Quantum Mechanics Chapter 1.5: An illustration using measurements of particle spin.

Transcription

1 I Introduction. Quantum Mecanics Capter.5: An illustration using measurements of particle spin. Quantum mecanics is a teory of pysics tat as been very successful in explaining and predicting many pysical penomenon, particularly tose tat occur on a molecular, atomic, and subatomic scale. Its pilosopical interpretation as been te subject of debate. In order to understand te teory of quantum mecanics we will explore its application to te pysical property of spin. Spin is an observable quantity of a particle (suc as an electron or atom). Altoug it is not strictly correct, one can tink of spin in terms of te particle spinning around its axis like te eart rotates (spins) around it axis. Tus measuring spin would be equivalent to measuring ow fast, and about wat axis (vertical, orizontal or oter) te particle is spinning. We use te observable Spin because te matematics involved is relatively simple. Neverteless, te teory of quantum mecanics applies to Spin te same way as it does to any oter observable suc as te position or velocity of te particle. II. Matematical Background. A. A x matrix. A matrix is an ordered array of numbers. Here we need to introduce te concept of a x matrix. Here is an example: Notice te following: () ) Te matrix consists of 4 numbers. ) Te numbers are arranged in a specific manner. We say tat te numbers 3 and 7 are in column. Likewise, and 5 are in column. Te numbers 3 and are in row and 7 and 5 are in row. Tus te number 3 is in row column and 5 is in row column. B. A x vector. A x vector is like a matrix wit two rows and column. Here is an example: 9 4. () Here we ave two numbers tat are arranged in a single column. C. Matrix Operations.

2 . Multiplication by a scalar. A scalar is just a number. Wen multiplying a matrix (or a vector) by a scalar, te scalar just multiplies eac number in te matrix. For example if we multiply te matrix in Equation above by te scalar (number) 4 we ave 3 (4)(3) (4)() 8 4. (3) 7 5 = = (4)(7) (4)(5) 8 0 Notice tat multiplying a matrix by a scalar yields anoter matrix. Here's wat appens wen we multiply te vector in Equation by te scalar 4: 9 (4)(9) = = (4)(4) 6. (4) Tus if we multiply a vector by a scalar we get anoter vector. Please complete te following exercises = = (5) 6 3 = 7 = 3. Operation of a x matrix on a vector. A x matrix can operate on a (x) vector to produce anoter vector. We say tat te x matrix is an operator. Te operation is illustrated below. 3 9 (3)(9) + ()(4) (6) = = (7)(9) (5)(4) 63 0 = Te element in te first row and first column of te resultant vector (63) comes from multiplying te first row of te matrix wit te first (and only) column of te vector. Te element in te second row and first column of te resultant vector (83) comes from multiplying te second row of te matrix wit te first (and only) column of te vector. Here is anoter example: 3 ()( ) + (3)() = 3 0 = = (3)( ) + (0)() (7) Please complete te following exercises:

3 3. Multiplication of two vectors = = 5 3 = (8) Two vectors can be multiplied using te dot or scalar product to produce a scalar (number). For example, = ()(4) + ()(3) = = 0 3. (9) Here we ave canged te column vector to row vector [ ]. Wenever we multiply two vectors we always cange te first one to a row vector. Ten te multiplication is te same as for matrices. Let 3 A, B, C 0 = = =. 4 (0). Find te scalar product of AB,BA,AC,BC. Te norm of a vector is its dot product wit itself. If a vector is complex, One takes te complex conjugate of te first, transposed vector to calculate te scalar product, so if [ ] 4 Find AD, DA, so tat AD = (DA)* i D =, () D. Probablility. Wen you toss a quarter tere is a 50% cance tat it will land on "eads" and a 50 % cance it will land on "tails". We can say tat te probability tat it will land on eads is 0.5 and te probability tat it will land on tails is also 0.5. Tis means tat if we toss a lot of quarters tat alf of tem will land on eads and te oter alf will land on tails. Tus we can predict wit some certainty tat if we make 000 tosses 500 will land on eads and 500 will land on tails. However, we can not predict te outcome of a single toss. Te probability tat te coin will land on eiter eads or tails during one toss is te sum of tese two probabilities ( ) =. A probability of means tat it is certain to occur. Te probability tat we will get two eads in a row is (0.5)(0.5) = 0.5. Tat is tere is a 5% cance of tis occurring. 3

