Note: we can take the Real and Imaginary part of the Schrödinger equation and write it in a way similar to the electromagnetic field. p = n!

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1 Quatum Mechaics //8 Staford class with Leoard Susskid. Lecture 5. The ext class i this series has the title Special Relativity ad icludes classical field theory, least actio, Lagragia techiques, tesor fields, E.M. fields. Note: we ca take the Real ad Imagiary part of the Schrödiger equatio ad write it i a way similar to the electromagetic field. Particles movig o periodic lies x + L x R Require * dx, * dx e px R ekx R p k Periodicity requires that k therefore k R Possible values of mometum p R. Mometum values are close together, cotiuous as R. Delta fuctio: R e kx e k x dx k k R This is a example of the orthoormality of eigefuctios of a Hermitia operator. k k k k Now stretch by a factor of R.??? I did t get all the otes here, but I thik the previous argumet demostrates the trasitio from discrete to cotiuous. Toss R ad use a itegral rather tha a summatio. Outer products: dyad Quatum Mech 8.doc 3/9/ Page of

2 B A C B A C Let basis vectors be the expad a fuctio i terms of the basis vectors. From previous results V V where V V V V The idetity operator Î discrete cotiuous Î dx x x V dx x x V x x Cosider a wave fuctio with a defiite mometum k. x k eikx k k Isert idetity k dx k x x Coectig positio ad mometum k dx e ikx x So that it s clear that k is the Fourier Trasform of x. The FT exposes the reciprocal relatioship betwee positio ad mometum. Similarly x x x dk x k k k k Quatum Mech 8.doc 3/9/ Page of

3 x dk e ikx k Mometum vs. Positio represetatio k x xk x ˆx x ˆk x i x x ˆk k k k ˆx k i k k Homework: work out the last lie from defiitios. Coservatio of phase space volume i Classical Mechaics becomes Uitarity of the time evolutio operator i Quatum Mechaics. Compatible Operators Exclusive, positio or mometum, but ot arbitrary precisio of measuremets of both. Icompatible observables do ot have simultaeous eigevectors ad eigevalues. Example: Polarized Photo Cosider a radio wave photos passig through a vertical grid of wires. The vertical E is shorted out so that a polarized wave passes through the grid. V for vertical polarizatio Quatum Mech 8.doc 3/9/ Page 3 of 3

4 H for horizotal polarizatio The polarizer is a device to prepare photos i a particular state of polarizatio. Each photo will either pass through or reflect. The probability of passig through depeds o the agle of polarizatio. The polarizer is a preparer of states as well as a detector. Aliged photos pass through a secod polarizer with probability. Use QM to describe polarizatio of photo. x y polarizatio alog x-axis polarizatio alog y-axis x y These are orthogoal states The polarizatio is aliged alog two mutually exclusive directios of polarizatio. x x y y Normalize where x so that Now discuss observables. Defie the polarizatio operator o the x-y plae. A photo passig through the horizotal polarizer cout as +. A photo passig through the vertical polarizer cout as -. x x y y Therefore is completely characterized. Quatum Mech 8.doc 3/9/ Page 4 of 4

5 Represetatio of the polarizatio operator The matrix represetatio of is so that Basis vectors represet states of defiite polarizatio. What is polarizatio alog 45 axis? This is ot a state represeted by the basis vectors. A photo polarized at 45 has goe through a 45 polarizer. If such a photo ecouters a horizotal polarizer, we ca guess that the photo has a 5-5 chace of passig through. Guess a wave fuctio with equal amouts of horizotal ad vertical polarizatio. x + y A 45 photo should be orthogoal ad ot pass through the detector. x y Note that the states are orthogoal. Quatum Mech 8.doc 3/9/ Page 5 of 5

6 x y x + y Side ote: polarizatio is a observable with possible measuremet outcomes beig the eigevalues. A observable is aythig you ca measure i a sigle experimet. Therefore Probability is ot a observable sice it requires may experimets to measure. Now sed the 45 photo to a vertical polarizer. What is the probability of it passig through? It should be. y y x + y ad y As calculated before, the states are orthogoal. Now use ad as ew basis vectors. Defie a ew polarizatio operator. so that Quatum Mech 8.doc 3/9/ Page 6 of 6

7 Quatum Mech 8.doc 3/9/ Page 7 of 7 Note that there are o commo eigevectors for ad, so these observables are icompatible. Ca t measure both. x x is the probability of fidig a x-polarized photo. y y is the probability of fidig a y-polarized photo. The probabilities must add to. Could choose ay pair of eigevalues, ot just ±. Next week, we ll look at polarizatios through agles other tha 45. Notice that oly real umbers are required to represet polarized photos. Next week we ll look at circularly polarized photos where complex umbers are required.

Physics 324, Fall Dirac Notation. These notes were produced by David Kaplan for Phys. 324 in Autumn 2001.

Physics 324, Fall Dirac Notation. These notes were produced by David Kaplan for Phys. 324 in Autumn 2001. Physics 324, Fall 2002 Dirac Notatio These otes were produced by David Kapla for Phys. 324 i Autum 2001. 1 Vectors 1.1 Ier product Recall from liear algebra: we ca represet a vector V as a colum vector;