Q8 Lecture. State of Quantum Mechanics EPR Paradox Bell s Thm. Physics 201: Lecture 1, Pg 1

Transcription

1 Physics 56: Lecture Q8 Lecture State of Quantum Mechanics EPR Paradox Bell s Thm Physics 01: Lecture 1, Pg 1

2 Question Richard Feynman said, [the double-slit experiment] has in it the heart of quantum mechanics; in reality it contains the only mystery. Can you tell what mystery Feynman is talking about? Physics 01: Lecture 1, Pg

3 Quantum Mechanics Even physicists who accept QM disagree about how to interpret the rules Physics 01: Lecture 1, Pg 3

4 Quantum Indeterminacy Even if we know all there is to know (wavefunction) we cannot predict the result of a simple experiment to measure position Question: Suppose I measure the position of a particle to be at position x i. Where was it prior to the measurement? A. Realist: It was really at x i (QM is incomplete) B. Orthodox (Copenhagen Interpretation): The particle wasn t anywhere. The act of measurement forced the particle to take a stand. Physics 01: Lecture 1, Pg 4

5 Einstein and Objective Reality Einstein was never comfortable with quantum mechanics He felt that properties of physical objects have an objective reality independent of measurement In the orthodox interpretation, we cannot say a quanton measured to be spin up had that property before the measurement In fact, we can only know one spin component of the particle, because measurement of one component disturbs knowledge of the others Therefore Einstein thought QM was an incomplete description of reality Physics 01: Lecture 1, Pg 5

6 Two Views of Quantum Mechanics Realist: Quantum mechanics must be incomplete. There is such a thing as objective reality. Orthodox/Copenhagen: There is nothing wrong with quantum mechanics. Objective reality does not exist. Physics 01: Lecture 1, Pg 6

7 Locality Locality (rough version): Only things close to me can affect me Locality: No influence may travel faster than the speed of light Physics 01: Lecture 1, Pg 7

8 EPR Paradox (1935) Source of Spin 0 decays into an electron and positron To conserve spin one must have spin up one must have spin down (quantons are entangled) The state vector is some linear combination of z A z B z A z B Physics 01: Lecture 1, Pg 8

9 Einstein-Podolsky-Rosen Paradox If you measure spin up at A you are determined to have spin down at B Locality: A cannot affect B (at least not faster than light) The particle at B must have been spin down all along, it must be an element of reality And QM is just not good enough to predict this (incomplete) Physics 01: Lecture 1, Pg 9

10 EPR Paradox Einstein: Since we cannot have a non-local influence, the outcomes must be pre-determined by objectively real properties (hidden variables) that t each electron carries with it Einstein: Since quantum mechanics does not describe these properties it must be incomplete Problem: The results of the EPR paradox are the same regardless of your interpretation Physics 01: Lecture 1, Pg 10

11 Bell s Theorem Provided a way to differentiate between a local hidden variable theory (LHVT) and QM Bell s Theorem: No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics. There are specific measurements that can be made to distinguish between QM and LHVT Experiments can now show whether QM is wrong, or if the world is inherently non-local Physics 01: Lecture 1, Pg 11

12 Bell s Theorem Bell s Theorem: No physical theory of local hidden variables (LHVT) can ever reproduce all of the predictions of quantum mechanics. There are specific measurements that can be made to distinguish between QM and LHVT Experiments can now show whether QM is incomplete, or if the world is inherently non-local Physics 01: Lecture 1, Pg 1

13 Experiment Source of Spin 0 decays into an electron and positron Now experimenters measure the spin along one of three axis at random (a, b or c: 10 degrees apart) Each measurement will return red (R) or green light (G) We will look at the combination of measurements The Same: RR or GG 1 Different: RG or GR 3 R G G R Physics 01: Lecture 1, Pg 13

14 Instruction Sets For the local hidden variable ab theories es (LHVT) the results s of the experiments are already known when the particles leave the source. There are eight ways this could be. 1 3 R G G R Physics 01: Lecture 1, Pg 14

15 Clicker Question 1: Here eeae are the eeg eight possbeco possible combinations o sof spins sthat coud could result from our decay (if there are hidden variables). What should the three highlighted boxes read? (a) (b) (c) (d) (e) SSS DSS SDS SDD DDS Physics 01: Lecture 1, Pg 15

16 Clicker Question : Here are the eight possible combinations of spins that could result from our decay (if there are hidden variables). If we run the experiment with a large number of electrons what is the hidden variable probability of getting Same? (a) Pr(S) = 1 (b) Pr(S) = 5/9 (c) Pr(S) 5/9 (d) Pr(S) 5/9 (e) P(S) Pr(S) 5/9 Physics 01: Lecture 1, Pg 16

17 LHVT: Probabilities Bell inequalities No matter how one adjusts the populations p the results will be consistent with these inequalities Pr 4 D 9 Pr S 5 9 Physics 01: Lecture 1, Pg 17

18 Quantum Mechanics Now we can calculate the probabilities according to QM Two possibilities: The two axis are the same. The two axis are 10 degrees off of each other Physics 01: Lecture 1, Pg 18

19 Clicker Question 3: If the two axis are the same what results will we have? (a) (b) (c) The Same Different It Depends A = 0 B = 0 R G G R Physics 01: Lecture 1, Pg 19

20 Clicker Question 4: The detector at A is in the z-direction and the detector at B is 10 degrees off from this, suppose you measure spin up at A right before the measurement at B is made. What is the spin state vector at B before a measurement is made? ( a) cos sin ( b) sin cos 1 0 (c ) (d) 0 1 A = 0 B = 10 ẑ ˆ xˆ R G G R Physics 01: Lecture 1, Pg 0

21 Clicker Question 5: If the two axis are 10 degrees off of each other, what is the probability you will get a Different result? (a) ( c) sin cos cos (b) sin cos 10 ( d) sin 10 A = 0 B = 10 cos sin sin cos R G G R Physics 01: Lecture 1, Pg 1

22 QM: Probabilities The angle between measurements will be 0 1/3 of the time and 10 /3 of the time Therefore QM makes the following predictions Pr Pr D S 1 3 sin 3 cos These predictions are inconsistent with those of the LHVT It is now an experimental question Physics 01: Lecture 1, Pg

23 Results These type of experiments have been carried out and consistently agree with the predictions of quantum mechanics and thus seem to rule out a local hidden variable theory Quantum mechanics is inherently non-local This non-localities that are at the foundations of quantum information and quantum computing How can measuring A collapse the wave function at B? Doesn t this violate relativity? No. Because no information can be transmitted via this collapse Physics 01: Lecture 1, Pg 3

24 Clicker Question 6: Experiments have been carried out and consistently agree with the predictions of quantum mechanics and thus seem to rule out a local hidden variable theory. Does this mean the Copenhagen interpretation is correct? A. Yes B. No Physics 01: Lecture 1, Pg 4

25 Schrodinger s Cat Physics 01: Lecture 1, Pg 5

26 The Collapse Problem Ordinary: Wave function evolves in a leisurely fashion according to the S.E. TISE leads to the time-evolution e o rule Measurement: Wave function suddenly collapses around one point. Physics 01: Lecture 1, Pg 6

27 PCQ: Which do you personally think is the weirdest part of QM and why? (a) (b) (c) (d) It does not support objectively real quanton properties It requires possibly superluminal connections It has two incompatible time evolution rules There seems to be some kind of unexplained disconnect between microscopic and macroscopic physics Physics 01: Lecture 1, Pg 7

28 And Still Some Loose Ends Physics 01: Lecture 1, Pg 8