperson (mean square dsplacement) of the dfferent measurements around the expectaton value, ψ A = = = [ ] ψ (A A ψ ) ψ [ ] A 2 ψ + A 2 ψ 2 A ψ [ A 2 ψ A ψ]

42 Measurements Expectaton value and uncertanty relatons The uncertanty of an observable n a pure state s zero f t s an egenvector of the observable. Check, for example, that n ψ = S z +, S z ψ = h/2, ψ S z = 0 but S x ψ = 0, ψ S x = h/2. It s easy to show that the product of the uncertantes of two observables A and B n a state ψ s ψ A ψ B 1 2 ψ [A, B] ψ. These uncertanty relatons are a generalzaton of the poston-momentum uncertanty relatons we wll fnd later. They have mportant consequences: If two observables commute, we can measure them smultaneously wth full precson n any state. That s why we call them compatble. If two observables do not commute (they are ncompatble) the product of ther uncertantes may not be zero (see example below). 65

43 Measurements Expectaton value and uncertanty relatons Usng that [S x, S z ] = hs y, check the uncertanty relaton of S x and S z n the state ψ = Sy + = 1 2 ( S z + + S z ): ψ S x ψ S z h 2. (Notce that, of course, f ψ s an egenvalue of S x or S z then the product of uncertantes n that state would be zero.) 66

44 Measurements Complete Set of Compatble Observables When two observables A and B are compatble ther correspondng self-adjont operators commute, [A, B] = 0. Then there exsts a bass of egenvectors { a b } of A and B that s common to A and B smultaneously, A a, b = a a, b, B a, b = b a, b. Two (or more) compatble observables defne a complete set (CSCO) f any par of egenvectors n the common bass dffers at least n one egenvalue. Then the egenvalues label unambguously (up to a complex phase) the vectors of the bass,.e. the states of the system that can be measured smultaneously by all the observables n the CSCO. A characterzaton of a CSCO s: () They are compatble (commute). () The bass of common egenvectors s unque (up to phases). () The set s mnmal. Then the descrpton of the system s not redundant. Ths condton was not assumed above but t s often mposed. 67

45 Measurements Complete Set of Compatble Observables Example 1: A. = 1 1, B = bass { 1, 1, 1, 0, 1, 0 } A and B are a CSCO. (The egenvalues of one of them break the degeneracy of the other.) Example 2: C. = 1 0, D = bass { 1, 1, 0, 2, 1, 2 } C and D are not a CSCO because t s not mnmal. (C s enough to label the bass states.) 68

46 Densty matrx The formalsm developed so far apples to pure states. We have seen that the quantum mechancal predctons are probablstc, they are understood as the results of the measurements over a collecton of dentcally prepared physcal systems, all descrbed by the same vector of a Hlbert space α. We wll now consder the most general case, a statstcal ensemble of N pure states { α } wth frequences 0 w 1 (there are N = w N n each pure state) and w = 1. The α do not need to be orthogonal and N s arbtrary (nothng to do wth the dmenson of the Hlbert space). A system chosen randomly from ths statstcal ensemble s sad to be n a mxed state. 69

47 Densty matrx The mxed state s descrbed by a densty matrx, ρ = w α α that gves the expectaton value (average) of an observable A measured over the statstcal ensemble. In fact, A ρ = N A α N = = a = a = a w α A α a a a w α a a A a a α w a α α a a A a ρ aa A a a = Tr(ρA) where a and a are egenvectors of A, that satsfy a a a = a a a = I. 70

48 Densty matrx Notce that a complex phase of α above s, of course, rrelevant. The densty matrx has the followng propertes: () ρ = ρ (self-adjont). () Tr(ρ) = 1, snce Tr(ρ) = = w a w = 1. a α α a = w a α a a α = w α α 71

49 Densty matrx () Tr(ρ 2 ) 1, snce Tr(ρ 2 ) = = = = j j j j j a a w w j a α α αj αj a w w j α αj αj a a α w w j α αj αj α w w j α αj 2 w w j = ( w ) 2 = 1. The equalty occurs when w = 0 = j and w j = 1 (pure state) ρ = αj αj. 72

50 Densty matrx (v) ψ ρ ψ 0, ψ H, snce ψ ρ ψ = w α ψ 2 0. On the other hand, the probablty to obtan a non-degenerate value a of the observable A n a random element of the ensemble descrbed by ρ s p a = w α a a α = w a α α a = a ρ a snce w s the probablty to choose α and α a a α s the probablty to obtan a f we have chosen α. 73