4 Te probability tat a die tat you trow will land wit 6 is /6. If you made 000 tosses 66 to 67 of tem would most likely land on 6. Te iger te number of tosses you make te more predictable will be te outcome. Tat is, if you make only one toss you know tat tere is /6 cance of getting 6 but can only make a weak prediction. If you make 00 tosses it is likely tat you will get 6 to 7 tat land on 6 but you migt get only 4 or 5. E. Te eigenvalues and eigenvectors of a matrix. We can refer to a x matrix as an operator in te sense tat it can operate on a vector and give a new vector. Ô V = V. () An operator is said to be Hermitian iff Ô = Ô, (3) were te dagger refers to te adjoint wic is te transpose conjugate. For all te matrices we will use in examining Spin, tere are associated two eigenvectors and two eigenvalues. An eigenvector is a vector (like in Equation ) and an eigenvalue is a number or scalar. Te eigenvectors and eigenvalues obey te following relation: (Matrix)(eigenvector) = (eigenvalue)(eigenvector) Let's define tree matrices, i σ x =, σ 0 z = 0 σ y =, i 0. (4) / / Te eigenvectors associated wit σ x are wit te eigenvalue, and / / wit te eigenvalue -. Te eigenvectors associated wit σ z are 0 0 wit te eigenvalue, and wit te eigenvalue -. / Let's verify tat is an eigenvector of σ x wit te eigenvalue : / We need to see if 4

5 Let's also verify tat 0 /? / 0 = / / (0)( / ) + ()( / )? ()( / ) = ()( / ) + (0)( / ) ()( / ) / / = / / 0 is an eigenvector of σ z wit an eigenvalue of -: 0 0? 0 0 = ()(0) + (0)( )? ( )(0) (0)(0) ( )( ) = ( )( ) = (5) (6) As an exercise, sow tat tat 0 / is an eigenvector of σ x wit te eigenvalue -, and / is an eigenvector of σz wit an eigenvalue of : One can use metods of linear algebra to calculate te eigenvectors and eigen values of a matrix. One usually cooses te normalized eigenvectors. Also, we require tat te eigenvectors be ortogonal. For ermitian operators, eigenvalues are obtained from te equation: Det( Ô ωî ) = 0 (7) Once tese ave been found, te ortonormal eigenvectors can be determined from te eignvalue equation. Exercise: Sow tat te above eigenvectors and eigenvalues are correct by using equation 7. All eigenvectors are only uniquely determined to witin a factor of e iφ, tat is to witin a pase factor. 5

6 III. Te postulates of Quantum Mecanics. () Te state of a system (tat is wat we know about it) is completely described by its state vector, wic is generally represented by ψ. Wen we are dealing only wit spin, te state vector will be a by vector suc as tose above (Equation ). () Every observable (a measurable quantity) is represented by an operator. For spin, te operators are x matrices like in Equation, We will generally represent an operator by Ŝ. (3) Te only possible outcomes of a measurement of an observable are te eigenvalues of te operator associated wit te observable. [We will describe wat an eigenvalue is below along wit a description of eigenvectors]. We will represent te eigenvalues as ω. (4) Te probability tat a measurement will result in a value of ω is Ψ. Ω, were Ω is te eigenvector associated wit ω. (5) After a measurement resulting in ω te state vector becomes (collapses) into te eigenvector Ω. (6) Te state vector evolves in time in a specific way. [For our purposes in tis capter, we can ignore tis postulate because te state vectors describing spin will be constant in time. In general, te state vector obeys an equation called te time-dependent Scrödinger equation.] III. Spin. A. Te operators. Te two observables we will consider are te spin of a certain type of particle measured in te x-direction and te spin measured in te z-direction. You can tink of tese observables as an extent to wic a particle is spinning around an axis oriented up and down (z) and oriented in a orizontal direction (x). Te operators associated wit tese observables are ˆ 0 ˆ 0 Sx = σ, Sz x = σ 0 = z = 0, ˆ 0 i S y =. (8) i 0 Here is planck's constant, = 6.63 x 0-7 erg-sec divided by π. B. Te eigenvectors and eigenvalues. A state vector, as well as an eigenvector, is usually written so tat its magnitude is one. Tat is te first term squared plus te second squared is one. Tis is necessary so tat te total probability of getting some value from a measurement be one. 6

