10 More general formulation of quantum mechanics

Transcription

1 TFY4250/FY2045 Tillegg 10 - More geeral formulatio of quatum mechaics 1 TILLEGG More geeral formulatio of quatum mechaics I this course we have so far bee usig the positio represetatio of quatum mechaics, ad we have briefly touched upo the mometum represetatio. I this Tillegg, we shall see that these formulatios are special cases of a more geeral formulatio. This ew formulatio is importat firstly because it offers a umber of techical advatages, simplifyig a umber of calculatios (oce we get used to the ew techique). The mai advatage, however, is that the ew formulatio allows us to treat systems which caot be described by wave mechaics, that is, with wave fuctios i x- or p-space. A cetral example is the descriptio of spi degrees of freedom. [You might wat to supplemet these otes with Hemmers chapter 6, chapter 3 i Griffiths, ad chapter 5 i B&J. Aother referece is J.J. Sakurai, Moder Quatum Mechaics, chapter 1.] 10.1 Vectors i Hilbert space 10.1.a Cetral results i the positio represetatio Let us briefly review some cetral results i the theory preseted so far: The state of a particle at a give poit i time ca be described by a spatial wave fuctio ψ(x), if we cosider a oe-dimesioal case. The value of ψ at the poit x gives the probability amplitude of fidig the particle i this positio, i the sese that ψ(x) 2 gives the probability desity. Thus, whe we kow the complete fuctio ψ(x), we kow the amplitudes of fidig the particle i all possible positios. We are free to expad the wave fuctio i terms of a arbitrary complete set of basis states. There are may such bases; we remember that for each observable there is a hermitia operator with a complete set of eigefuctios. Examples are p x, Ĥ ad x, with eigevalue equatios ad eigefuctios as follows: p x ψ p (x) = pψ p (x) ; ψ p (x) = (2π h) 1/2 e ipx/ h ; Ĥψ (x) = E ψ (x) ; ψ (x) depeds o the potetial V (x); (T10.1)

2 TFY4250/FY2045 Tillegg 10 - More geeral formulatio of quatum mechaics 2 xψ x (x) = x ψ x (x) ; ψ x (x) = δ(x x ). Here, we are assumig a discrete ad o-degeerate eergy spectrum {E }, so that we ca use the orthoormality relatio ψ k, ψ ψ k (x)ψ (x)dx = δ k. (T10.2) All these sets of states are complete, i the sese that a arbitrary state ψ(x) may be expaded i ay of these sets. The expasio i terms of eergy eigefuctios the is a sum, while the expasios i mometum ad positio eigefuctios become itegrals, sice the spectra {x } ad {p} are cotiuous: ψ(x) = dp φ(p)ψ p (x); ψ(x) = c ψ (x); (T10.3) ψ(x) = dx c(x )ψ x (x). The expasio coefficiets φ(p), c ad c(x ) i these formulae characterize the state perfectly, like ψ(x) itself, i the same way as the compoets of a vector i a give basis gives a perfect represetatio of the vector. The expasio coefficiets i (T10.3) are the projectios of ψ oto the respective basis vectors ψ p, ψ ad ψ x (x) = δ(x x ) : φ(p) = (ψ p, ψ) = ψ p (x)ψ(x)dx; c = (ψ, ψ) = ψ (x)ψ(x)dx; c(x ) = (ψ x, ψ) = δ(x x)ψ(x)dx = ψ(x ). (T10.4) Let us repeat the proof for the formula for c, by projectig ψ(x) = c ψ (x) o ψ k. Iterchagig the order of itegratio ad summatio, we fid that ψ k, ψ = ψ k (x) c ψ (x) = c ψ k (x)ψ (x)dx = c δ k = c k, q.e.d. (T10.5) A small exercise: Fid the mometum wave fuctio φ(p) by projectig ψ o ψ p. Hit: Chage the order of the itegratios ad use the delta-fuctio ormalizatio ψ p (x)ψ p(x)dx = δ(p p). I a similar maer you ca derive the formula for c(x ) usig the relatio δ(x x )δ(x x )dx = δ(x x ). But it is eve simpler to use the idetity ψ(x) = ψ(x )δ(x x )dx.