51 Densty matrx Usng P A,a = a a, another way to wrte ths result s p a = w α P A,a α = = a = a w α P 2 A,a α = Tr(P A,a ρp A,a ) = Tr(ρP A,a ) w α P A,a a a PA,a α w a PA,a α α P A,a a Ths expresson s also vald f a s degenerate wth egenvectors { a(j) }, P A,a = j p a = a(j) a(j) w j α P A,a α = j a(j) ρ a(j) = Tr(ρP A,a ). 74

52 Densty matrx If we measure A to all the elements of the ensemble and select those wth egenvalue a, what s the densty matrx of the resultng ensemble? Accordng to postulate IV, f we pck up α and obtan a the state collapses nto one egenstate a(j), wth probablty α a(j) = P A,a α P A,a α α a(j) 2. 75

53 Densty matrx Therefore, after the measurement: If a s non degenerate then α a and we get a preparaton n the pure state a. But, n general, we get an ensemble of pure states a(j) (mxed state) descrbed by the densty matrx: ρ = w α α w P A,a α α P A,a P A,a α 2 j α a(j) 2 = P A,a ρp A,a where we have used P A,a α 2 = j a(j) α 2. We must normalze the densty matrx so that Tr(P A,a ρp A,a ) = Tr(ρP A,a ) = 1. So fnally, ρ A,a = P A,a ρp A,a Tr(ρP A,a ). 76

54 Densty matrx Puttng together prevous results we get a generalzed verson of the postulates: Postulate I In QM a physcal system s assocated to a complex Hlbert space and any state of the system s descrbed by a lnear operator ρ, called densty matrx, that satsfes ρ = ρ, Tr(ρ) = 1, ψ ρ ψ 0, ψ H. Postulate II (same as Postulate II) Every observable of a physcal system s represented by a self-adjont lnear operator actng on the assocated Hlbert space, whose egenvalues are the only possble values of the observable. 77

55 Densty matrx Postulate III If a physcal system s n state descrbed by the densty matrx ρ, the probablty of obtanng an egenvalue a of an observable A s p a = Tr(ρA). Postulate IV If a physcal system s n a mxed state descrbed by the densty matrx ρ and one flters the egenvalue a of an observable A, the system s left n a mxed state descrbed by the densty matrx ρ A,a = P A,a ρp A,a Tr(ρP A,a ). 78

56 Densty matrx Pure states are specal cases of mxed states. A state s pure f ts densty matrx has the form ρ = ψ ψ for some ψ H. A pure state s characterzed by ρ 2 = ρ and Tr(ρ 2 ) = 1. Otherwse, t s not pure. If the state s not pure, t s specfed by the set of frequences where more than one w s dfferent from zero. Then the decomposton s not unque: For example, the followng densty matrces are the same (same ψ ρ ψ, ψ ) but they are made of a mxture of dfferent pure states: ρ = a u u + (1 a) v v, 0 < a < 1, { u, v } orthonormal, ρ = 1 2 x x y y, wth x = a u 1 a v, y = a u + 1 a v. Hence, we do not have a maxmal nformaton of the state snce we do not know what the mxture s made of. 79

57 Densty matrx Let us llustrate wth an example the dfference between a coherent superposton of pure states (another pure state) and a ncoherent mxture of pure states (mxed state). Consder the followng two states: The pure state S x +, that can be wrtten as superposton of egenstates of S z, S x + = 1 S z S z = ρ 1 = S x + S x +. = n the bass { S z +, S z }. The densty matrx ρ 1 s an alternatve way of descrbng ths state. Notce that t corresponds to a pure state because ρ 2 1 = ρ 1, Tr(ρ 2 1 ) = 1. 80

58 Densty matrx The mxed state ρ 2 = 1 2 S z + S z S z S z. = (ρ 2 2 = ρ 2, Tr(ρ 2 2 ) = 1). In both states the probablty to fnd ether S z = ± h/2 s the same,. P Sz,+ = S z + S z + = 1 0., P Sz, = S z + S z + = ρ 1 : p Sz + = Tr(ρ 1 P Sz,+) = 1 2, p S z = Tr(ρ 1 P Sz, ) = 1 2, ρ 2 : p Sz + = Tr(ρ 2 P Sz,+) = 1 2 p Sz = Tr(ρ 2 P Sz, ) =