7 Te eigenvectors for S ˆx are te same as tose for σ x but now te eigenvalues are + / / / for and - / for. / / Te eigenvectors for Sˆz are te same as tose for σz but now te eigenvalues are + / 0 for, and - / for. 0 C. Direct interpretation of te postulates for te observable spin. () Te first postulate says tat everyting we know about a system will be given by its state vector. In tis case, for spin, tat is a vector suc as: 0.6 ψ = 0.8. (9) () Te two operators are already given above in equation 8. Te eigenvectors and eigenvalues of tese operators give us te power to make predictions about te outcome of a measurement. (3) Te only possible values of a spin measurement are te eigenvalues + / and - /. Tat means tat if we were to measure te spin of a particle along a given axis, we could only get eiter one of tese values. Tis is like saying tat if we were to measure ow ig we can trow a ball, we would only get certain values like0 feet or one undred feet, but not any value in between. Tis is wat is meant by quantization. Te possible values of spin are not any value between + / and - / but only tese exact values. Tis is completely different tan anyting classical mecanics predicts. (4) Let's take as our state vector tat given in equation 9. As stated in postulate () all our knowledge of te system is contained in te state vector. Let's calculate te probability tat if we make a measurement of te spin of te particle along te x-axis tat we will get te value of + /. Tis probability is given by te product of te state vector and te S ˆx eigenvector associated wit te eigenvalue + /. Tus we ave tat te probability of obtaining a value of + / for te spin along te x-axis is / (0.6 [ 0.8 ] ) / = ((0.6)( / ) + (0.8)( / )) = ((0.6)(0.707) + (0.8)(0.707)) = ( ) = (0.990) = were I ave rounded off at te tird decimal point. Likewise te probability of obtaining a value of - / for te spin along te x-axis is, (0) 7

8 (0.6 [ 0.8 ] ) / / = ((0.6)( / ) + (0.8)( / )) = ((0.6)(0.707) + (0.8)( 0.707)) = ( ) = ( 0.4) = 0.00 Taken togeter, tis means tat if a particle begins in a state given by equation 6, we will ave a % cance of measuring - / for its spin in te x-direction and a 98% cance of measuring + /. If we ad many particles, all prepared in te same state given by equation 6, ten, upon measuring te spin along x, 98% of tem would ave + / and % would ave - /. Tese results are totally different tan wat one gets from classical mecanics. Notice tat we can only predict te probabilities of te outcome of a given measurement. In classical mecanics, if we know enoug about te system, we can predict te exact outcome of every measurement. Here, in Quantum Mecanics, tere is an inerent indeterminism in measurement. As an exercise, calculate te probabilities tat we will measure + / and - / for te spin along te z-axis, if a particle starts in a state described by te state vector of equation 9. (5) Let's say tat we make a measurement of te spin along te x-axis as we did above and get te result + /. According to postulate 5, our state vector now becomes te / eigenvector associated wit te operator S ˆx and te eigenvalue + /:. Tis / result is independent of te state of te system before te measurement. Tat is te 0.6 system could ave been in te state described in equation 9, ψ =, or some 0.8 oter state but after measuring + / for te spin along te x-axis te state vector / collapses into. / Tis result is totally different tan anyting encountered in Classical Mecanics. Classically, te act of measuring someting, does not alter te ting you are measuring. Here, in Quantum Mecanics, te state of te system is drastically and fundamentally altered.. () IV. Quantum Weirdness. A. Experimental measurements of Spin: Te Stern-Gerlac device 8