3 TFY4250/FY2045 Tillegg 10 - More geeral formulatio of quatum mechaics 3 We must also remember the physical iterpretatio of the expasio coefficiets: ψ(x ) is (as already metioed) the probability amplitude of fidig the particle at the positio x (i the sese that ψ(x ) 2 is the probability desity i positio space). I the same maer, φ(p) 2 is the probability desity i mometum space, ad c 2 is the probability of measurig the eergy E (ad leavig the particle i the state ψ ) b State vectors i Hilbert space. Dirac otatio The most cetral aspect of the ew formulatio is that: With each wave fuctio ψ(x) we may associate a abstract vector which is called a state vector. For such vectors we shall use the symbol..., which was iveted by Dirac. (T10.6) For this so-called ket vector we may choose a suitable label to idicate which state we have i mid. As a example, we may deote the vector correspodig to the wave fuctio ψ(x) by ψ. The vector correspodig to the eergy eigefuctio ψ (x) may be deoted by ψ (or or E ). The vector correspodig to the positio eigefuctio ψ x (x) = δ(x x ) may suitably be called x. The vector correspodig to the mometum eigefuctio ψ p (x) = (2π h) 1/2 e ipx/ h might be labeled p or ψ p, ad so o. Thus (o matter which label we choose to use) we have a oe-to-oe correspodece betwee wave fuctio ad state vector: ψ(x) ψ, ψ (x) ψ = = E, ψ x (x) = δ(x x ) x, (T10.7) ψ p (x) p = ψ p, ad so o. Here, we should ot be cofused by the fact that also wave fuctios are vectors (cf Tillegg 2, or Griffiths); The ket vector ψ ad the wave fuctio ψ(x) are differet objects; they belog to two differet vector spaces, with a oe-to-oe correspodece betwee the vectors i the two spaces. The space spaed by the ket vectors will from ow o be deoted the Hilbert space H, (eve if also the wave fuctios ψ(x) are vectors i a Hilbert space, mathematically speakig). 1 A importat property of Hilbert space (shared with all liear, complex vector spaces) is that a arbitrary liear combiatio of two vectors is also a vector i this space: c 1 ψ 1 + c 2 ψ 2 = ψ 3. (T10.8) 1 As you may recall from the discussio of the mometum eigefuctios ψ p (x), these are ot quadratically itegrable. They therefore do ot belog to the mathematical Hilbert space L 2 (, ) of quadratically itegrable fuctios, ad either do the positio eigefuctios ψ x (x) = δ(x x ). Correspodigly, the ket vectors ψ p ad x do ot really belog to the Hilbert space H. However, these vectors fuctio perfectly well as basis vectors for expasios of vectors i Hilbert space. Thus the sets { ψ p } ad { x } are complete, together with the set { ψ }. I quatum mechaics oe therefore works with a kid of elarged Hilbert space, where the vectors x ad ψ p p may largely be treated i the same way as the proper Hilbert-space vectors.

4 TFY4250/FY2045 Tillegg 10 - More geeral formulatio of quatum mechaics 4 I particular c ψ gives a ew vector ψ 1 : 2 c ψ = ψ c = ψ 1. (T10.9) Here, ψ correspods to the wave fuctio ψ(x), ad ψ 1 correspods to cψ(x) = ψ(x)c. As you will remember, ψ(x) ad cψ(x) describe the same physical state. The same the holds for the ket-vectors ψ ad ψ 1 = c ψ c The dual Hilbert space. Scalar product To get a ew formulatio that is equivalet with the old oe, we must postulate the existece of a salar product (ier product) betwee ket vectors. Before we defie this product, it may be istructive to review how the scalar product is defied for ordiary complex vectors A ad B. I physics literature this product is usually defied as A, B = A B A i B i. (T10.10) i From this defiitio it follows that the (usually) complex scalar product satisfies the relatio A, B = B, A. (T10.11) Furthermore it is easy to see that, with c a complex umber, the scalar product has the properties A, cb = c A, B ad ca, B = c A, B. (T10.12) We the say that the scalar product V 1, V 2 is liear with respect to vector o 2 (V 2 ) ad ati-liear with respect to vector o 1 (V 1 ). Notice that the scalar product betwee fuctios has exactly the same properties: f, g = f g dτ = g, f ; f, cg = c f, g ; cf, g = c f, g. (T10.13) To obtai a coveiet otatio for the scalar product (ier product) ψ 1, ψ 2 betwee two ket-vektorer i Hilbert space, Dirac itroduced the so-called dual Hilbert space H. We create this space by associatig with every ket vector ψ i H a so-called dual vector ψ i H : dual of ψ ψ. (T10.14) Employig these ew vectors..., which Dirac called bra vectors, we ca formulate rules of calculatio ad establish the Dirac otatio, which are i fact easier to use tha the wave-fuctio formalism (ad which may also be applied to problems for which there does ot exist ay wave fuctios, like e.g. for spi). The mai reaso for itroducig the bra vector is that it allows us to write the scalar product betwee ψ 1 ad ψ 2 as a simple product of the bra vector ψ 1 ad the ket vector ψ 2 : ψ 1, ψ 2 = ψ 1 ψ 2 ψ 1 ψ 2. (T10.15) We ca ow uderstad why Dirac itroduced the ames bra ad ket: The scalar product ψ 1 ψ 2 ψ 1 ψ 2 2 Some authors also use the otatio c ψ = ψ c = cψ.