59 Densty matrx And the expectaton value (average) of S z s also the same,. S z = h S z ρ1 = Tr(ρ 1 S z ) = 0, ρ1 S z = Sz 2 ρ1 S z 2 ρ 1 = h 2 S z ρ2 = Tr(ρ 2 S z ) = 0, ρ2 S z = Sz 2 ρ2 Sz 2 2 ρ 2 = h 2. But n contrast to ρ 2, the state ρ 1 has a well defned spn orentaton (along the x-axs),. P Sx,+ = S x + S x + = , P Sx, = S x S x = ρ 1 : p Sx + = Tr(ρ 1 P Sx,+) = 1, p Sx = Tr(ρ 1 P Sx, ) = 0 ρ 2 : p Sx + = Tr(ρ 2 P Sx,+) = 1 2, p S x = Tr(ρ 2 P Sx, ) =

60 Densty matrx In fact,. S x = h S x ρ1 = Tr(ρ 1 S x ) = h 2, ρ 1 S x = S 2 x ρ1 S x 2 ρ 1 = 0 S x ρ2 = Tr(ρ 2 S x ) = 0, ρ2 S x = S 2 x ρ2 S x 2 ρ 2 = h 2. Actually ρ 1 represents a polarzed beam (along the x-axs) and ρ 2 an unpolarzed beam. The slver atoms extng the furnace n the Stern-Gerlach experment are n the mxed state ρ 2 (unpolarzed), but those fltered by SG ˆx are n the pure state ρ 1 (polarzed). 83

61 Densty matrx We could also prepare a partally polarzed beam along the z-axs,. ρ 3 = w 1 S z + S z + + w 2 S z S z = w w 2 wth w 1 + w 2 = 1 (w = 0, w 1 = w 2 ). Ths s also a mxed state (ρ 2 3 = ρ 3, Tr(ρ 2 3 ) = 1) that has ρ 3 : p Sz + = Tr(ρ 3 P Sz,+) = w 1, p Sz = Tr(ρ 3 P Sz, ) = w 2 p Sx + = Tr(ρ 3 P Sx,+) = 1 2, p S x = Tr(ρ 3 P Sx, ) = 1 2 S z ρ3 = Tr(ρ 3 S z ) = h 2 (w 1 w 2 ), ρ3 S z = S x ρ3 = Tr(ρ 3 S x ) = 0, ρ3 S x = Sz 2 ρ3 S z 2 ρ 3 = h w 1 w 2 S 2 x ρ1 S x 2 ρ 3 = h 2. 84

62 Densty matrx Of course, S z +, Sy + = 1 2 ( S z + + S z ) and n general (θ, ϕ) = cos(θ/2) S z + + e ϕ sn(θ/2) S z, wth θ [0, π], ϕ [0, 2π], are other examples of pure states, polarzed along the drecton ˆn(θ, ϕ). Check that ther correspondng densty matrces fulfll ρ 2 = ρ and Tr(ρ 2 ) = 1. The spn along ˆn can be determned wth full precson: we have maxmal nformaton about them. In contrast, the spn cannot be determned wthout uncertanty when measured over the mxed states ρ 2 or ρ 3. 85

63 Composte systems. Entanglement A composte system of two subsystems wth Hlbert spaces H 1 and H 2 s assocated the Hlbert space H = H 1 H 2 (tensor product). Ths space conssts of all the ordered pars u v u v uv, wth u H 1, v H 2, and ther lnear combnatons. By defnton, f c C, c( u v ) = (c u ) v = u (c v ) ( u 1 + u 2 ) v = u 1 v + u 2 v u ( v 1 + v 2 ) = u v 1 + u v 2. The states that can be wrtten as the drect product of one vector u H 1 and one vector v H 2 are called separable states. The lnear combnaton of two or more separable states are called entangled states. If { u } and { vj } are bases of H1 and H 2, respectvely, then { u vj }, = 1,..., n, j = 1,..., m, s a bass of H 1 H 2 (that has dmenson m n), ψ = j α j u v j, ψ H1 H 2. 86

