aa r X i v : . [ phy s i c s . g e n - ph ] M a y Many worlds interpretation for double slit experiment
Zinkoo Yun ∗ Department of Physics and Astronomy University of Victoria, Canada
As is well known, the double slit experiment contains every key concepts of quantum mechanicssuch as phase effect, probability wave, quantum interference, quantum superposition. In this article,I will clarify the meaning of quantum superposition in terms of phase effect between states. Afterapplying standard quantum theory, it leads to serious questions about the unitary process of anisolated system. It implies that non collapsing interpretations including many worlds may not bejustified. This also could explain that there is no such boundary between classical and quantumdomains.
PACS numbers: 03.65.Ta, 03.65.Yz, 04.60.Bc, 04.70.Dy
I. INTRODUCTION
In early 1800s, Thomas Young has proposed an exper-iment which provides a strong evidence of wave theory oflight. Figure 1 illustrates the experiment he proposed. Astream of photons starts from x and passes through twoholes x and x on the slit and finally lands at spots onthe screen. If the stream contains enough number of pho-tons, we can observe interference pattern on the screen.We have no problem of predicting exactly what this in-terference pattern looks like. However, though more than200 years have passed now since its discovery, it appearsthat we still do not completely understand this phenom-ena in quantum perspective, especially about the mean-ing of quantum collapse and measurement.In this paper I will clarify about the meaning of x ’2 x x x ψ( x , t ) t t ’ x ’1 FIG. 1: Double slit experiment in a box. Interference patternappears as particles land at each spot x ′ i on the screen. Twoposition eigenstates | x i and | x i on slits are in quantumsuperposition, because there is non zero phase effect betweenthem. ∗ Electronic address: [email protected] quantum superposition in terms of quantum phasebetween states. Then after applying standard quantummechanics, we will see what it implies to the unitaryprocess of isolated quantum system and currentlypopular interpretations of quantum mechanics such asthe many worlds interpretation.Since its first introduction by Hugh Everett[1, 2] in1957, and being ignored for quite a while before redis-covered by Bryce DeWitt[3], the theory of many worldsinterpretation becomes popular and currently it is oneof main interpretations of quantum mechanics acceptedwidely among many physicists because of its attractivefeature of unitary evolution without collapsing wave func-tion.In order to see how the many worlds view interpretsthe double slit experiment, let’s change the experimentsetup little bit as shown in figure 1. First, suppose weplace a device at x which emits a photon roughly everyhour. This photon passes the slits and hits the screeneventually, where “screen” means the apparatus measur-ing position eigenstate of photon. Second, we put a stopwatch on the screen, so that if the photon hits any posi-tion on the screen it records the time of the event. By thismethod, we can measure when the photon has landed onthe screen. Finally let’s put this whole system and appa-ratus in a box in order to make everything isolated. Afteran hour, suppose we open the box to check the positionsof photon landed on the screen.According to many worlds interpretation, the wavefunction of photon never collapses to one spot on thescreen before opening box. We can express the state ofsystem+apparatus+observer as a superposition of coex-isting many worlds by X i a i | x ′ i i | A x ′ i i | OA x ′ i i (1)for each spot x ′ i on the screen, where | x ′ i i stands forthe state of photon landed on the screen at x ′ i .[6] | A x ′ i i stands for the states of particles of screen at x ′ i when thephoton landed at x i . | OA x ′ i i represents the observerwho observes the state of photon and the state of particlesof screen after the photon landed at x ′ i ; When we open thebox, this many worlds split and we are subjected to oneof them. Any non collapsing interpretation of quantummechanics implies the entangled joint state similar to (1).Thus we cannot say that the photon has landed at anyspecific spot on the screen before we open the box. It hasbeen in quantum superposition of all spots on the screen before observation. II. THE STATE OF PHOTON
