Introduction to Bells theorem: the theory that solidified quantum mechanics Jia Wang Department of Chemistry, University of Michigan, 930 N. University Ave., Ann Arbor, MI 48109 (Received November 30, 005; accepted December 14, 005) I. Introduction The classical world as we know it can be described definitively. As an example, this paper that you are holding has a measurable length, width, and thickness. If one were to throw this paper out the window, one could monitor the velocity (v) with which it falls and from this measurement, one would be able to find its momentum (p=mass*v). This indisputable, definite identification of objects that we can see and touch has been known for centuries. However, one might ask whether it is possible to extend the relationships between measurable quantities of everyday objects to atomic or subatomic particles. This question led to the development of quantum mechanics. Quantum mechanics as presented by Schrödinger, Heisenberg, Dirac, and others provided the best mathematical description of experimental findings. The wavefunction,, that describes a particle (or particles) contains all measurable information, including the position and momentum. However, a key result of quantum mechanical calculations is that certain observable quantities cannot be exactly and simultaneously known. This Uncertainty Principle, is mathematically generalized in Eq. 1. 1 A B 1 i [ A,B] (1) In this equation, is the standard deviation of the possible outcomes of the observable, A or B. The quantity on the right hand side is related to the average value of the commutator of the operators A and B. For instance, using position and momentum (in the position representation) as: Position Operator (x): x; Momentum Operator (p): Eq. 1 becomes d x, i = i h dx h. d ih. dx The ih term is a constant, so its expectation (average) value is itself. Thus, reintroducing the above into Equation 1, one will find x p i = i 4 h h, and after taking the square-root to both sides, the equation becomes x p h. * Here, the product of the standard deviation of measuring x and p is be greater than or equal to h. This means that if the position of the particle is known with absolute certainty, then the position probability distribution collapses to a delta function at that point; the momentum distribution, in turn, will have a deviation that is infinitely large, meaning that it becomes impossible to find the exact momentum. This concept seems to defy the classical logic that an object s position and momentum can be found with certainty simultaneously. Results such as the above from quantum mechanics generated a list of critics, including Albert Einstein, who was * Note that only the positive value to the square-root is used, since standard deviation is always positive. 3
determined to prove that quantum mechanics is an incomplete theory. II. The EPR Paradox God does not play dice with the universe, was a famous quote from Einstein. In 1935, Einstein, Podolsky, and Rosen 3 introduced what became known as the EPR Paradox. Their argument can be shown in a simple example. Suppose there is a particle that can be split into a positron and an electron. Physicists call such a particle a pion, and it is neutral with spin zero: 0 + e + e. Hypothetically, the pion is initially at rest and the electron and positron fly away in opposite directions. Angular momentum is conserved; thus, if there were parallel detectors (oriented in the same direction; i.e. both up or both down) on either side, they would find one particle with spin up and the other with spin down. Since there are two possible degenerate spin pairings for the two particles (positron spin up, electron spin down, and vice versa) and that they are fermions, the following linear combination of the possible configurations shows the overall description: 1 + + ( ee ee. () These two particles are related by the fact that they arose from a single pion. If the results at only one detector are known, one would be able to conclude that the other particle must have the opposite spin. The spin states of the two particles are thus correlated, or entangled. Suppose that the electron and positron are allowed to fly further and further away before detection, perhaps 100 km or even 100 light years. Knowing the spin state of one particle will still allow one to know Fermions are particles with non-integer spin. They occupy different states; hence there is a negative sign in Equation. Bosons, particles with integer spin, have a positive combination, for they can occupy the same state. the spin state of the other particle instantaneously. Quantum mechanics is a tool to formulate the possible combinations, as shown in Eq.. It does not, however, reveal which configuration the two particles take immediately following their release. Different interpretations of the actual physical phenomenon have been presented. In one case, one may think that the instant the electron and positron leave their starting point, they have assumed a definite spin; their spin merely remains unknown until measurement is taken. Another explanation states that the particles do not have a definite spin until a measurement is made. In other words, the positron and electron leave their starting point without deciding on a spin. This view would mean that if a measurement is taken after 100 light years on one particle, the outcome would have to travel at 00 times the speed of light to reach its complimentary particle so that it would know which spin state is opposite to occupy. The latter view has troubled numerous scientists, including Einstein, Podolsky, and Rosen. Einstein called it spooky action-at-adistance. Einstein, Podolsky, and Rosen (EPR) concluded that this interpretation of the physical phenomenon is absurd and must be invalid. EPR argued that information (or a signal) cannot travel faster than the speed of light. This idea is based on the principle of locality, which asserts that a particle s state is inherently defined, or localized, since information cannot travel faster than the speed of light to reach a partner particle. In addition, the possibility of reaching the corresponding particle with a non-relavistic speed (a speed that is much slower than the speed of light) would mean that it is possible to measure either spin up or spin down states on the second particle while the information from the first particle is traveling across space to reach the second. The latter assumption would mean that while the information is traveling to the second particle, the second particle may have either value of spin. Being able to detect either spin states on the second particle after the first is known would violate the conservation of angular momentum The particles will remain entangled (even for an eternity) until a force acts upon either one or both particles. 4
