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Lecture 7: Expectation Values The material in this lecture covers the following in Atkins. 11.5 The informtion of a wavefunction (d) superpositions and expectation values Lecture on-line Expectation Values (PDF) Expectation value (PowerPoint) handouts Assigned problems for lecture 7 Tutorials on-line Reminder of the postulates of quantum mechanics The postulates of quantum mechanics (This is the writeup for Dry-lab-II)( This lecture has covered postulate 5) Basic concepts of importance for the understanding of the postulates Observables are Operators - Postulates of Quantum Mechanics Expectation Values - More Postulates Forming Operators Hermitian Operators Dirac Notation Use of Matricies Basic math background Differential Equations Operator Algebra Eigenvalue Equations Extensive account of Operators Historic development of quantum mechanics from classical mechanics The Development of Classical Mechanics Experimental Background for Quantum mecahnics Early Development of Quantum mechanics Audio-visuals on-line Postulates of Quantum mechanics (PDF) (simplified version from Wilson) Postulates of Quantum mechanics (HTML) (simplified version from Wilson) Postulates of quantum mechanics (PowerPoint ****)(simplified version from Wilson) Slides from the text book (From the CD included in Atkins ,**) Operators and Expectation Values Re view of average calculations Consider a large number N of identical boxes with identical particles all described by the same wavefunction (x,t) : Let us for each system at the same time meassure the property F let the outcome of this meassurement be f1, f2 , f3 , ........, fN the average value for F is given by N fk <F > = k N k runs over number of meassurements Operators and Expectation Values Re view of average calculations Since N is large many experiments might give the same result. Let n i be the times f i was observed. In this case we might also wrire < F > as : <F> = 1 fj Nj 1 = nifi Ni j runs over all values i runs over different values We might also write: ni < F > = ( )fi Pifi i N i ni Here Pi = ( ) is the probability of measuring the N value f i for F Operators and Expectation Values New apl. of Born interp. Let us now consider the x - coordinate in our N systems. We have from the Born interpretation probability of finding particle Pi P(x ) (x,t)* (x,t)dx between x and x + x Thus the average value of x is given by x = P(x)x = (x, t)x * (x, t)dx x - Operators and Expectation Values New apl. of Born interp. For a physical property that depends on the x, y,x coordinates only : F(x,y,z) The average value is given by F = * (x,y, z,t)F(x,y,z)(x,y,z, t)dxdydz - - - This is a simple extension of the Born postulate which is part of Operators and Expectation Values New postulate 5. A general property will depend on x,y,z as well as the linear momenta px , py , pz . F = F(x,y, z,px ,py ,p z ) We postulate : F = * (x,y, z,t)Fˆ (x,y, z,t)dxdydz - - - ˆ x,p ˆ y , pˆ z ) Where Fˆ = Fˆ (x,y, z, p Note : operator Fˆ is "sandwiched" between * and . the average value < F > is also called an expectation value Operators and Expectation Values New postulate 5. Consider the special case where (x) is a ˆ and Fˆ simultanious eigenfunction to H ˆ (x) = E(x) H ; Fˆ (x) = k(x) In this case <F> = * (x)Fˆ * (x)dx - 1 = k * (x) * (x)dx =k - In this case a meassurement of F will always give k as an answer Operators and Expectation Values New postulate 5. Consider next the more general case where (x) as a statefunction is an eigenfunction to ˆ but not to Fˆ H ˆ (x) = E (x) ; Fˆ (x) k (x) H In this case the meassurement of F will give one of the eigenvalues of F Fi k i i The average value from a large number of meassurements will be ni F ( )fi * (x )Fˆ (x ) N i statistics (logic) Postulate 5 Operators and Expectation Values ni F ( )fi * (x )Fˆ (x ) N i Good question about postulate 5. What is the probability n Pi ( i ) N That the meassurement will have the outcome f i ? the eigenfunctions i (i = 1,2,..) Fi k i i forms a complete set on which we can expand our statefunction (x) : (x) = aii (x ) : ai f * (x )i (x ) i Operators and Expectation Values ni F ( )fi * (x )Fˆ (x )dx N i Long answer to good question about postulate 5. Now substituting the expression for the expansion of the state function (x ) in terms of the eigenfunctions i to Fˆ < F > = ( ai*i* )Fˆ ( a j j )dx i j Or after working with Fˆ on the sum to the right of Fˆ , and remember that Fˆ j k j j < F > = ( ai*i* )( a j k j j )dx i j Operators and Expectation Values < F > = ( ai*i* )( a j k j j )dx i Long answer to good question about postulate 5. j Now multiply each term in the right hand sum with each term in the left hand sum < F > = (ai* i*a j k j j )dx i j Interchanging next order of integration and summation, which is allowed for 'well behaved sums' : < F > = (ai* i*a j k j j )dx i j Operators and Expectation Values Long answer to good question about postulate 5. < F > = (ai* i*a j k j j )dx i j Taking constant factors outside integration sign <F > = * ai a j k j i* j dx i j Making use of th orthonormality of eigenfunctions < F > = ai*a j k j i* j dx i j < F > = ai*ai k i | ai |2 k i i i ij * i j dx ij Operators and Expectation Values By comparing < F > = ai*ai k i | ai |2 k i i Long answer to good question about postulate 5. i with ni F ( )fi * (x )Fˆ (x ) N i ni we note that | ai | N 2 probability of obtaining ki from a meassurement of F in state with state function (x) We have that ai * (x)i (x)dx Thus the chance of obtaining k i from a meassurement of F for a system with state function (x) is large if the 'overlap' between (x) and i (x) is large Operators and Expectation Values We have that (x) is normalized Long answer to good question about postulate 5. * * (x ) (x )dx [ a i i (x ) ][ a j j (x )]dx 1 * - i - j or after multiplying out the sum and interchange summation and integration * * (x ) (x )dx a i i (x )a j j (x )dx 1 - * i j - Operators and Expectation Values finally using the orthonormality properties of the set {i ,i 1,2..} i j ai* * i (x )a j j - (x )dx i j ai*a j * i (x )j (x )dx 1 - or : | ai |2 1 sum of all probabilities i ij Thus the sum of the individual probabilities ai (i = 1,2,..)for obtaining the values fi (i = 1,2,..) in a meassurement of F for a system with the statefunction (x) is one as it should; if (x) is normalized Operators and Quantum Mechanics ikx ikx (x) exp exp is a linear combination of two eigenfunctions to pˆ x px k How can we find px in this case ? 50 % chance to measure p = k 50 % chance to measure p = - k Px 0 px k 2k 2 p2 E 2m 2m What you should learn from this lecture 1. Postulate 2 (Review) For any observable (x,y, x , px , py ,pz ) that can be expressed in classical physics in terms of x, y, x and px , py , pz . We can construct the corresponding ˆ ( xˆ , yˆ , xˆ , pˆ x , pˆ y , pˆ z ) quantum mechanical operator operator from the substitution: Classical Mechanics Quantum Mechanics x px y py z pz ˆ (x,y,z, as xˆ x ; pˆ x i x yˆ y ; pˆ y i y zˆ z ; pˆ z i z d d d , , ) i dx i dy i dz What you should learn from this lecture 2. Postulate 3 (Review) ˆ The meassurement of the quantity represented by has as the o n l y outcome one of the eigenvalues n n = 1,2, 3 .... ˆ n n n to the eigenvalue equation : 3. Postulate 5. For a system in a state described by (x, y, z, t) the average value meassured for will be ˆ ˆ (x,y, z, t)dxdydz = * (x,y, z, t) - - - We call that the expectation value. 4. For a system in a state described by (x, y, z, t) the probability to obtain the value n in a meassurement of 2 is | a n | where a n = * (x,y, z, t) ndxdydz - - - ˆ n n n and n Here n is an eigenvalue to the corresponding eigenfunction