5 TFY4250/FY2045 Tillegg 10 - More geeral formulatio of quatum mechaics 5 has the form bra ket, derived from the word bracket. Notice that ψ 1 plays a role similar to A i the dot product A, B = A B, ad to ψ 1 i the scalar product ψ 1, ψ 2 = ψ 1 ψ 2 dτ. I (T10.15), we have a additioal rule: I this ier product, the bra vector is always stadig o the left ad the ket vector is stadig o the right. The physical iterpretatio of ψ 1 ψ 2 = ψ 1, ψ 2 the projectio of ψ 2 o ψ 1 is exactly the same as for the correspodig wave fuctios: The projectio is the probability amplitude that a measuremet fids the system i the physical state correspodig to ψ 1 (ad to ψ 1 ) whe the system is i the state correspodig to ψ 2 (ad to ψ 2 ) before the measuremet. Thus ψ 1 ψ 2 ψ 1 ψ 2 = ψ 1, ψ 2 ψ 1 ψ 2 dτ. (T10.16) From the relatio ψ 1, ψ 2 = ψ 2, ψ 1 it the follows that ψ 2 ψ 1 = ( ψ 1 ψ 2 ) = complex umber, or We the see that ψ 1 ψ 2 = ψ 2 ψ 1. (T10.17) ψ 1 c ψ 2 = ψ 1, cψ 2 = c ψ 1, ψ 2 = c ψ 1 ψ 2. (T10.18) Thus the scalar product is liear i the ket vector (vector o 2). With ψ 1 = c φ we see i the same maer that Thus the dual of ψ 1 = c φ is c φ : ψ 1 ψ 2 = ψ 2 ψ 1 = (c ψ 2 φ ) = c φ ψ 2. c φ dual of c φ, (T10.19) i aalogy with (ca) = c a. We shall use this relatio frequetly. Notice that this meas that the scalar product is ati-liear with respect to vector o 1, just as f, g = f g dτ. The relatio betwee ψ ad ψ(x) While the wave fuctio is ow a well-kow ad hopefully dear fried, you are probably havig difficulties i imagiig what kid of object the abstract state vector ψ is. However, as you get more acquaited with the ew formalism, you will lear to live with the abstractess, ad you will probably ed up appreciatig the coveiece of the ew formalism, oce you lear how to use it. We shall ow attempt to preset the cetral parts of these methods, ot ecessarily i the most systematic way. Two importat aspects have already bee metioed: The dual correspodece (T10.14) ad (T10.19), together with the fact that the scalar product ψ 1 ψ 2 is the same as betwee the correspodig wave fuctios. (See (T10.16)). Thus the physical iterpretatio of the ew scalar product is the same as for the old oe: For example, the scalar product ψ ψ is the probability amplitude of fidig the particle i the state described by the vector ψ

6 TFY4250/FY2045 Tillegg 10 - More geeral formulatio of quatum mechaics 6 (i.e., with the eergy E ) if it was i a state desrcibed by the vector ψ (correspodig to the wave fuctio ψ(x)) before the measuremet. I the same maer p ψ is the amplitude of fidig the particle described by the vector p (i.e., with mometum p). Aother example: We remember that the vector x describes a state where the particle has the positio x (correspodig to the wave fuctio ψ x (x) = δ(x x )). The scalar product x ψ therefore is the amplitude of fidig the particle at x whe it is i the state ψ. But this amplitude is precisely what the wave fuctio ψ(x ) is. Thus the relatio betwee the wave fuctio ψ(x) ad the vector ψ is x ψ = ψ(x). (T10.20) Notice also that ψ x = x ψ = ψ (x). (T10.21) 10.1.d Completeess Thus the wave fuctio ψ(x) = x ψ is the compoet of the Hilbert vector ψ i the x -directio, or the projectio of the vector ψ o the basis vector x. I the same maer the scalar product ψ ψ is the projectio of ψ o aother basis vector, ψ. As the term basis idicates, these sets are complete, as are the correspodig sets of wave fuctios (cf (T10.1)). This meas that a arbitrary vector ψ i Hilbert space may be expaded i terms of these basis sets, i aalogy with the expasios (T10.3): ψ = dp φ(p) p, ψ = c ψ, ψ = dx c(x ) x. (T10.22) Here we must of course expect that the coefficiets are the same as i (T10.4), c = ψ, ψ = ψ ψ, etc, but let us check it, to get a little experiece with the ew formalism: By projectig the secod of equatios (T10.22) o ψ k (that is, by multiplyig from the elft by ψ k ), we fid ψ k ψ = c ψ k ψ = c δ k = c k, q.e.d.. Here we have used (T10.16), which implies that the set ψ k is orthoormal, just like the set ψ k (x): ψ k ψ = ψ k, ψ = δ k. (T10.23) Isertig ito (T10.22) ow gives ψ = ψ ψ ψ. (T10.24)