64 Composte systems. Entanglement The scalar product n H 1 H 2 s defned by ( j α j u vj, j β j u ) vj = αj β kl u u k v vl j jkl = α j β j (f both are orthohormal bases). If A, B are operators actng on H 1 and H 2, respectvely, we defne the operator A B actng on H 1 H 2 by (A B)( u v ) = (A u ) (B v ). In fact, every lnear operator C on H 1 H 2 can be wrtten as C = j c j A B j, wth A and B j operators on H 1 and H 2, respectvely. 87

65 Composte systems. Entanglement Consder an observable A actng just on the subsystem H 1. Then t s of the form A I H2 on H 1 H 2 and A u v j = vj A u. We can wrte the expected value of A n a state of densty matrx ρ of the composte system as ts expected value n the subsystem H 1 wth reduced densty matrx ρ H 1, Tr(ρA) = j u v j ρa u v j = j = u v j ρ vj A u u ( j vj ρ vj ) A u = Tr H1 (ρ H 1 A) where we have ntroduced the partal trace of ρ (or any other operator) as ρ H 1 Tr H2 (ρ) = vj ρ vj. j 88

66 Composte systems. Entanglement We see that the reduced densty matrx, defned as the partal trace of the densty matrx of a composte system, descrbes the state of a subsystem when we gnore the nformaton about the rest of the system. Snce, n prncple, we lose part of the nformaton, the reduced densty matrx of a pure state may be a mxed state. Ths happens n partcular when the state of the composte system s an entangled state. 89

67 Composte systems. Entanglement For example, consder a four-dmensonal system S = S 1 S 2 composed of two subsystems of bases {, } and { +, }. Assume the system s n an entangled state ψ = = that we have expressed by convenence n the bass { +,, +, }. 90

68 Composte systems. Entanglement The densty matrx descrbng the composte system n that state s ρ = ψ ψ = that, of course, fulflls Tr(ρ 2 ) = Tr(ρ) = 1, because ψ s a pure state. The reduced densty matrx of subsystem S 1 s the partal trace ρ S 1 Tr S2 (ρ) = + ρ + + ρ = + ψ ψ + + ψ ψ = Notce that ρ S 1 does not descrbe a pure but a mxed state (Tr[(ρ S 1) 2 ] < Tr(ρ S 1) = 1): half of the tmes the subsystem s n the state and the other half n the state, but never n a coherent superposton. 91

69 Composte systems. Entanglement The coherence s lost, just because we gnore (have not measured) all the detals of the complementary system(s). In practce, ths s always what happens when we measure an observable n a non-solated system: the system s entangled wth the measurng apparatus, trllon trllons atoms whose state s mpossble to determne. Schrödnger s cat s ether dead or alve. In general, a bpartte pure state ρ s entangled f and only f ts reduced states are mxed rather than pure. 92

70 Quantum dynamcs: the Schrödnger equaton Postulate V How does a quantum system change wth tme? Postulate V In the tme nterval between two consecutve measurements (closed system), pure states reman pure, and tme evoluton s descrbed by the Schrödnger equaton, h d dt ψ(t) = H(t) ψ(t), where H(t) s an observable called the Hamltonan of the system. The Schrödnger equaton s determnstc. Gven the quantum state at a tme t 1 t s known at any later (or earler) tme t 2. Notce that n QM tme s not an observable, t s a parameter. In contrast, the poston s an observable. Ths s at odds wth the theory of Specal Relatvty, where space and tme are treated on an equal footng. 93

71 Quantum dynamcs: the Schrödnger equaton Postulate V An mportant property of the Schrödnger equaton s that, durng the evoluton between two measurements, the norm of the states does not change, h d [ dt ψ(t) ψ(t) = h d ψ(t) ] [ ψ(t) + ψ(t) h d ψ(t) ] dt dt = ψ(t) H(t) ψ(t) + ψ(t) H(t) ψ(t) = 0 where we have used that H(t) s Hermtan. On the other hand, the Schrödnger equaton s lnear. Therefore, the tme evoluton must be descrbed by a untary operator a ψ(t) = U(t, t 0 ) ψ(t 0 ), U U = UU = I. a If U s untary and ψ = U ψ then the norm s preserved, ψ ψ = ψ U U ψ = ψ ψ. 94