(i.e. the two particles cannot have the possibility of having the same spin). Therefore, the collapse of the wavefunction for the second particle must be instantaneous. The question left unanswered is that how one can determine the state of the particle before a measurement is taken. This very critique lead to Bell s famous theorem. III. Bells Inequality Some of quantum theory s harshest critics, including Einstein, Podolsky, and Rosen, did not refute the results obtained from quantum theory; they merely believed that additional information providing a more complete explanation of nature was possible. Quantum mechanics affords a good description of the possible states and their corresponding probabilities, such as the probability of one spin up electron and one spin down positron from a pion decay, or a range of possible positions and momentum values for a given particle. It does not, however, reveal which state the electron occupies upon immediate creation from the pion nor does it reveal a particle s position and momentum at the same instant. A large array of theories arose claiming that the missing details in knowing a state definitively (prior to experimentation) can be explained with variables that cannot be detected or have not yet been found. If these variables are defined, then the principle of locality is satisfied. The collective name given to these unknown parameters is local hidden variables. If these variables are found, then it would prove that quantum mechanics is indeed incomplete, as EPR suggested. On the other hand, if a set of experiments can show that no more further knowledge about a system is obtainable, then quantum mechanics is a complete theory. In the year 1964, John S. Bell proposed a theorem that would prove that any local hidden variable theory is incompatible with quantum mechanics. 4 Bell starts his theorem by looking at a generalized pion experiment. He supposed that instead of fixing the electron and positron detectors parallel to each other, they would be allowed to rotate so that they can become antiparallel or parallel randomly. Then, (the projection of) the angular momentum along the unit vectors parallel to each detector can be reported. For simplicity, the values are presented in units of h. A representative list of results from multiple pion decay trials are shown below: Electron Positron - Product (Spin e *Spin + e ) +1-1 -1 +1 +1 +1-1 +1-1 -1-1 +1 Next, Bell proposed to find the average value of the spin products. If the first detector has a unit vector a and the second detector has a unit vector b that is parallel to then b = a and the probability of the average outcome would be the product of the normally expected values of +1 and -1: Paa (, ) = 1 (3) If the detectors were in the anti-parallel alignment, then b = -a and the two possible outcomes are +1*+1 or -1*-1, where both would equal +1: Pa (, a) =+ 1 (4) The generalized result as predicted by quantum mechanics (from Eq. 3 and 4) is the negative dot product of the two vectors P( = a b. (5) Bell was able to show that Eq. 5 is incompatible with any local hidden variable theory. The localized theory states that the spin state of each particles is determined when the pair is created, and these states are independent of each Anti-parallel means having one detector up the other down; vs. parallel means having both detectors in the same direction. 5
other. ** Also, a hidden variable (or a set of hidden variables),, is included following the hypothesis that everything about the system can be known. Then there would be functions in the form of A and B (Eq. 6) that would predict the results at the detectors, which are represented by their vectors a and b. These functions can only have values of +1 or -1 for spin fermions (in units of h ): A ( = ± 1; B ( = ± 1. (6) When the detectors are parallel, the result is that a = -b: = B( ; = B(. (7) Equation 7 holds true for any, since one particle must be spin up and the other spin down. The average of the product of the measurements is shown in Eq. 8, where ( is the probability density of the hidden variable(s). = ( B( d (8) Using the second equation in Eq. 7, Eq. 8 becomes = ( d. (9) Next, suppose another detector vector c is introduced, then the difference between the probability of the product of a and b and a and c is the following: P( P( = ( [ c, ]d (10) = Since [ A ( ] 1, one can safely multiply this quantity into Eq. 10 to obtain ** This means that if one disturbs the spin state of the positron, the spin state of the electron remains unchanged. If one were to detect the spin state of the electron, one would be able to find the state of the positron before it was disturbed. In other words, they are not actually entangled. All notation follows closely to that found in Griffiths Introduction to Quantum Mechanics. P( P( = (. (11) c, ]d Rearranging Eq. 11 by factoring out A ( yields P( P( = ( [ 1 c, ] d. (1) From Eq. 6 and 7, it can be seen that [ ( ] 1 1 A + (13) since A ( and A ( can only be either +1 or -1. The product of A ( and A (, hence, has a maximum at +1 and a minimum at -1. The Ab (, Ac (, ) quantity also has a maximum at +1. Combined with the knowledge that the probability density of the hidden variable(s), ( ), is greater than or equal to +1, the following is true: [ ( c, ] 0 ( 1 A. (14) The Pab (, ) Pac (, ) term can be either positive or negative, depending on which probability is greater. If the absolute value of Pab (, ) Pac (, ) is taken, it must be greater than or equal to zero. Then, taking the maximum value that the combination of Eq. 13 and 14 produces, the absolute value of the probability difference must be [ [ c, ] d P( ( 1 ). (15) Notice that the negative sign from Eq. 1 is gone in Eq. 15. This is allows the = sign to become a sign. One can next separate the integral into P( ( d ( c, d. Prior to integration, it can be greater than 1. 6