7 TFY4250/FY2045 Tillegg 10 - More geeral formulatio of quatum mechaics 7 The correspodig expasios i terms of mometum ad positio vectors are ψ = dp p ψ p, ψ = dx x ψ x. (T10.25) This ca be show by projectig (T10.22) o x ad ψ p, respectively. Both these are cotiuum sets, with δ-fuctio ormalizatio, so that istead of (T10.23) we have 3 p p = p, p = δ(p p ), Hece so that x x = ψ x, ψ x = δ(x x ). (T10.26) x ψ = dx c(x ) x x = dx c(x )δ(x x ) = c(x ), ψ = dx c(x ) x = dx x ψ x = dx x ψ x, I the same maer, p ψ = dp φ(p) p p = dp φ(p)δ(p p ) = φ(p ), Notice that p ψ = φ(p) q.e.d.. q.e.d.. (T10.27) is the amplitude of fidig the mometum p, that is, the mometum wave fuctio. Notice also that the formulae (T10.24) ad (T10.25) are aalogous to expasios of a ordiary vector i alterative basis sets ê i ad ê i, A = A i ê i = A iê i ; the compoets (A i ) deped o our choice of basis, while the vector A itself is basisidepedet; it is a absolute geometric object. The same ca be stated about the Hilbert vector ψ. The completeess relatio Equatios (T10.24) ad (T10.25) rely o the fact that the three basis sets are complete. A slight re-writig of (T10.24) gives 4 ψ = ψ ψ ψ = ψ ψ ψ ( ) = ψ ψ ψ. (T10.28) 3 Vectors with δ-fuctio ormalizatio (ad the correspodig wave fuctios) strictly speakig do ot belog to the respective Hilbert spaces, as already metioed. I quatum mechaics we prefer to iclude these vectors ito our formalism. This meas that we are workig with a exteded Hilbert space, ofte simply called the Hilbert space. 4 Notice that the scalar product ψ ψ is a complex umber, which may be moved wherever it suits us.

8 TFY4250/FY2045 Tillegg 10 - More geeral formulatio of quatum mechaics 8 Sice this holds for a arbitrary vector ψ, it follows that ψ ψ = 1. (the completeess relatio) (T10.29) This is the completeess relatio i Dirac otatio (for a discrete vector set). As we shall see below, both sides of this equatio are operators. The quatity o the right-had side is the uit operator; 1 ψ = ψ. I a similar maer we fid for the two cotiuum sets: ( ) ψ = dx x ψ x = dx x x ψ, etc. Thus the completeess relatios for these sets are dx x x = 1, dp p p = 1. (completeess) (T10.30) I what follows we shall see that these relatios are easier to use tha the correspodig relatios for the wave fuctios (see e.g. B&J p 205): ψ (x)ψ (x ) = δ(x x ), dp ψ p (x)ψ p (x ) = δ(x x ), etc. (T10.31) As a example we ca re-write the square ψ 2 ψ ψ of the orm of ψ i several ways: ( ) ψ ψ = ψ 1 ψ = ψ dx x x ψ = dx ψ x x ψ = dx ψ (x)ψ(x) = ψ, ψ, ( ) ψ ψ = ψ ψ ψ ψ = ψ ψ ψ ψ = ψ ψ 2 ( ) ψ ψ = ψ dp p p ψ = dp p ψ 2 dp φ(p) 2. c 2, (T10.32) These relatios are geeralized Parseval relatios. Notice that the scalar product ψ ψ is always a real o-egativeumber, so that we ca defie the orm of ψ as ψ = ψ ψ 1/2 0. (T10.33) The techique applied above, of isertig the uit operators (T10.29) or (T10.31) (or a similar expressio for aother complete set) where it is suitable, is a trick that we shall use

9 TFY4250/FY2045 Tillegg 10 - More geeral formulatio of quatum mechaics 9 frequetly i what follows. Let us as a example use this trick to show the equivalece of (T10.29) ad (T10.31): From the ormalizatio coditio (T10.26) we have: δ(x x ) = x x = x 1 x = x ψ ψ x (T10.34) = x ψ x ψ = ψ (x)ψ (x ), q.e.d Operators, eigevectors, expectatio values, etc 10.2.a Geeral defiitio of operators A operator applied to a fuctio has the property operator fuctio =ew fuctio. (T10.35) The operators appearig i the ew formulatio have the same kid of property: Whe a operator  acts o a Hilbert vector b, the result is a ew vector i Hilbert space, c =  b. I other words: Operators i Hilbert space map the space ito itself. Notice that the operator is stadig to the left of a ket vector. Example: Projectio operator The expressio P ψ ψ is a operator. Applied to a arbitrary vector ψ it gives P ψ ( ψ ψ ) ψ = ψ ψ ψ c ψ. (T10.36) Thus the operator P ψ ψ projects out the part of the vector ψ which poits i the ψ directio. It is therefore called a projectio operator. Notice that P = ψ ψ = 1. (completeess relatio) (T10.37) Thus, as already metioed, the expressio o the left i the compleess relatio is a sum of operators, addig up to the uit operator. Applyig the trick of isertig this uit operator wherever it suits us, we must therefore always let the uit operator act from the left o a ket vector. It may also stad to the right of a bra vector. Some times we shall apply the uit operator to the right of a isolated bra vector ; the resultig expressio bra vector 1 may the be applied from the left o a ket. (See sectio 10.3.b below.) 10.2.b Operators ad eigevectors As you ca see e.g. i sectio 6.4 i Hemmer book, we may use the same postulates of quatum mechaics i the ew formulatio as i the old oe, apart from the otatio: Thus to each physical observable (e.g. x, p x, E, etc) there correspods a Hermitia operator. The problem with these Hilbert-space operators is that they are as abstract as the vectors but agai we shall see that we ca live with the abstractess. For example, the oly thig we

