The material in this lecture covers the following in Atkins The informtion of a wavefunction (d) superpositions and expectation values

Transcription

1 Lecture 7: Expectaton Values The materal n ths lecture covers the followng n Atkns The nformton of a wavefuncton (d) superpostons and expectaton values Lecture on-lne Expectaton Values (PDF) Expectaton value (PowerPont) handouts Assgned problems for lecture 7

2 Tutorals on-lne Remnder of the postulates of quantum mechancs The postulates of quantum mechancs (Ths s the wrteup for Dry-lab-II)( Ths lecture has covered postulate 5) Basc concepts of mportance for the understandng of the postulates Observables are Operators - Postulates of Quantum Mechancs Expectaton Values - More Postulates Formng Operators Hermtan Operators Drac Notaton Use of Matrces Basc math background Dfferental Equatons Operator Algebra Egenvalue Equatons Extensve account of Operators Hstorc development of quantum mechancs from classcal mechancs The Development of Classcal Mechancs Expermental Background for Quantum mecahncs Early Development of Quantum mechancs

3 Audo-vsuals on-lne Postulates of Quantum mechancs (PDF) (smplfed verson from Wlson) Postulates of Quantum mechancs (HTML) (smplfed verson from Wlson) Postulates of quantum mechancs (PowerPont ****)(smplfed verson from Wlson) Sldes from the text book (From the CD ncluded n Atkns,**)

4 Operators and Expectaton Values Consder a large number N of dentcal boxes wth dentcal partcles all descrbed by the same wavefuncton Ψ( xt, ): Let us for each system at the same tme meassure the property F let the outcome of ths meassurement be f 1, f 2, f 3,...,fN the average value for F s gven by N fk <F> = k N k runs over number of meassurements Revew of average calculatons

5 Operators and Expectaton Values Revew of average calculatons Snce N s large many experments mght gve the same result. Let n be the tmes f was observed. In ths case we mght also wrre < F > as : 1 1 < F > = = N f N nf We mght also wrte : <F> = Here P = ( n ) N value f for F runs over all values ( n )f = P N f s the probablty of measurng the runs over dfferent values

6 Operators and Expectaton Values New apl. of Born nterp. Let us now consder the x - coordnate n our N systems. We have from the Born nterpretaton probablty of fndng partcle between x and x + x P = Px ( ) = * Ψ(x, t) Ψ (x, t)dx Thus the average value of x s gven by < x > = P(x)x = Ψ( x, t)xψ ( x, t)dx x - *

7 Operators and Expectaton Values New apl. of Born nterp. For a physcal property that depends on the x,y,x coordnates only : F(x, y, z) The average value s gven by < F > = Ths s a smple extenson of the Born postulate whch s part of Ψ * (x,y,z,t)f(x,y,z) Ψ(x,y,z,t)dxdydz

8 Operators and Expectaton Values New postulate 5. A general property wll depend on x,y,z as well as the lnear momenta p x, p y, p z. F = F(x,y,z,p,p,p We postulate : x y z ) < F > = Ψ (x,y,z,t)f Ψ(x,y,z,t)dxdydz * ˆ Where F ˆ = F(x, ˆ y, z,p ˆx,p ˆy,pˆ z) Note : operator F ˆ s " sandwched" between * Ψ and Ψ. the average value < F > s also called an expectaton value

9 Operators and Expectaton Values New postulate 5. Consder the specal case where ψ( x) s a smultanous egenfuncton to H ˆ and F ˆ Ĥψ(x) = E ψ(x) In ths case < F > = ψ (x)f ˆψ (x)dx - * * ˆFψ(x) = k ψ(x) In ths case a meassurement of F wll always gve k as an answer 1 = k ψ (x) ψ (x)dx =k - * *

10 Operators and Expectaton Values Consder next the more general case where ψ( x) as a statefuncton s an egenfuncton to H ˆ but not to Fˆ Hˆψ(x) = E ψ(x) ; Fˆψ(x) k ψ(x) In ths case the meassurement of F wll gve one of the egenvalues of F Fξ = k ξ The average value from a large number of meassurements wll be n < F >= = N f * ( ) ψ ( x ) F ˆ ψ( x ) statstcs (logc) Postulate 5 New postulate 5.

11 Operators and Expectaton Values n < F >= = N f * ( ) ψ ( x ) F ˆ ψ( x ) What s the probablty n P = ( ) N That the meassurement wll have the outcome f? the egenfunctons ξ ( = 1,2,..) Fξ = k ξ forms a complete set on whch we can expand our statefuncton ψ(x) : ψ(x) = a ξ ( x) : a = f ( x) ξ ( x) * Good queston about postulate 5.