ON SCREEN
In quantum mechanics, a superposition state meansthere is a phase effect between component states. In theexample of double slit experiment in figure 1, we canmeasure it by observing interference effect between twoquantum paths passing two slits. Thus we can say twoposition eigenstates | x i and | x i of photon on slit arein quantum superposition.Then, is the quantum state of photon on the screen also in quantum superposition? In Feynman path inte-gral technique calculating the probability amplitude tomeasure the final position eigenstate ( x f , t f ) from initialstate ψ ( x, t ), h x f , t f | ψ ( x, t ) i = Z dx h x f , t f | x, t i ψ ( x, t ) (2)we interfere all quantum paths ending up ( x f , t f ). Wedon’t count any quantum path ending up other than( x f , t f ) because they do not interfere with paths endingup ( x f , t f ). Namely, two quantum paths ending up twodifferent space time points do not interfere each other.That is, in figure 1 experiment, any quantum path endingup at x ′ and any quantum path ending up at x ′ do notinterfere each other. In other words, there is no phaseeffect between position eigenstates | x ′ i and | x ′ i ofphoton on the screen, so they are not in quantum super-position contrast to non collapsing interpretation. Thatis, the state of photon on the screen should be express byeither | x ′ i or | x ′ i not by their superposition | x ′ i and | x ′ i . Therefore even before opening box, the state ofphoton must be expressed by one of position eigenstateon the screen not by their quantum superposition like(1).Of course, if there is no screen there, then the quantumstate of photon can be expressed by quantum superposi-tion of position eigenstates in there, P i a i | x i i , becausequantum paths of photon do not end up there, so we maymeasure interference effect between them. III. MEANING OF SUPERPOSITION STATE
What exactly does it mean by “quantum superposi-tion”? Let’s remind the definition of this word. Figure2(a) describes typical double slit experiment and we puta divider in figure (b) experiment. In figure 2(a) experi-ment, we can express the wave function of photon at slit x x x ψ( x , t ) t t ’ x ’1 x x x φ( x , t ) t t ’ x ’1 x ’2 FIG. 2: (a) Two quantum paths end up at the same point x ′ .They interfere, so two position eigenstates | x i and | x i arein quantum superposition; Two quantum paths did not endup at | x i and | x i . They end up on the screen. (b) Twoquantum paths could not end up at the same point. Theycould not interfere, so | x i and | x i are not in quantumsuperposition. as a quantum superposition of two position eigenstates | x i and | x i , | ψ ( x, t ) i = a | x i + a | x i (3)For given configuration of experiment, if the calculationof the relative phase effect between two states | φ i and | φ i could be non zero, then they are in quantum su-perposition. For example, suppose two quantum paths h f | φ ih φ | i i and h f | φ ih φ | i i pass twostates | φ i and | φ i . For arbitrary change of phasefrom | φ ih φ | to e iθ | φ ih φ | (from | φ i to e iθ | φ i ,if | φ i is an initial or final state), if there exist quantumpaths resulting in non zero change of (magnitude of) am-plitude of measurement, we can say two quantum states | φ i and | φ i are in quantum superposition. If thereis no quantum paths resulting in non zero change of am-plitude by arbitrary phase change, then two states | φ i and | φ i are not in quantum superposition.In figure 2(a) experiment, if we change the phase alongone path from | x ih x | to e iθ | x ih x | arbitrary, thechange of amplitude due to interference with anotherquantum path passing | x i could be non zero. Thus twoquantum states | x i and | x i are in quantum super-position in figure 2(a) experiment; However, the changeof amplitude by arbitrary phase change is zero in figure2(b) experiment. So | x i and | x i are not in quantumsuperposition in figure 2(b) experiment. Thus the con-figuration of experiment could be a factor in determiningwhether two given states are in quantum superpositionor not.The reason is that because two quantum paths passing | x i and | x i could end up at the same point in (a),so they could interfere each other, while they couldnot end up at the same point in (b). This reminds usthe claim of Feynman path