10 TFY4250/FY2045 Tillegg 10 - More geeral formulatio of quatum mechaics 10 eed to kow about the operator p x is that it is defieed i such a way that it satisfies the eigevalue equatio p x p = p p. (T10.38) Similarly, the operator x is defied by x x = x x. (T10.39) Thus the positio vector x is a eigevector of the operator x with the eigevalue x. We shall see that (T10.38) ad (T10.39) are sufficiet to determie the actio of the operators x ad p x o a arbitrary vector ψ. Usig the completeess relatio for the set { p } of eigevectors of p x we have ψ = dp p p ψ. (T10.40) Usig (T10.38) we the fid that the operator acts as follows: p x ψ = p x dp p p ψ = dp p x p p ψ ( ) = dp p p p ψ. Sice this holds for all ψ, we have a well-defied expressio for the operator p x : p x = dp p p p. (T10.41) (T10.42) Here, the umber p ca be placed i ay positio i the itegrad. This expressio is the so-called spectral represetatio of the operator p x ; otice that the itegral over p rus over the complete spectrum of the operator. Usig the same method we fid x = dx x x x. (T10.43) A similar procedure ca obviously be carried through for all Hermitia operators (with complete sets of eigevectors). A operator F with a cotiuous spectrum {f} ad eigevectors f ca be expressed as F = df f f f. (T10.44) A operator  with a discrete eigevalue spectrum {a } ad eigevectors a ca be writte as  = a a a, (T10.45) ad so o. The same recipe ca also be used to obtai a expressio for a arbitrary power of a Hermitia operator. Applyig p x oce more i (T10.41) we fid for example that p 2 x = dp p p 2 p, ad similarly for higher powers of p x. This way we ca also fid a expressio for a operator-valued fuctio F ( p x ). Such operator fuctios are defied by a Taylor expasio of F i powers of the argumet. Thus the operator F ( p x ) ca be writte as F ( p x ) = dp p F (p) p, (T10.46)

11 TFY4250/FY2045 Tillegg 10 - More geeral formulatio of quatum mechaics 11 where F (p) is the fuctio F of the variable p. A example is the potetial V (x) for which the abstract potetial operator is give by V ( x) = dx x V (x ) x. (T10.47) 10.2.c Adjoit We have bee discussig Hermitia operators, but without defiig Hermiticity i Dirac otatio. The ew defiitio must of course be equivalet with the old oe, that is, we must require that the operator is self adjoit. The old defiitio of the adjoit of the operator  was (Âψ 1) ψ 2 dτ ψ 1  ψ 2 dτ, (T10.48) which may also be writte as ψ ( 1  ψ 2 dτ = ψ ) 2 Âψ 1dτ, or The equivalet defiitio i Dirac otatio is (ψ 1,  ψ 2 ) = (ψ 2, Âψ 1). a  b = b  a, (def.  ) (T10.49) where we have deoted the two arbitrary vectors by a ad b. Deotig the vector  a by c, we have from (T10.49): a  b = b c = c b. Thus the dual of the vector c =  a is a  : a  dual of  a. (T10.50) For a self adjoit (i.e. Hermitia) operator  =  we see that a  b = b  a. ( Hermitia) (T10.51) Later we shall also see that the expectatio values of Hermitia operators ad their eigevalues are real, as i the old formalism. The equivalece betwee the old defiitio (T10.48) of the adjoit ad the ew oe, (T10.49), implies that the rules of calculatio for obtaiig a adjoit (foud i Tillegg 2) apply equally well for Hilbert-space operators. Thus ( B) = B  ; ( BĈ) = Ĉ B Â, etc; (T10.52) (c F ) = c F, (c a complex umber).