72 Quantum dynamcs: the Schrödnger equaton Postulate V From the relatons one gets ψ(t 3 ) = U(t 3, t 2 ) ψ(t 2 ), ψ(t 2 ) = U(t 2, t 1 ) ψ(t 1 ), U(t, t) = I, U(t 3, t 1 ) = U(t 3, t 2 )U(t 2, t 1 ), U(t 2, t 1 ) = U 1 (t 1, t 2 ) = U (t 1, t 2 ) U(t 2, t 1 )U(t 1, t 2 ) = I. Notce that, as antcpated above, the tme evoluton of a state of a closed system s reversble. If t > t 0, ψ(t) = U(t, t 0 ) ψ(t 0 ), ψ(t 0 ) = U (t, t 0 ) ψ(t). There s no loss of nformaton. 95

73 Quantum dynamcs: the Schrödnger equaton Postulate V In contrast, the measurement process (collapse of the state) s a not untary, not reversble process. Snce ths s produced by the nteracton wth an external apparatus, the system wll be no longer closed. But, as we have seen, one can nclude the measurng apparatus as a part of the (composte) system. Then the tme evoluton wll be untary and reversble and there s no need to ntroduce the bzarre collapse. The evoluton of a mxed state ρ(t) = w α (t) α (t) also follows from the Schrödnger equaton, h dρ(t) dt = = [ ] d α (t) w { h α dt (t) + α (t) [ h d α ]} (t) dt w {H(t) α (t) α (t) α (t) α (t) H(t)}, assumng tme-ndependence of the frequences, and hence dρ(t) dt = ī h [ρ(t), H(t)]. 96

74 Quantum dynamcs: the Schrödnger equaton Postulate V In general, the expectaton values change wth tme, [ ] [ ] d d ψ(t) d ψ(t) ψ(t) A ψ(t) = A ψ(t) + ψ(t) A dt dt dt = ī A ψ [A, H] ψ + ψ h t ψ. + ψ(t) A t ψ(t) The Hermtan operator H s called Hamltonan, but n QM there s no prescrpton to obtan t. It has clearly the dmensons of energy, thanks to the ntroducton of the dmensonful constant h n the Schrödnger equaton. In systems wth a quantum analog one can usually (not always) nfer ts form from the correspondng classcal Hamltonan (see Quantzaton Rules). 97

75 Quantum dynamcs Tme evoluton operator Substtutng ψ(t) = U(t, t 0 ) ψ(t 0 ) we get the Schrödnger equaton for U (the tme evoluton operator), where we have used that because ψ(t 0 ) does not depend on t. h d dt U(t, t 0) = H(t)U(t, t 0 ) d dt {U(t, t 0) ψ(t 0 ) } = d dt U(t, t 0) ψ(t 0 ) 98

76 Quantum dynamcs Tme evoluton operator Then, usng the propertes of U, du(t, t 0 ) = ī h H(t)U(t, t 0)dt U(t + dt, t 0 ) U(t, t 0 ) = ī h H(t)U(t, t 0)dt and takng t 0 = t, we obtan U(t + dt, t) = I ī h H(t)dt. Ths s the expresson for an nfntesmal tme evoluton. It reveals that H/ h s the generator of tme translatons. 99

77 Quantum dynamcs Tme evoluton operator Let us fnd the evoluton operator for an arbtrary tme nterval. If H = H(t), the Schrödnger equaton for U(t, t 0 ), wth U(t 0, t 0 ) = I, s easy to solve, U(t, t 0 ) = exp { ī } h H(t t 0). If H = H(t) one can check that the soluton s the Dyson seres, ( U(t, t 0 ) = I + ī ) n ˆ t ˆ t1 ˆ tn 1 dt 1 dt 2... dt n H(t 1 )H(t 2 )... H(t n ). h n=1 t 0 t 0 t 0 (t 0 < t 1 < t 2 < < t n 1 < t n ) If [H(t), H(t )] = 0 t smplfes to U(t, t 0 ) = I + n=1 1 n! [( ī ) ˆ t ] n { dt H(t ) = exp ī h t 0 h } dt H(t). t 0 ˆ t 100

78 Quantum dynamcs Statonary states and constants of moton Consder a tme-ndependent Hamltonan H = H(t). Snce H s self-adjont t can be dagonalzed, H E n = E n E n, E n R. The egenvalues E n are the allowed energes or energy levels and the E n the energy egenstates of the system. The tme evoluton of the energy egenstates s trval, U(t, t 0 ) E n = e ī h H(t t 0) E n = e ī h E n(t t 0 ) E n. The only change s an rrelevant global phase, so the state remans the same. Hence, the energy egenstates are statonary. 101