integral. Feynman pathintegral insists that if two quantum paths ends up atthe same space time point, they interfere each other.Thus two states along each path could be in quantumsuperposition. If two quantum paths do not end up atthe same space time point, they do not interfere eachother. Thus two states along each path could not be inquantum superposition.With this standard in mind, let’s consider whether twoposition eigenstates of photon on the screen could be inquantum superposition in figure 1 experiment. The am-plitudes of measuring | x ′ i and | x ′ i on the screen are h x ′ | ψ ( x, t ) i = a h x ′ | x i + a h x ′ | x i (4) h x ′ | ψ ( x, t ) i = a h x ′ | x i + a h x ′ | x i (5)For arbitrary change of phase from | x i to e iθ | x i ,the amplitudes do change. Thus | x i and | x i are inquantum superposition.On the other hand, these amplitudes do not change aswe change the phase of one quantum path arbitrary from | x ′ i to e iθ | x ′ i or from | x ′ i to e iθ | x ′ i . Thus | x ′ i and | x ′ i are not in quantum superposition as expectedfrom the fact they end up at different space time points.Thus the phase difference between them does not meananything physically. i.e., It is not measurable quantity inprinciple.This fact does not matter how photon interacts withscreen. Someone may argue thatThe photon landing at each position x ′ i in-teracts with the screen in complicate way. Ifwe count all this complications, then quan-tum state | x ′ i | A x ′ i and the quantumstate | x ′ i | A x ′ i may show phase effectbetween them through the entangled state P i a i | x ′ i i | A x ′ i i . (6)It is possible to argue against (6) byAll experimental data says that a photonphysically interacts with particles of screenat only one position. No data reveals that aphoton physically interacts with particles ofscreen at multiple positions. Thus we don’tneed to consider interference between parti-cles of screen at multiple positions. (7)First of all, according to (6), two quantum paths in fig-ure 2(b) also interfere each other, so that two eigen states | x i and | x i are in quantum superposition; But mostimportantly, no matter which one is true, this is a nullargument for the purpose of falsifying the other view: (6)already assumes non collapsing of wave function and (7)already assumes collapsing view. Regardless which one istrue, we are not supposed to argue in this way to disprovecollapsing or non collapsing view. In disproving collaps-ing or non collapsing model, we are not allowed to useany claim which presumes collapsing or non collapsingview.[7] We have to argue by the criterion of definitionof “superposition” itself. x x x t ψ( x , t ) x ’1 x ’2 x "3 t ’ t " ψ ’ ( x ’ , t ’ ) FIG. 3: After making two holes at x ′ and x ′ (or removing thefirst screen completely), these two position eigenstates | x ′ i and | x ′ i become in quantum superposition. We can measuretheir phase difference effect on the second screen. In the analysis above, from the fact the quantum paths of photon end up on screen, we have proved that | x ′ i and | x ′ i on the screen are not in quantum superposition bythe definition of it. In order for (6) to be true, | x ′ i and | x ′ i on the screen must be in quantum superposition.However two position eigenstates | x ′ i and | x ′ i inthere can be in quantum superposition, if the quantumpaths do not end up there. Suppose we make two holes at | x ′ i and | x ′ i and put another screen at further distanceas shown in figure 3. Then the amplitude of measuringthe particle at | x ′′ i is h x ′′ | ψ ′ ( x ′ , t ′ ) i = c h x ′′ | x ′ i + c h x ′′ | x ′ i (8)Changing phase | x ′ i to e iθ | x ′ i arbitrary results in thechange of amplitude. Thus in this configuration of ex-periment, two position eigenstates | x ′ i and | x ′ i are inquantum superposition. It implies that if we completelyremove the first screen in figure 3 experiment, the wavefunction in there is in quantum superposition of all po-sition eigenstates | x ′ i i . This is expected because thequantum paths do not end up there. They end up at thesecond screen at further distance.According to non collapsing interpretation of quantummechanics including many worlds interpretation, in dou-ble slit experiment, before observation (before openingbox in figure 1 experiment), the state of photon