12 TFY4250/FY2045 Tillegg 10 - More geeral formulatio of quatum mechaics 12 We may also iclude the rule which follows by applyig (T10.51) twice: ( ) = Â, (T10.53) a ( ) b = b  a = a  b. The bra vector a, which is dual to the ket vector a, is ofte called the adjoit of a (ad vice versa): ( a ) = a, ( a ) = a. (T10.54) We may the obtai the adjoit of a arbitrary expressio cataiig costats, operators ad ket ad bra vectors usig the followig substitutios: c c ; (costat)   ; (operator) ; (ket bra) (T10.55) ; (bra ket) while the order of the terms is reversed. Note, however, that costats ad scalar products like a b ad a  b ca be moved aroud freely, because they are all complex umbers. Examples: (i) The adjoit of the operator c u  v φ ψ is ψ φ v  u c = c v  u ψ φ. (ii) The adjoit of the ket vector c u v w is the bra vector 10.2.d Expectatio values w v u c = c w v u. The expectatio value of a observable F is i Dirac otatio give by F ψ = ψ F ψ. (expectatio value) (T10.56) Let us check that this defiitio is equivalet with the old oe (cf Postulate C i Tillegg 2): We take F = x as a example ad use the stadard trick: x ψ = ψ x 1 ψ = ψ x dx x x ψ = dx ψ x x x ψ = dx ψ (x ) x ψ(x ), q.e.d. (You ca derive the same result more directly by usig (T10.43)). Furthermore we have from (T10.27) ad (T10.42): p x ψ = ψ p x ψ = dp ψ p p p ψ = dp φ (p) p φ(p) = dp φ(p) 2 p, (T10.57)

13 TFY4250/FY2045 Tillegg 10 - More geeral formulatio of quatum mechaics 13 where p ψ = φ(p) is the amplitude of fidig the mometum p (i.e., the mometum wave fuctio) ad φ(p) 2 is the probability desity i mometum space. More geerally, we have for a (Hermitia) operator  with eigevalues a ad eigevectors a (cf (T10.45)): A ψ = ψ  ψ = ψ a a a ψ = a ψ a a ψ = a ψ 2 a P (a )a, (T10.58) where P (a ) = a ψ 2 is the probability of measurig a. Notice that the Hermiticity of  implies that A ψ is real for all ψ (cf (T10.51)). I particular, if we set ψ = a, equatio (T10.58) gives A a = a, showig that all eigevalues a are real, as we have see before Equatio of state. Positio ad mometum represetatios 10.3.a Equatio of state So far we have ot discussed the time depedece of state vectors. A time-depedet wave fuctio Ψ(x, t) must correspod to a time-depedet Hilbert vector Ψ(t) : Ψ(x, t) = x Ψ(t). (T10.59) We shall postulate that the quatum-mechaical equatio of motio for Ψ(t), or the equatio of state, geerally has the same form as the Schrödiger equatio, i h d Ψ(t) = Ĥ Ψ(t), dt (T10.60) also for systems that ca ot be described by ordiary wave fuctios (cf spi degrees of freedom). We may call this equatio a geeralized Schrödiger equatio. I this equatio, Ĥ is the Hamiltoia of the system, which determies how Ψ(t) develops as a fuctio of time. Supposig that Ĥ is Hermitia, we may say that Ψ(t) rotates i Hilbert space, sice the orm is the coserved. This follows from (T10.60): sice Ĥ = Ĥ whe Ĥ is Hermitia. d dt Ψ(t) Ψ(t) = ( d d Ψ ) Ψ + Ψ dt dt Ψ = 1 i h Ψ Ĥ Ψ + Ψ 1 Ĥ Ψ i h = ī h Ψ (Ĥ Ĥ) Ψ = 0, q.e.d., (T10.61)

14 TFY4250/FY2045 Tillegg 10 - More geeral formulatio of quatum mechaics b Positio ad mometum represetatios For systems which ca be described by wave fuctios, equatio (T10.60) must of course be equivalet to the Schrödiger equatio, as is also idicated by the form of the proposed equatio (T10.60). We shall see that the Schrödiger equatio ad the old formulatio of quatum mechaics i positio space, which we call the positio represetatio, follows simply by projectig (T10.60) oto the x basis. I the same maer, the mometum represetatio will follow whe we project (T10.60) o the mometum basis p. We start by establishig some simple formulae, which are very hady tools both here ad i other coectios: x x = x x ; (T10.62) x p x = h i x x ; p p x = p p ; p x = h i p p. (T10.63) (T10.64) (T10.65) These formulae ca be geeralized, with the operators x ad p x replaced by powers of these operators, ad hece also with operator-valued fuctios which ca be expaded i powers: x F ( x, p x ) = F (x, h i x ) x ; p F ( x, p x ) = F ( h i p, p) p. (T10.66) (T10.67) All these expressios become meaigful whe they are applied from the left o a chose ket vector. Proof of the relatios above: Equatios (T10.62) ad (T10.64) follow by takig the adjoit of the eigevector equatios (T10.39) ad (T10.38) (cf the rules (T10.55)). Formula (T10.63) is show as follows: From the relatio x p = ψ p (x ) = (2π h) 1/2 e ipx / h = p x (T10.68) we have the idetities p x p = h i x x p ad x p x = h i p p x. Usig the stadard trick with the completeess relatio, we the have x p x 1 = x p x dp p p = x dp p p p = dp x p p p = dp h i x x p p = h i x x dp p p = h i (T10.69) x x 1, q.e.d. (T10.70)

