H 45 Quantum Measurement and Spn Wnter 003 Ensten-odolsky-Rosen aradox The Ensten-odolsky-Rosen aradox s a gedanken experment desgned to show that quantum mechancs s an ncomplete descrpton of realty. The authors were uncomfortable wth the quantum mechancal noton that we can only know certan propertes of an atom (e.g., only one of the spn components, not all three). The gedanken experment attempts to prove that the unknown propertes are really there (they are elements of realty n the authors words). The expermental stuaton s depcted below (ths verson of the ER experment s due to Davd Bohm). A spn 0 source decays nto two spn / partcles, whch by conservaton of angular momentum must have opposte spn projectons and by conservaton of momentum must head n opposte drectons. Observers A and B are on opposte sdes of the source and each has a Stern-Gerlach apparatus to measure the spn projecton of the partcle headed n ts drecton. Whenever one observer measures spn up along a gven drecton, then the other observer measures spn down along that same drecton. The quantum state of the two partcle system can be wrtten as ψ = + + [ ], where the subscrpts label the partcles and the relatve mnus sgn ensures that ths s a spn-0 state. Because of the correlaton between the measurements, f observer A measures spn up along a gven drecton, for example S z =+h /, then we can predct wth 00% certanty what the result of observer B s measurement wll be ( S z = h /), wthout performng the measurement or dsturbng partcle n any way. ER contend that f we can predct a measurement result wth 00% certanty, then that result must be a real property of the partcle -- t must be an element of realty. Snce the partcles are wdely separated, ths element of realty must be ndependent of what observer A does, and hence must have exsted all along. Observer A could have chosen to measure S x or S y, whch by the same reasonng means that S x and S y for partcle must also be â â B Spn 0 Source A ĉ ˆb artcle artcle ĉ ˆb
H 45 Quantum Measurement and Spn Wnter 003 elements of realty. Quantum mechancs mantans that we can only know one spn component at a tme for a sngle partcle. Snce t thus does not descrbe all the elements of realty of the partcle, quantum mechancs must be an ncomplete descrpton of physcal realty. If ER are correct, then the elements of realty, whch are often called hdden varables or nstructons sets, are really there but for some reason we cannot know all of them at once. Thus one can magne that there are dfferent types of partcles wth dfferent nstructons sets that determne the results of measurements. One can also magne that the populatons or probabltes of these dfferent sets can be properly adjusted n a hdden varable theory to produce results consstent wth quantum mechancs. Snce quantum mechancs and a hdden varable theory cannot be dstngushed by experment, the queston of whch s correct s then left to the realm of metaphyscs. For many years, ths was what many physcsts beleved. In 964, John Bell showed that there are specfc measurements that can be made to dstngush between a hdden varable theory and quantum mechancs. By consderng measurements that observers A and B make along three dfferent drectons (all n a plane as shown above), he derved an nequalty that could be tested by experment. He derved a very general relaton, but we wll deal wth a specfc one here to make lfe easy. Consder three drectons n a plane as shown, each 0 from any of the other two. Each observer makes measurements of the spn projecton along one of these three drectons, chosen randomly. Any sngle result can only be + or -, and we calculate the probablty that the results from a correlated par (.e., one decay from the source) are the same (++ or --) or opposte (+ or +), where +-, for example, means observer A recorded a + and observer B recorder a -. We know that when both observers measure along the same drecton, then only a +- or a -+ s possble. To reproduce ths aspect of the data, a hdden varable theory would need 8 nstructon sets as shown n the table. We don t yet know what the probabltes are for cases where the observers do not measure along the same drectons, so we do not assgn any populatons (or weghts or probabltes) to the dfferent nstructon sets. resumably we can adjust these as needed to make sure that the hdden varable theory agrees wth the actual (or quantum mechancal results). Now use the nstructon sets to calculate the probablty that the results are the same ( same = ++ + -- ) and the probablty that the results are opposte ( opp = +- + -+ ), consderng all possble measurements. There are 9 dfferent combnatons of measurement drectons for the par of observers. If we consder partcles of type, then for each of these 9 possbltes, the results wll be opposte (+-). The results can never be the same. The same argument holds for type 8 partcles. For type partcles, there wll be 4 possbltes of recordng the same results and
H 45 Quantum Measurement and Spn Wnter 003 Instructon Sets (Hdden Varables) opulaton artcle artcle N (ˆ a+, b ˆ +, cˆ + ) (ˆ a, b ˆ, cˆ ) N (ˆ a+, b ˆ +, cˆ ) (ˆ a, b ˆ, cˆ + ) N 3 (ˆ a+, b ˆ, cˆ + ) (ˆ a, b ˆ +, cˆ ) N 4 (ˆ a+, b ˆ, cˆ ) (ˆ a, b ˆ +, cˆ + ) N 5 (ˆ a, b ˆ +, cˆ + ) (ˆ a+, b ˆ, cˆ ) N 6 (ˆ a, b ˆ +, cˆ ) (ˆ a+, b ˆ, cˆ + ) N 7 (ˆ a, b ˆ, cˆ + ) (ˆ a+, b ˆ +, cˆ ) N 8 (ˆ a, b ˆ, cˆ ) (ˆ a+, b ˆ +, cˆ + ) 5 possbltes for recordng opposte results. We thus arrve at the followng probabltes for the dfferent partcle types: opp same = = 0 types & 8 opp same 5 = 9 4 = 9 types 7 Now average over all the possble partcle types to fnd the probabltes of recordng the same or opposte results n all the measurements. The probablty of any partcular partcle type, for example type, s smply N / N (recall we wll adjust the actual values later as needed). Thus the averaged probabltes are: opp = N N + N + 5 9 N + N + N + N + N + N ( ) 5 9 8 3 4 5 6 7, same 4 = ( N + N + N + N + N + N ) N 9 4 9 3 4 5 6 7, where the nequaltes follow smply because the sum of all the probabltes for the partcular partcle types must sum to one. In summary, we can adjust the populatons all we want but that wll always produce probabltes of the same or opposte measurements that are bound by the above nequaltes. That s what s meant by a Bell nequalty. 3