(16) The maximum value of Eq. 16 is obtained when either A ( or A ( c, is negative, making the second integral positive. Also, the integral of ( ) is 1, so the expression can be rewritten as P( 1+ P( (17) This final form is known as Bell s inequality. It is possible to show that the quantum mechanical prediction is incompatible with Bell s inequality. Suppose that the system described above is composed of three vectors with the orientation shown in Figure 1. b c 45 o 45 o a FIG. 1. Sample orientation of the detectors used in proving Bell s hypothesis. Quantum mechanics (Eq. 5) would predict that = 0 ; = P( = 0. 707. When these quantities are substituted into Bell s inequality, one would obtain 0.707 (1 0.707) 0.707 0.93, which is incorrect. This shows that results from quantum mechanics are incompatible with theories involving local hidden variables. If local hidden variables exist, then quantum mechanics must be incomplete, if not wrong. On the contrary, if quantum mechanics is correct, then according to Bell s findings, there is no additional information that would provide locality for any given particle. IV. Experimental Proof Soon after Bell s paper, numerous experiments were performed to provide insight to the validity of quantum mechanics. The key results are summarized by Aspect, Grangier, and Roger in 198. 5 The systems examined in the Aspect et al. experiments were two-photon atomic transitions instead of pion decays, but the results show the same principle. The faint possibility of having one detector sensing the orientation of the corresponding detector is accounted by allowing the two detectors to orient quasirandomly after the photons are already in flight. Each set of results from the experiments concur that predictions made using quantum mechanics are correct. Consequently, this showed that hidden variables are non-existent and that a particle could not be localized. Hence, quantum mechanics is a complete theory. V. A World of Non-Locality The completeness of quantum mechanics proved that Einstein s worst nightmare about nature is true: the world, as we know it, is inherently non-local. This fact defies our every perceived notion that we can eventually determine everything about nature at all times. It is genuinely perplexing that reality remains purely probabilistic before a measurement is taken, at which time the probability of the measured state become one and that of all other states become zero instantaneously. Returning to the EPR paradox, which asks the question: Does information travel faster than the speed of light? If information is defined as a wave or a particle, then the answer is no, it does not. The message between two entangled particles is not made from matter, so it can travel faster than the speed of light. The connection between the two particles is a function, where the definite determination of one state would lead to the instantaneous collapse of the other possible state(s), because this is the behavior of the function. Quantum mechanics is introduced to explain the behavior of matter on the atomic scale. However, it is non-relavistic, meaning that it is not related to Einstein s theory of relativity, which is currently the most accurate description They used a random number generator. Random number generators are never truly random. 7
of the behavior of matter on the order of large molecules to galaxies. The next daunting task for theorists is to unify the theory of large and small objects to produce a theory of everything in the universe. In the recent years, there has been much excitement about the development of M-theory, which claims to finally connect quantum mechanics to relativity such that we would have a theory that can explain matter at any level of composition. Certain results to M-theory cannot yet be proven, such as the existence of 11 dimensions or that the big bang was a time when two parallel dimensions briefly collided and that it may reoccur. These claims cannot be directly proven, so there is much work in finding methods to indirectly prove the validity of M- theory. Bell s theorem silenced many doubts surrounding quantum theory. Few remaining scientists question the completeness of quantum mechanics, but this small minority s concern is centered on the possible existence of non-local hidden variables. The non-local hidden variables, even if found, would not refute the non-localized behavior of nature. Suppose that quantum mechanics is indeed complete, does this mark the end of the work for theoretical chemists and physicists? Despite having the knowledge to use quantum mechanics to describe atomic and subatomic systems, formulating the exact functions for each specific system is an extremely complicated task and solving these functions, if they can be found, is an entire field worthy of decades of research alone. Quantum mechanics provides us the tools to unravel the mysteries of atomic and subatomic matter. Some of the results are perplexing, since we are used to observing the behavior of much larger objects. If we were born the size of atomic or subatomic matter, then the probabilistic nature of those particles would be normal and the degree of certainty in the behavior of much larger things will seem spooky. Eventually, we will accept the quantum mechanical result that we cannot know everything about the Universe at all times.. http://www.upscale.utoronto.ca/generalinterest/h arrison/bellstheorem/bellstheorem.html 3. Einstein, A.; Podolsky, B.; Rosen, N. Phys. Rev. 1935, 47, 777. 4. Bell, J.S. Physics, 1964, 1, 195. 5. Aspect, A.; Grangier, P.; Roger, G. Phys. Rev. Lett. 198, 49, 91. 1. Griffiths, D.J. Introduction to Quantum Mechanics. Pearson Prentice Hall. 005. nd Ed. 8