landedon screen is considered as a quantum superposition ofall position eigenstates | x ′ i i on the screen . This su-perposition is accomplished through the entanglement P i a i | x ′ i i | A x ′ i i between photon and screen which actsas a measuring apparatus. The analysis above provesthat this cannot be true. The analysis proves that twoposition eigenstates | x ′ i and | x ′ i on the screen infigure 1 experiment are not in quantum superposition.Thus before opening box, the state inside box is one of | x ′ i i | A x ′ i i , not their quantum superposition. Thus wehave proved that non collapsing interpretation of quan-tum mechanics cannot be justified. IV. QUANTUM ENTANGLEMENT ANDBOUNDARY
We have proved that the state of photon in double slitexperiment cannot be expressed by the entangled statebetween system and measuring apparatus like | x ′ i i | A x ′ i i . Let’s discuss its implication on boundary betweenquantum and classical world.The state expressed by (1) is called the quantum en-tangled state. It is well accepted that if the apparatus(in initial state | A i ) measuring eigenstates | S i i of sys-tem is in contact with the quantum superposition state P i a i | S i i of a system, then the resulting state beforeobservation is expressed by the entanglement betweensystem and the measuring apparatus, X i a i | S i i | A i → X i a i | S i i | A i i (9)where | A i i stands for the state of apparatus measured | S i i state of the system. This entangled state is pro-posed in order to hold the unitary evolution of quantumtheory. For example, (9) implies the quantum state be-fore observation, is P i a i | x ′ i i | A x ′ i i for photon andscreen which we have disproved. Thus we know that gen-erally the type of entanglement like (9) is not possible.The entanglement between objects is possible only veryspecial cases like decay of elementary particles. Besides,we can see that the process (9) itself is a kind of nonunitary process.The entanglement process (9) between system andapparatus cannot be justified as we have demonstratedin double slit experiment. “Why the measuring device apply classical theory notquantum mechanics.” This has been the unansweredquestion for a long history of quantum mechanics. Know-ing that entanglement like (9) does not occur, we mayanswer this question; The truth is, in fact macroscopicapparatus do follow quantum mechanics exactly the sameway as electron does. The difference is, as the mass in-creases, the dispersion of wave packet of Gaussian distri-bution becomes smaller according to Schrodinger’s equa-tion. For electron in static state, the dispersion speed isas the speed of light and for 1kg object the dispersionspeed is 1m for hundred thousand years, so its dispersioncan be ignored for our life time scale measurement. Itmeans if we use 1kg object as an indicator of an exper-iment, we can almost trust our reading, but if we use amicroscopic object as an indicator, we cannot trust ourreading. That explains why Bohr insisted that measuringapparatus has to follow classical theory. In fact measur-ing apparatus also follows quantum mechanics, just itsquantum effect is almost ignoble in our life time scale.This shows that there is no boundary between quantumand classical realm. V. CONCLUSION
We have proved that non collapsing interpretation ofquantum mechanics including many worlds cannot becompatible with standard quantum mechanics. In or-der to show it, the meaning of quantum superposition isclarified in terms of phase effect between states. Follow-ing the logic of standard quantum mechanics, it leads tothe conclusion “In general, the type of entanglement like(9) is impossible in principle.” With this fact in mind,we could understand that there is no boundary betweenquantum and classical world. [1] Hugh Everett,
The theory of the universal wave function
Princeton Ph.D thesis , (1957).[2] Hugh Everett,
Relative state formulation of quantum me-chanics
Reviews of modern physics , , 454–462 (1957).[3] Bryce DeWitt, R. Neill Graham, The Many-Worlds In-terpretation of Quantum Mechanics, Princeton Series inPhysics
Princeton University Press , , 3–140 (1973).[4] John Wheeler, Law without law: Quantum theory andmeasurement
Princeton University Press , 182–213 (1983). [5] Richard P. Feynman, A. R. Hibbs,
Quantum Mechanicsand Path Integrals