H 45 Quantum Measurement and Spn Wnter 003 What does quantum mechancs predct for these probabltes? For ths smple system of spn / partcles, we can easly calculate them. Assume that observer A records a + along some drecton (of the three) and defne that drecton as the z-axs (no law aganst that). Then we know that the quantum state of partcle s. The probablty that observer B records a + along a drecton at some angle θ wth respect to the z-axs s = + = θ θ + + = θ cos sn sn, φ same n e where + n s the egenstate for spn up along the drecton of measurement. The same result wll be obtaned f you assume A records a - and ask for the probablty that B records a - also. The probablty that observer B records a - along ths drecton, when A records a +, (hence opposte results) s = = θ θ + = θ sn cos cos. φ opp n e Snce the angle θ wll be 0 n /3 of the measurements and 0 n /3 of the measurements, the average probabltes wll be 0 opp = + 0 = + = 3 3 3 cos cos, 3 4 0 same = + 0 = + = 3 3 3 0 3 sn sn. 3 4 These predctons of quantum mechancs are nconsstent wth the nequaltes derved from hdden varable theores. Snce these probabltes can be measured, we can do experments to test whether hdden varable theores are possble. The results agree wth quantum mechancs and hence exclude the possblty of hdden varable theores. The ER paradox also rases ssues regardng collapse of the quantum state and how a measurement by A can nstantaneously alter the quantum state at B. However, there s no nformaton transmtted nstantaneously and so no volaton of relatvty. What observer B measures s not affected by any measurements that A makes. They only notce that when they get together and compare results that some of the measurements (along the same axes) are correlated. 4
H 45 Quantum Measurement and Spn Wnter 003 Schrödnger Cat aradox The Schrödnger cat paradox s a gedanken experment desgned by Schrödnger to llustrate some of the problems of quantum measurement, partcularly n the extenson of quantum mechancs to classcal systems. The expermental apparatus conssts of a radoactve atom, a Geger counter, a hammer, a bottle of cyande gas, a cat, and a box. The atom has a 50% probablty of decayng n one hour. The components are put together such that f the atom decays, t trggers the counter, whch causes the hammer to break the bottle and release the posonous gas, kllng the cat. Thus, after one hour there s 50% probablty that the cat s dead. We can descrbe the quantum state of the atom as [ ] ψ = ψ + ψ atom undecayed decayed The apparatus has been desgned such that there s a one-to-one correspondence between the undecayed atomc state and the lve-cat state and between the decayed atomc state and the dead-cat state. Though the cat s macroscopc, t s made up of mcroscopc partcles and so should be descrbable by a quantum state (albet a complcated one). Thus we expect that the quantum state of the cat after one hour s ψ = ψ + ψ [ ] cat alve dead Both quantum calculatons and classcal reasonng would predct 50/50 probabltes of observng an alve or dead cat when we open the box. However, quantum mechancs would lead us to beleve that the cat was nether dead nor alve before we opened the box, but rather was n a superposton of states and the quantum state only becomes the sngle alve or dead state when we open the box and make the measurement by observng the cat. But our classcal experences clearly run counter to ths. We would say that the cat really was dead or alve, we just dd not know t yet. (Imagne that the cat s wearng a cyande senstve watch -- the tme wll tell us when the cat was klled f t s dead!) The man ssues rased by ths thought experment are () Can we descrbe macroscopc states quantum mechancally? and () What causes the collapse of the wave functon? The Copenhagen or standard nterpretaton of quantum mechancs champoned by Bohr and Hesenberg mantans that there s a boundary between the classcal and quantum worlds. We descrbe mcroscopc systems (the atom) wth quantum states and macroscopc systems (the cat, or.. 5
H 45 Quantum Measurement and Spn Wnter 003 even the Geger counter) wth classcal rules. The measurement apparatus causes the quantum state to collapse and yeld the sngle classcal or meter result. Where to draw ths lne s not clear and wll depend on the problem. Others have argued that the human conscousness s responsble for collapsng the wave functon, whle some have argued that there s no collapse, just bfurcaton nto alternate, ndependent unverses. Snce most of these ponts of vew are untestable, t s often left as a phlosophcal debate. More recent dscussons and experments have focused on the ssue of usng quantum mechancs to descrbe macroscopc systems. By studyng mesoscopc systems that are small enough to control precsely, yet large enough to have macroscopcally dstngushable states, one can probe the regon between the quantum and the classcal worlds. In recent experments n quantum optcs, t has been shown that the relatve phase between two parts of a superposton state becomes randomzed very quckly, yeldng a mxture state, whch s not dstngushable from a classcal probablty mxture. It has been shown that ths coherence decay proceeds more quckly as the sze of the system (and hence ts complexty) becomes larger. It may not be long before we conclude that the wave functon collapse s not dstnct from ordnary Schrödnger tme evoluton, but rather just a consequence of the decoherence of large mult-component states. 6