9 Te device used to measure Spin in te real world is called a Stern-Gerlac device. Wen a particle is injected into te device moving orizontally, say in te y-direction, and te device is oriented in te z-direction (vertically) ten te particle will be deflected vertically as illustrated in Figure below. A stream of particles is sown incident on te device. + / Incident particles Stern-Gerlac Device - / Figure Te particles are deflected eiter up or down a certain amount, but not in between. Tis is because tere are only two values of te Spin, + / and - /. If a screen were detected beind te device te particles would sow up at only two positions. Classical mecanics predicted tat tey would sow up as all possible positions. B. A series of measurements Let us now see wat would appen if we make as series of tree measurements using Stern-Gerlac devices. Let us first measure te Spin along te x-direction, ten (selecting tose wit Spin, x = - /) make a measurement along te z-direction, and finally (selecting te particles wit Spin z = + /), make a measurement anoter measurement in te x-direction. Tis series of measurements is illustrated in te Figure. + / + / Incident particles + / - / Stern- Gerlac Device, x - / - / Stern- Gerlac Device, x Stern- Gerlac Device, z Figure 9

10 One migt predict (in accordance wit a classical view) tat since we ave already selected particles tat ave Spin in te x-direction of - / after te first device, tat all of te particles coming out of te tird device will also ave - / Spin in te x-direction. Classically, if we measure a property of someting, say its color or its Spin along x, ten it possesses tis property. Furtermore, classical mecanics olds tat te act of measuring it or anoter property, suc as Spin along z, does not cange its properties. As sown ere, Quantum Mecanics predicts tat only alf of te particles coming out of te tird Stern-Gerlac device ave Spin - / along te x-direction. After te first Stern Gerlac device, te state vector of te particles wit Spin = - / / along x is te eigenvector associated wit tis eigenvalue :. Now let's / calculate te percentage of tese particles tat will ave a Spin along z = + /. Tat fraction is: / ( [ 0 ] ) / = (()( /) + (0)( /)) = (()(0.707) + (0)( 0.707)) = (0.707) = 0.5 So alf te particles tat enter te second Stern-Gerlac device also go into te tird one. Now let's calculate te quantum mecanical prediction of wat fraction entering te tird device will ave Spin = - /. Remember classically tat fraction sould be (or 00%). According to Quantum mecanics te state vector of te particles entering te tird Stern-Gerlac device is te eigenvector associated wit a value of + / along z: 0. According to Quantum Mecanics te fraction of particles leaving te tird device wit Spin - / along x is / ( 0 ) / [ ] = (()( /) + (0)( /)) = (()(0.707) + (0)( 0.707)) = (0.707) = 0.5 Tus te Quantum Mecanical prediction is different tan te classical one. Tis is largely because te measurement of Spin by te second device alters te system we are measuring. () (3) 0