15 TFY4250/FY2045 Tillegg 10 - More geeral formulatio of quatum mechaics 15 (Notice that the use of (T10.42) for the operator p x would save us from the first steps i this derivatio). Equatio (T10.65) is proved i the same maer. The proof (T10.70) of equatio (T10.63) ca easily be geeralized to arbitrary powers of p x, so that a power p x to the right of x ca be replaced by the correspodig power ( h ) i x to the left of x. Similar statemets are valid for all four of the relatios (T10.62) (T10.65). This also holds for sums of powers, that is, for Taylor expasios. This prooves (T10.66) ad (T10.67). It is ow a trivial matter to derive the Schrödiger equatio from the geeralized equatio (T10.60). Projectig the latter o x we fid from (T10.66): i h d dt x x Ψ(t) = x Ĥ( x, p x) Ψ(t) = Ĥ(x, h i x ) x Ψ(t), or i h t Ψ(x, t) = Ĥ(x, h i x )Ψ(x, t), (T10.71) which is the Schrödiger equatio i the positio represetatio. It is ow ot really ecessary to show that also the mometum represetatio follows from (T10.60), because we kow that it follows from (T10.71) (which follows from (T10.60)). But let us do it ayway, to get some more practice: By projectig (T10.60) o p we have from (T10.67) i h d p Ψ(t) = dt p Ĥ( x, p x) Ψ(t) = Ĥ( h p i p, p) p Ψ(t). Here, p Ψ(t) is the amplitude of fidig the mometum p at the time t, which is of course idetical to the mometum-space wave fuctio Φ(p, t). Thus we have i h t Φ(p, t) = Ĥ( h i, p)φ(p, t), p (T10.72) which is the Schrödiger equatio i the mometum represetatio (cf Tillegg 2). The derivatio of (T10.71) ad (T10.72) show that x ad p x (ad hece also Ĥ) i the positio ad mometum represetatios ca be read straight out from the right-had sides of equatios (T10.62) (T10.67). This is because if we let e.g. p x act o a vector ψ, we get a ew vector which we may call ψ. This vector correspods to the wave fuctio ψ(x ) = x ψ = x p x ψ (T10.63) = = h i x ψ(x ). h i x x ψ (T10.73) This shows that the mometum operator i the positio represetatio is h ; which ca i x be read out from the right-had side of (T10.63). As a little exercise, let us show that the Hilbert-space operators x ad p x satisfy the commutatio relatio [ x, p x ] = i h 1. (T10.74)

16 TFY4250/FY2045 Tillegg 10 - More geeral formulatio of quatum mechaics 16 There are several ways to show this. together with (T10.62) ad (T10.63): Here we do it by usig the completeess relatio x p x p x x = = = = = = dx x x ( x p x p x x) dx x ( x x p x x p x x) dx x dx x dx x ( ( ( x x p x h i x h i x h i x x h i x x h i dx x i h x = i h 1, ) x x x ) x x x ( ) x x x h x i q.e.d. ) x x Notice the order of the factors: x x p x = x x p x = x h i x p x x = h i 10.4 Matrix mechaics x x ; x x x = h i x x x. I the preceedig sectio we saw that the positio ad mometum represetatios follow whe we project the equatio of state (T10.60) o the x - ad p -bases, respectively. These are two of may possible represetatios. I this sectio we shall see what kid of represetatio that results from the projectio o a discrete basis a Matrix represetatios of vectors ad operators We cosider a discrete set of basis vectors 1, 2,, which are orthoormalized; k = δ k. (T10.75) A vector a = 1 a = k k k a = k k a k may the be represeted by the compoets (or coordiates ) which we may call a k ; k a a k ; a = k a k k. (T10.76)

17 TFY4250/FY2045 Tillegg 10 - More geeral formulatio of quatum mechaics 17 These compoets may coveietly be collected i a colum matrix a. Similarly, the bra vector a = a k k k may be represeted by the row matrix a = (a 1, a 2,...). We ote that this row matrix is the adjoit (or Hermitia cojugate) of the colum matrix a. 5 The scalar product may the be writte as b a = b 1 a = b k k a = k k a 1 a 2 = (b 1, b 2,...). b k a k b a. (T10.77) A operator  is ow represeted by a matrix. This follows whe we cosider the compoets of the vector c =  a, which are k c = k  a = k  a, that is, c k = A k a, or i matrix form: c 1 c 2. = A 11 A 12 A 21 Similarly, employig two uit operators, we have b  a = k, b k k  a = (b 1, b 2,...). a 1 a 2 A 11 A 12 A a 1 a 2. (T10.78) b Aa. (T10.79) A product of two operators is represeted by the product of the two matrices: that is, ( B) k = k  B = k  m m B = A km B m, m m (AB) = (A)(B). For the adjoit of a operator  we have from (T10.49) ( ) k = k  =  k = A k, A = Ã. or (T10.80) (T10.81) Thus the operator  is represeted by the adjoit (or Hermitia cojugate) of the matrix. 5 Takig the adjoit, or Hermitia cojugate, of a matrix correspods to takig the traspose cojugate; M M.