12 Operators and Expectaton Values n < F >= = N f * ( ) ψ ( x ) F ˆ ψ( x ) dx Now substtutng the expresson for the expanson of the state functon ψ( x ) n terms of the egenfunctons ξ to F ˆ <F> = ( a ξ )ˆ F( a ξ ) dx * * Or after workng wth F ˆ on the sum to the rght of F, ˆ and remember that Fˆξ = k ξ <F> = * * ( a ξ )( a k ξ ) dx Long answer to good queston about postulate 5.

13 Operators and Expectaton Values <F> = ( a ξ )( a k ξ ) dx * * Now multply each term n the rght hand sum wth each term n the left hand sum Long answer to good queston about postulate 5. <F> = * * ( a ξ a k ξ ) dx Interchangng next order of ntegraton and summaton, whch s allowed for ' well behaved sums' : <F> = ( a ξ a k ξ ) dx * *

14 Operators and Expectaton Values <F> = ( a ξ a k ξ ) dx * * Takng constant factors outsde ntegraton sgn Long answer to good queston about postulate 5. * * <F> = aak ξξdx Makng use of th orthonormalty of egenfunctons * * <F> = aak ξξdx <F> = aak = a k * 2 δ ξξ dx = * δ

15 perators and Expectaton Values By comparng * <F> = aak = a k th n < F >= = N f * ( ) ψ ( x ) F ˆ ψ( x ) we note that a 2 = n N ψ * We have that a = 2 (x) ξ ( x)dx probablty of obtanng k from a meassurement of F n state wth state functon ψ(x) Thus the chance of obtanng k from a meassurement of F for a system wth state functon ψ(x) s large f the 'overlap' between ψ(x) and ξ (x) s large Long answer to good queston about postulate 5.

16 Operators and Expectaton Values We have that ψ(x) s normalzed Long answer to good queston about postulate 5. * * * - - ψ ( x ) ψ ( x ) dx = [ a ξ ( x )][ a ξ ( x )] dx = 1 or after multplyng out the sum and nterchange summaton and ntegraton * * * - - ψ ( x ) ψ ( x ) dx = a ξ ( x ) a ξ ( x ) dx = 1

17 Operators and Expectaton Values fnally usng the orthonormalty propertes of the set {ξ, = 12,..} * * * * - - a ξ ( x ) a ξ ( x ) dx = a a ξ ( x ) ξ ( x ) dx = 1 or : a 2 = 1 sum of all probabltes δ Thus the sum of the ndvdual probabltes a ( = 1,2,..)for obtanng the values f ( = 1,2,..) n a meassurement of F for a system wth the statefuncton ψ(x) s one as t should; f ψ(x) s normalzed

18 Operators and Quantum Mechancs kx kx ψ( x) = exp + exp s a lnear combnaton of two egenfunctons to pˆ x px = h k p x = hk How can we fnd p x n ths case? 50 % chance to measure p = hk 50 % chance to measure p = -hk < P x >= 0 E p h k = = 2m 2m

19 What you should learn from ths lecture 1. Postulate 2 (Revew) For any observable Ω( x,y,x,p x, py, pz) that can be expressed n classcal physcs n terms of x,y,x and p x, py, pz. We can construct the correspondng quantum mechancal operator operator Ωˆ (ˆ x,y,x ˆ ˆ,pˆ x, pˆ y, pˆ z) from the substtuton : Classcal Mechancs Quantum Mechancs h δ x px xˆ > x ; pˆx > δx h δ y py yˆ > y ; pˆy > δy h δ z pz zˆ > z ; pˆ z > δz as ˆ h d h d h d Ω(x,y,z,,, ) dx dy dz

20 What you should learn from ths lecture 2. Postulate 3 (Revew) The meassurement of the quantty represented by Ωˆ has as the o n l y outcome one of the egenvalues ϖn n = 1,2,3... to the egenvalue equaton : Ωˆ ψn = ϖnψn 3. Postulate 5. For a system n a state descrbed by Ψ(x,y,z,t) the average value meassured for Ω wll be < Ωˆ > = Ψ (x,y,z,t) ΩΨ ˆ (x,y,z,t)dxdydz We call that the expectaton value. 4. For a system n a state descrbed by Ψ(x,y,z,t) the probablty to obtan the value ϖ n a meassurement of Ω s a where a = Ψ (x,y,z,t) ψ dxdydz n * n Here ϖn s an egenvalue to Ωˆ ψ = ϖ ψ and ψ the correspondng egenfuncton * n n n n n n