11 Exercise: Repeat tis exercise wit a sequence of Stern-Gerlac devices as follows: z, x, z, finding te particles tat ave first up ten down ten up spin. IV.5 Heisenberg Uncertainty Principle (HUP) Te HUP says tat tere is a minimum ontological indeterminacy in conjugate attributes. For example X P /, were X and P are te position and momentum. Te delta refers to te indeterminacy. Tis relation says, for example, tat if we ave no uncertainty in te position ( X = 0) ten te indeterminacy of te momentum (and ence velocity) is infinite. Tus, if we know exactly were someting is at some moment, we will ave no idea were it will be afterwards. Similarly, we cannot know bot te spin in te x-direction and in te y-direction. Tis is illustrated in our example of Figure. After te first detector te spin in te x-direction, for te particles tat are deflected downwards is, - /. At tis point we can say tat S is zero, so we know wat te spin in te x-direction is. Now if we were to measure ˆx Sˆz, quantum mecanics tells us tat te best we can do is predict te probability of getting up or down and tat tis is te same (0.5/0.5). Tis is wat we saw above. So if we know te spin in te x-direction, we cannot know it s value in te z-direction. V. Te EPR Paradox A. Conservation of angular momentum and Spin. In order to describe te important argument made by Einstein, Podolsky and Rosen (Pys. Rev. 47, 777, 935) we need to expand our description of Spin to include tat of a system of two particles. Classically, and quantum mecanically, te total angular momentum or Spin of a system, along any direction, is conserved. Te angular momentum is a measure of ow muc matter is spinning around a certain axis at a certain speed. Tis is also, ow we migt tink of Spin. If a system starts out wit a total Spin of zero, ten te total spin of te system will be zero for all time afterwards. Tus if we prepare two of our particles (tat can ave Spin = + / or Spin = - /) in a state tat starts wit total Spin of 0, ten wen one as Spin + / in a certain direction, te oter will ave Spin - / along tat direction (so te total Spin is (+ /) + (- /) = 0). Te situation of interest is sown below in Figure 3.

12 Detector Detector Pair of particles prepared in Spin zero state. Stern- Gerlac Device, x Figure 3 Stern- Gerlac Device, x Spin is conserved in eac direction. Wenever a pair of particles are produced and one is incident on Detector (oriented along x) and te oter is incident at Detector (also oriented along x) ten te two detectors must measure opposite spins. In order to simplify our discussion, from now on wen a particle as a Spin of + / we will simply say its Spin is + or tat it is up. Wen a particle as a Spin of - / we will say tat it as a Spin of - or tat it is down. If te two detectors are not oriented along te same direction ten te sum of te detected Spins is not necessarily equal to zero. Tat is, if Detector is oriented along x and Detector along z ten te sum of te detected Spins may be oter tan zero. In fact, in tis case, te probability tat te two spins will be te same is / and te probability tat tey are different is also /. So, for example, if Detector measures Spin + (up) along x ten tere is a 50/50 cance tat Detector will measure eiter + or - if it is oriented along z. Tus if we measure Spin up at Detector along x, Quantum Mecanics cannot predict wit certainty te outcome of a measurement along z. Quantum Mecanics can only predict te probability tat it will be up or down (in tis case te probability for eac is /). B. EPR's argument Einstein, Podolsky and Rosen devised a tougt experiment to sow tat Quantum Mecanics is an incomplete description of reality. Te argument presented ere is similar to tat made by EPR but is based on te situation depicted in Figure 3. EPR defined a teory to be complete if and only if it includes all real quantities. A quantity was defined to be real if it could be predicted wit certainty, witout disturbing te system. If Detector is oriented along x and we measure te Spin to be up ten we can predict (wit certainty) tat te Spin along x at Detector is down. Tus, te Spin at Detector along x can be said to be real quantity. EPR went on to argue tat we could ave made a measurement along z in Detector along te z-direction, if we liked. If we ad done so, we could ave predicted (wit certainty and witout disturbing te system) te outcome