18 TFY4250/FY2045 Tillegg 10 - More geeral formulatio of quatum mechaics b Chage of basis *** 6 The freedom i the choice of basis makes it iterestig to see how a chage of basis works. Suppose that we wat to chage from the origial basis k (labeled by lati letters) to a ew discrete set α (α = 1, 2,...), labeled by greek letters (to distiguish betwee old ad ew compoets). The ew basis vectors ca be expressed i terms of the old oes: α = k k k α k k S kα. (T10.82) Here, S kα k α is the projectio of the ew vector o α o the old vector o k. Thus the umbers S kα are the compoets of the ew vectors i the old basis. These complex umbers may be collected i a matrix S. Thus we have k α = S kα, If the ew basis is to be orthoormal we must have α k = k α = S kα = (S ) αk. (T10.83) δ βα! = β α = k β k k α = k S βk S kα = (S S) βα. (T10.84) I the same maer, δ k = k = α k α α = α S kα S α = (SS ) k. Whe these relatios are satisfied, the matrix S is said to be uitary: S S = SS = 1; S = S 1, (uitarity). (T10.85) We may call S the trasformatio matrix, because it gives the coectio betwee vector compoets i the old ad ew bases, ad similarly for the operator matrices. Thus the ew matrix elemets of the operator  are A βα β  α = k, β k k A α = k, S βk A ks α, or o matrix form: A y = S AS. (T10.86) Furthermore, the vector a = k k k a is i the ew basis represeted by the compoets a α α a = k α k k a = k S αk a k, 6 ot compulsory

19 TFY4250/FY2045 Tillegg 10 - More geeral formulatio of quatum mechaics 19 or o matrix form: For a bra vector we fid similarly a y = S a. (T10.87) b β = b β = b β = b β = b S β, or o matrix form b y = b S. (T10.88) Such a uitary trasformatio (chage of basis) of course does ot alter the physics of a give problem. For example, all scalar products are coserved uder the trasformatio: By combiig (T10.75) (T10.78) we fid that b ya y a y = (b S)(S AS)(S a) = b Aa. This is how it should be, because the scalar product may also be writte as is basis idepedet. b  a, which Diagoalizatio Some times oe wishes to fid a ew basis set { α } which are eigevectors of a give operator Â,  α = a α α, (T10.89) so that the matrix A y becomes diagoal: A βα = β  α = a αδ βα. (T10.90) Thus the task of fidig the eigevalues of the operator  is equivalet to diagoalizig the A-matrix. This diagoalizatio problem, i.e., to fid the compoets k α (equal to S kα ) of the vectors α, ca i priciple be solved startig with the o-diagoal matrix A k (i the old basis). This procedure is relevat whe workig with a fiite-dimesioal sub-space of the Hilbert space. From (T10.89) we the have: or o matrix form: N k  α = a α k α, =1 that is, A k S α = a α S kα ; α = 1, 2,, N; A 11 A 12 A 1N A 21. A N1 A NN S 1α S 2α S 1α S 2α. α. S Nα S Nα. (T10.91)

20 TFY4250/FY2045 Tillegg 10 - More geeral formulatio of quatum mechaics 20 Colum umber α i the matrix S the is a eigevector of the A k matrix. By solvig this eigevalue problem for the N eigevalues a α, oe fids the N colums of the matrix S. Notice that the N eigevalues are foud by settig the system determiat equal to zero: det[a a α 1] = 0. (T10.92) Sice the determiat is a polyomial i a α of degree N, it follows that this so-called characteristic equatio has N roots. These roots are the eigevalues a α ; α = 1, 2,, N. If oe is iterested oly i the eigevalues, it suffices to fid these roots. If oe also wishes to fid the vectors α, the (T10.91) must be solved for the N eigevalues a α, so that the compoets S α = α of α ca be determied c Equatio of state o matrix form The projectio of the equatio of state (T10.60) o the k basis gives the equatio i h d dt k Ψ(t) = k Ĥ Ψ(t) = k Ĥ Ψ(t), or i h d dt Ψ k(t) = H k Ψ (t). (T10.93) The time-depedet k -compoets Ψ k (t) of Ψ(t) thus are give by a coupled set of first-order differetial equatios i t. O matrix form: i h d dt Ψ 1 (t) Ψ 2 (t). = H 11 H 12 H 21. Ψ 1 (t) Ψ 2 (t).. (T10.94) Note that if oe diagoalizes this Hamiltoia matrix, the the diagoal elemets (i.e., the eigevalues of the H matrix) are the eergies E α of the statioary states of the system at had. These eergies are determied by equatio (T10.92): det[h E α 1] = 0. (T10.95) Usig the method described above, oe ca also fid the correspodig eigevectors α of Ĥ, which correspod to the eergy eigestates of the system. The matrix formulatio of quatum mechaics is kow as matrix mechaics, ad is the formulatio that was actually foud first, by Werer Heiseberg i 1925, a few moths before Schrødiger preseted his wave mechaics.