79 Quantum dynamcs Statonary states and constants of moton One can wrte the tme evoluton operator n the bass of energy egenstates (spectral resoluton of U) as U(t, t 0 ) = m n E m E m e ī h H(t t 0) E n E n = n The tme evoluton of a generc state ψ = ψ(t) = U(t, t 0 ) ψ = n c E s c e ī h E n(t t 0 ) E n E n E = Snce the components change by dfferent phases, c c e ī h E (t t 0 ), the state ψ s not statonary unless t s an energy egenstate. e ī h E n(t t 0 ) E n E n. c e ī h E (t t 0 ) E. 102

80 Quantum dynamcs Statonary states and constants of moton On the other hand, accordng to d dt ψ(t) A ψ(t) = ī h ψ [A, H] ψ + ψ A t ψ we say that a tme-ndependent observable A that commutes wth H s a constant of moton snce ts expectaton value n any state ψ does not change wth tme, A t = 0, [A, H] = 0 h d dt A ψ = 0. In partcular, snce [H, H] = 0, a tme-ndependent Hamltonan s a constant of moton, and the average energy H ψ does not change wth tme even f ψ t s not a statonary state. 103

81 Quantum dynamcs Tme evoluton pctures So far, we have consdered that states evolve wth tme and observables (unless explctly dependent on tme) stay constant, α t U α, A = A(t). Ths s called the Schrödnger pcture. However, snce after all we just deal wth the results of our observatons (measurements), we could vew thngs n an alternatve way. The tme evoluton of the expected value α A β t α U AU β can also be nterpreted as f the states do not evolve but the observable does, α t α, β t β, A t U AU. Ths s the Hesenberg pcture. 104

82 Quantum dynamcs Tme evoluton pctures To dstngush both pctures, when necessary, we denote The predctons are dentcal: α H = α(0) S = U α(t) S A (H) (t) = U A (S) (t)u, A (H) (0) = A (S) (0). H α A(H) (t) β H = S α(t) A (S) (t) β S and the hamltonan H has the same form n both pctures, H = U HU. 105

83 Quantum dynamcs Tme evoluton pctures An observable A n the Hesenberg pcture may change wth tme because of the dynamcs of the system or because of ts explct dependence wth tme. Then, usng h d dt U(t, t 0) = H(t)U(t, t 0 ) we obtan the Hesenberg equaton of moton, da (H) [ du ] [ ] = A (S) U + U A (S) du + U A(S) U dt dt dt t where one usually wrtes = ī h U [A (S), H]U + U A(S) t = ī h [A(H), H] + A(H) t U A(S) t U A(H) t. U 106

84 Quantum dynamcs Tme evoluton pctures The densty matrx changes wth tme n the Schrödnger pcture accordng to dρ (S) (t) dt = ī h [ρ(s) (t), H(t)] but t s constant n the Hesenberg pcture, ρ (S) (t 0 ) = α (t 0 ) α (t 0 ) ρ (S) (t) = U(t, t 0 )ρ(t 0 )U (t, t 0 ) ρ (H) (t) = U (t, t 0 )ρ (S) (t)u(t, t 0 ) = ρ (S) (t 0 ) = ρ (H) (t 0 ) dρ(h) (t) dt =

85 Quantum dynamcs Tme evoluton pctures The Hesenberg pcture s more smlar to the usual descrpton n Classcal Mechancs, where the observables (poston, momentum,... ) change wth tme. Actually, the Hesenberg equaton of moton has the same form as the Hamlton s equaton for a classcal varable A = A(x 1,..., x N, p 1,..., p N ; t), replacng the Posson bracket, by a commutator, [A, B] P da dt = [A, H] P + A t ( A B A ) B x p p x (classcal) [, ] P ī [, ] (quantum). h Ths analogy renforces the dea that the operator H ntroduced n the Schrödnger equaton s n fact the Hamltonan of the system. 108

86 Quantzaton rules Postulate VI How to buld quantum operators that represent the physcal observables? Next, we wll dscuss the canoncal quantzaton rules. Postulate VI For a physcal system n whch the cartesan coordnates are x 1, x 2,..., x N, wth correspondng momenta p 1, p 2,..., p N, the operators X r and P s, whch represent these observables n QM, must satsfy the commutaton relatons [X r, X s ] = 0, [P r, P s ] = 0, [X r, P s ] = hδ rs I. If the system has an observable wth classcal expresson A(x 1,..., x N, p 1,..., p N ; t) then the correspondng operator can be obtaned by convenently substtutng he varables x r and p s by the operators X r and P s, respectvely. Here, convenently means the followng: 109