13 of te measurement of Spin along te z-direction for te particle at Detector. Tus, EPR argued, we can predict wit certainty te value of te Spin of te particle incident upon Detector along bot x and z directions simultaneously. However, according to Quantum Mecanics, if we know te spin along x ten we can only predict a probability of te outcome of a measurement of te Spin of te particle along z. So Quantum Mecanics, according to te Copenagen Interpretation, says tat we cannot simultaneously know (wit certainty) bot te Spin in te x-direction and te Spin in te z-direction. So, according to EPR, Quantum Mecanics does not include te two real quantities Spin along x and z simultaneously, so quantum Mecanics is incomplete. C. Te Copenagen interpretation Te realistic view of EPR was contrary to te widely eld Copenagen interpretation of Quantum Mecanics, promoted largely by Niels Bor. Bor argued tat altoug te statement "I could ave measured " makes no sense. For im properties of systems are only defined wit respect to teir measurement devices. VI Bell's Inequality A. Generalization of Spin measurements on a pair of particles. Te coice of axes is arbitrary. We can define x and z owever we like, as long as tey are perpendicular to eac oter. Furtermore, we need not orient our Stern-Gerlac device along eiter of tese axes. Consider tis generalization depicted in Figure 4. Detector A Detector B Pair of particles prepared in Spin zero state. Stern- Gerlac Device, θ A Figure 4 Stern- Gerlac Device, θ B We now depict te two detectors by Detector A (oriented along θ A ) and Detector B (oriented along θ B ). Quantum Mecanics predicts tat te probability tat te Spin measured at A is equal tat at B as 3

14 P(A=B) = [ cos( θa θb)]. (4) Notice tat if θ A = θ B ten P(A=B)= (-cos(0))/ = 0, wic says tat wen bot detectors are oriented along te same direction tey never give te same result. Wen one is up te oter cannot also be up. If θ A is 0 o and θ B is 80 o ten P(A=B) = (-cos(- 80))/ =. Wen we say tat te detector is oriented at 80 o we ave flipped te detector so Spins tat were measured up are now down and tose tat were down are now up. Tus our result for θ A is 0 o and θ B is 80 o giving P(A=B) = means tat wen one is up te oter is down. If θ A is 90 o and θ B is 0 o ten P(A=B) = (-cos(90))/ = 0.5. Tis says tat wen te detectors are oriented perpendicular to eac oter tere is a 50/50 cance tat tey will bot ave te same value. Te probability tat te Spin measured at A is not equal tat at B is just P(A B) = - P(A=B) = - [ cos( θa θb)] = [ + cos( θa θb )] To ease te notation in following discussions, let us define AB = P(A B). (5) B. Background Te completeness of quantum mecanics was debated a lot after EPR's paradox was proposed. Some pysicists claimed tat te indetermism of quantum mecanics was an indeterminism in our knowledge of reality. If only we knew more about te system ten we could predict everyting wit certainty. Some of tese pysicists developed oter teories tat made te same predictions as quantum mecanics but added a "idden" variable tat determined te outcome te measurement. Tese ad oc idden variable teories were consistent wit a realistic metapysics. Tey added noting new in te way of prediction (since te added variable was idden) but were attractive to many wo wanted to preserve a realistic view. In 965 Jon S. Bell proved tat any local, realistic teory is inconsistent wit te predictions of quantum mecanics. Locality refers to te inability of any cause or information to be able to travel faster tan te speed of ligt. Einstein's teory of special relativity sows tat if a causal element can go faster tan te speed of ligt, cause and effect as we know it is undermined. A realistic teory is one tat assumes tat quantities exist even wen we are not observing tem. Tus, in EPR's tougt experiment, a realistic teory would old tat eac particle as Spin bot in te x and z direction. Tat quantum mecanics cannot predict bot of tese quantities sows tat it is incomplete. If only we knew more about te system, te value of some idden variable, ten we would be able to predict te Spin in bot directions. Bell sowed tat suc a train of tougt, coupled wit locality, is inconsistent wit quantum mecanics. C. An example of Bell's Inequality 4

