Hidden Variable Theory

Transcription

1 Hidden Variable Theory Quantum Mechanics II February 7th, 2018 Maine Christos Department of Physics and Astronomy 1

2 In non-deterministic quantum mechanics... Both QM and classical physics seek to compute the future state of a system conditioned on past events which affected the system. In classical mechanics, we can accurately predict the future state of the system. We can throw away the probability distribution which disagrees with our observations. In this version of QM, we CANNOT assume this since we cannot know the initial state of every variable. Uncertainty principle. HOWEVER, quantum probabilities conditioned on macroscopic collective coordinates obey Bayes Law with fluctuations on the order of [2]. We may apply Bayes rule to collapsed wavefunction only if we are talking about almost classical values. In non-deterministic QM, there is no escaping randomness (i.e. no hidden variables) 2

3 Local vs. non-local hidden variables Hidden Variable Theories would allow for a deterministic interpretation of QM. There are some hidden variables which if measured would allow us to predict the histories of the system. Local Variables Theory Only variables at points which are close (time-like separated) can interact. Non-local Variable Theory Necessarily allow points which are space-like separated to interact. Operators which describe particles which are space-like separated will always commute [1]. Bell s Inequality Definitively rules out any local theory of hidden variables. Any local realistic theory of quantum mechanics disagrees with the statistical predictions of QM. 3

4 De Broglie-Bohm mechanics We begin with Schro. Eq: And choose a (through a change in variables) to write our wave equation as: We can then introduce a set of n trajectories. Every possible trajectory is represented by a different Q k with 0<k<=n. Combining with the Schro. Eq., we have: 4

5 De Broglie-Bohm mechanics This equation satisfies Hamilton-Jacobi equation (after setting hbar=1): So then these equations describe a probability distribution over all possible solutions Q k (0) of S. HOWEVER, these equations give us non-normalizable probability functions when we plug in real-world initial conditions. One possible solution is to assume the wave functions are real wave functions in space, the Q k (t)s describe Madelung trajectories of quantum hydro-dynamics and particle surfs wave of one of these trajectories. But what places particle on a specific trajectory Q k? The only way to consistently pick the trajectory which matches QM is by choosing according to the absolute square of a wave function! 5

6 De Broglie-Bohm mechanics 6

7 The case against non-local hidden variables QM predictions are already correct. Any hidden variable theory would have to match the accuracy of the predictions of quantum mechanics. Theories like Bohmian Mechanics criticized as artificial and contrived De Broglie-Bohm Mechanics was specifically formulated to meet the observational requirements of QM. Most natural interpretation of De Broglie-Bohm Mechanics doesn t predict particles. Forcing a theory of hidden variables leads to a non-intuitive interpretation of QM [2]. Several classes on non-local hidden variables have been ruled out. Tend to be more complicated Requires bizarre non-local interactions 7

8 References [1] Gröblacher, S., Paterek, T., Kaltenbaek, R., Brukner, Č., Żukowski, M., Aspelmeyer, M., & Zeilinger, A. (2007). An experimental test of non-local realism. Nature, 446(7138), [2] Banks, T. (2017). QM Fall 2017 [3] Contopoulos, G., Delis, N., & Efthymiopoulos, C. (2012). Order in de Broglie Bohm quantum mechanics. Journal of Physics A: Mathematical and Theoretical, 45(16),