87 Quantzaton rules Postulate VI Snce X and P are noncommutng observables, one should wrte classcal varables lke xp as an equvalent combnaton whose quantum analog s a Hermtan operator. In fact, X = X, P = P and [X, P] = XP PX = hi (XP) = (PX) hi = XP hi = XP. However, xp = 1 2 (xp + px) 1 (XP + PX) 2 s a Hermtan operator wth the same classcal expresson. Ths postulate wll look less bzarre when we see n next chapter that dentfyng the momentum wth an operator P that satsfes the commutaton relatons above s the rght way to understand P/ h as the generator of spatal translatons. 110

88 Superselecton rules Suppose we have an observable whose operator Q commutes (s compatble) wth all other operators assocated to observables n H, [Q, A] = 0, A. Then for any par of egenstates of Q wth dfferent egenvalues, Q ψ 1 = q 1 ψ 1, Q ψ 2 = q 2 ψ 2, we have that A 0 = ψ 1 [Q, A] ψ 2 = ψ 1 QA ψ 2 ψ 1 AQ ψ 2 = (q 1 q 2 ) ψ 1 A ψ 2 ψ 1 A ψ 2 = 0 f q 1 = q 2. Ths means there are no transtons between whatever two egenstates wth dfferent egenvalues of Q. As a consequence, let us see that n H there s no pure state that s a superposton of states wth dfferent values of Q. 111

89 Superselecton rules Suppose that such a pure state ψ exsts. Then, snce the egenvectors of Q s a bass of H, ψ = c ψ wth Q ψ = q ψ. Usng that ψ A ψj = 0 f ψ = ψ j, the expectaton of any observable A n ψ s ψ A ψ = c 2 ψ A ψ = Tr(ρA) wth ρ = c 2 ψ ψ. We see that unless ψ has a well-defned value of Q (there s just one c = 0) ρ descrbes a mxed state (ncoherent superposton of pure states). Any observable Q wth these propertes s called a superselecton observable and gves rse to superselecton rules: one can prepare only states wth well defned values of Q. States wth dfferent values of Q lve n separate Hlbert spaces H q. For example, the electrc charge, the party, the baryon and lepton number,

90 No-clonng theorem We have already emphaszed that a quantum state can not be understood as an element of realty but as a collecton of smlarly prepared systems. But how to make dentcal state preparatons of a state? Notce that the state, n prncple, maybe even unknown. Sometmes thngs are easy: t s possble to prepare the lowest energy state of a system by smply watng for the system to decay to ts ground state. Another way s flterng, the technque used n the Stern-Gerlach experment. But we would really lke to have a procedure to make exact replcas or clones of a prototype of the state, provded t exsts. Ths s a common method n classcal physcs: the duplcaton of a key or the copyng of a computer fle. However, surprsngly, let us see that clonng quantum states s mpossble. 113

91 No-clonng theorem Suppose we want to buld a machne to copy a quantum state. There are only two permssble quantum operatons wth whch we may manpulate the composte system: If we perform an observaton, the orgnal state wll rreversbly collapse nto some egenstate of the observable, corruptng the nformaton contaned n the qubt(s). Ths s obvously not what we want. Instead, we should use untary operatons, as the followng: Gven ψ and a blank pece of paper b, ψ b U( ψ b ) = ψ ψ. (Imagne we are so wse as to control the Hamltonan to make the state evolve ths way.) And the same wh another state φ, φ b U( φ b ) = φ φ. 114

92 No-clonng theorem Ths looks perfect but, f we take the scalar product of both resultng states, we see that ths s only possble f ( φ b )U U( ψ b ) = φ ψ = ( φ φ )( ψ ψ ) = φ ψ 2, φ ψ = 0 or ± 1, namely, f ψ and φ are ether the same state or they are orthogonal. Therefore, a sngle unversal U cannot clone a general quantum state (arbtrary superpostons of the orthogonal qubts 0 and 1 ). Notce that states whch are classcally dfferent wll certanly be orthogonal, so the no-clonng theorem for quantum states s not n conflct wth the well-known possblty of copyng classcal states. 115