15 Consider te experiment depicted in Figure 4. In addition to making measurements at Detector A at angle θ A, let us be able to make a measurement at Detector A at anoter angle θ A '. We will denote tis measurement as A'. Likewise let us be able to make anoter measurement Detector B at a new angle θ B ' tat we will denote as B'. Tus we can make two measurements at Detector A, denoted by A and A', tat differ in ow we ave oriented our detector. We can also make two measurements at detector B, denoted by B and B', tat differ in ow we ave oriented detector B. Note tat we cannot make any two measurements at one detector at te same time, te detector can only ave one orientation at any time. In a realistic model, owever, we assume tat a particle possesses Spin in bot directions at once so we can talk about tese properties belonging to te particle. We ave, according to quantum mecanics AB = P(A B) = [ + cos( θa θb)] A'B = P(A' B) = [ + cos( θa' θb)] (6) AB' = P(A B') = [ + cos( θa θb' )] A'B' = P(A' B') = [ + cos( θa' θb' )] Assuming te reality of te Spin of eac particle along eac direction we will now sow tat AB' + A'B+A'B' AB (7) To arrive at equation 7 we will treat eac measurement pair as a circle wose area represents its probability. For example let us consider AB, wic is te probability tat te spin measured at Detector A at angle θ A is not equal to tat measured at Detector B oriented along θ B. Tis is illustrated in Figure 5, below. 5

16 AB Figure 5. Te area of te square is taken to be. Te area inside te circle represents P(A B). Tus te area outside of te circle but inside te square is - P(A B) wic is P(A=B), since eiter A=B or A B. Te sum of tese possibilities must be. Let s illustrate tis wit an analogous situation. Let say tat instead of measuring Spin at two positions (A and B) we ave two people (at stations A and B) flipping pennies. Te result AB corresponds to te situation wen person A does not get te same result as person B (tat is wen A as eads, B as tails or wen A as tails, B as eads). Tis analogy is not perfect in te sense tat in quantum mecanics tere are defined relations between te results at A and B tat are not applicable to te flipping of pennies, but to illustrate te use of diagrams like Figure 5, te analogy is adequate. To figure out te probability of AB we can figure out te fraction of possibilities tat correspond to tis result. All togeter we ave four possibilities: HH, HT, TH, TT, were te first value is tat at A and te second one is tat at B. Ten AB= /4 = 0.5. Tis is illustrated in Figure 6. 6

17 HH And TT = 0.5 AB = HT and TH = 0.5 Figure 6. Now let us consider te probabilities of te tree measurements AB', A'B and A'B', depicted in Figure 7. AB' A'B' A'B A=B Figure 7 Wen A' B and A B' ten te circles representing tese overlap. We are considering a general case ere, were te intersections of te outcomes tese tree sets of measurements may overlap or not. For our penny analogy, let s introduce te prime measurements (A' and B') as wen, instead of using pennies, we use nickels. Ten te A' measurement is wen a nickel is flipped at position A and a B' measurement is wen a nickel is flipped at position B. AB' is wen te person at A gets a different value for is or er penny tan te person at B gets for is or er nickel (TH or HT). An example of overlap for te measurements A'B 7

18 and A' B' would be wen you ave H n T p and H n T n., were te subscripts denote weter a penny or nickel was flipped. Te area outside te circle for AB' is te probability tat A = B', tat outside A'B is P(A'=B), and tat outside A'B' is P(A'=B'). Terfore, te area tat is completely outside all 3 circles satisfies A = B', A' = B', and A' = B. Tus, in tat area outside all tree circles, A=B. Ten te area inside te area covered by te tree circles satisfies - P(A=B) = P(A B) = AB. Tus we get Equation 7. If te tree circles do not intersect at all ten AB' + A'B + A'B' = AB. If tey do intersect some, AB' + A'B + A'B' > AB. Now we sow tat Equation 7 can be inconsistent wit Quantum Mecanics. Let θ A = 90 o, θ B = 90 o, θ B' = 0 o, θ A' 330 o (wit all tese angles being measured counterclockwise from te orizontal direction). Ten Let θ A - θ B = 0 o, θ A' - θ B = 40 o, θ A - θ B' = -0 o, and θ A' - θ B' = 0 o. Ten, from equation 6, AB = [ + cos(0)] = A'B = [ + cos(40)] = 0.5. (8) AB' = [ + cos( 0)] = 0.5 A'B' = [ + cos(0)] = 0.5 Now putting tese into te relation obtained from our realistic model (Equation 4), we get (9) 8