Bell's theorem/Introduction
Bell's theorem/Introduction Bell's theorem/Probability Bell's theorem/Inequality
What is a Bell's theorem experiment? edit
Bell's theorem can be investigated by studying a pair of two particles that are simultaneously created by the decay of one object that we shall call an "atom"^{[1]}. Properties of both particles are measured some distance from where they were created. Experiments to investigate Bell's theorem have been performed on protons, neutrons, and electrons, over distances up to a few kilometers. We shall consider a single atom that simultaneously emits two photons that are detected in a way that also determines whether the photon would pass through a (linear) polarizing filter oriented at a given angle.^{[2]}. In some experiments the orientation of the polarization axis is rapidly and randomly switched in order to preclude the possibility of any "communication" between the photons (i.e., one photon cannot "know" what measurement is being made on the other). Some say this experiment fundamentally changes our notion of cause and effect, others argue that such challenges have been raised since 1909.
 See also A brief introduction to Bell's theorem (Wikipedia) (made brief by omitting most of the article). See here for a classroom activity on lede writing.
Entangled photons and the meaning of "remote" and "local" edit
The word entangled is popular and descriptive word for two photons that are in a mixed quantum state. It is convenient for us to label them as the "remote" and "local" photon, although these are arbitrary labels designed to make it easier to sort out the algebra. The analysis is unchanged if the "remote" and "local" measuring devices are interchanged.
Polarized light and photons edit
The internet has many excellent resources on linearly polarized light, for example:
 Wikipedia:Polarization_(waves)#Polarization_state
 Khan Academy: Polarization of lightlinear and circular
 Physics Classroom  Polarization
 What is a photon?
 A photon is a quantum, or "bundle" of energy associated with light. See physics.about.com.
 What is the component of a vector along a line?
 This is also called the projection of a vector onto a line. Vector A projected onto line b is Acosθ, as shown in the figure. This projection is also called a "component" of the vector along the line.
 How do photons behave when they strike a linear polarized filter?
 The filter blocks the component perpendicular to the filter's orientation and passes the component parallel to the filter's orientation
 How can energy be used to calculate a photon's probability of passing a filter?^{[3]}
 The total energy is the energy of one photon times the number of photons (often measured as energy per second, or power). We use this fact to calculate the probability that a single photon with a given orientation will pass through a filter with a different orientation. This probability depends only on the angle between the photon's orientation and the filter's orientation (this analysis is structured in such a way that only linearly polarized light needs to be considered^{[4]}.
 Such calculations of probability are usually associated with quantum mechanics, but in this case energy arguments give us almost^{[5]} everything we need.
 Key idea: We can use energy to convert knowledge about how a classical wave loses intensity as it passes through a polarizing filter into knowledge about the probability that an individual photon will pass through the filter.
Lab activity edit
This lab requires three low cost linear filters. First play with two until you understand how the electric field is reduced as you rotate the filters. Then, take the third filter and use vector diagrams (not just Mathus formula) to explain what happens when you insert a filter at 45° between two crossed filters as shown in the figure to the right. Start by drawing a rather long vector, and take a projection along a dotted line at 45 degrees to that arrow. Then, make another dotted line that is perpendicular to the first vector, and take that projection to find the electric field associated with the light emerging from the third filter. Measure the lengths of the first and third vectors and calculate the percent error between your drawing and the length that you calculate using geometry. It's faster and more fun if draw everything freehand, without a ruler.
Students in a physics class looked at Wikipedia:Polarization (waves) and found the diagrams illustrating the effect of a polarized filter a bit "excessive". The effort above and to the left represents the best effort made so far. If you wish to make a better illustration please do so and "publish" the improved drawing on commons. By copyright law, all use of your drawing and it's derivatives must be accredited to you. Click here if you need help.
Gallery recent images that need to be explained and/or improved edit
Wikiversity encourages student contributions, even images. The images shown below are svg and therefore editable. The students who sketched these will share credit with those who edit and improve them.

Summary on Commons. Note release to public domain

Shows reduction in the number of photons passing two filters

Recent commons file highlights the paradoxical nature of fasterthanlight information flow.
Using classical wave energy to calculate a quantum probability edit
It is possible to calculate most of the relevant quantum probabilities using conservation of energy.
Once you have the classical theory understood, use an energy argument to calculate the probabilities. Let denote probability:
where "pass" refers to the probability of a photon being transmitted through the filter.
Electromagnetic wave energy is proportional to the electric field squared (see Wikipedia: Simple harmonic motion and Electric field). The fraction of energy that passes a polarizing filter can be found by taking the projections described above. Since these photons carry identical units of energy, the probability of a photon passing a filter is directly proportional to the fraction of wave energy that passes.
Probability of passing at 30 and 60 degrees edit
For our purposes all we need is the probability that a photon passes a filter with only two orientations relative to the photon's polarization: 30° and 60°. Using the Pythagorean theorem and squaring the ratio of wave amplitudes, we have:
 ...and...
These equations will play a key role in our derivation of Bell's inequality for the special case of three polarization measurements taken 60° apart. This result can also be obtained using quantum theory.
The (linearly) polarized filter reduces the electric field, by a factor of cosθ, so that the number of photons per second is reduced by a factor of cos^{2}θ Hence,
where is the angle between the polarized filter and the polarization of the incident (incoming) light. If the incident light is unpolarized, then P(pass) = 1/2
Conditional probability edit
We need to calculate the probability that a photon of random polarization passes through two consecutive filters. The formula is sometimes called Bayes' theorem. This probability is:
where P(A) is the probability of passing filter A, and P(BA) is the probability of passing filter B, given that it has already passed filter A. Also it is necessary to know that a linear polarizing filter passes half the light energy if the incoming light is unpolarized, and that the entangled photons have perpendicular polarization whenever they are measured.^{[6]}
 See also Wikibooks:Probability/Conditional Probability#Even_and_prime_numbers ... (or this permalink)
 See Wikipedia:Probability#Summary_of_probabilities for this and other basic formulas of probability.
If you think you understand most of this you are ready for Bell's theorem/Probability edit
Bell's theorem/Introduction Bell's theorem/Probability Bell's theorem/Inequality
References and footnotes edit
 ↑ The "atom" might be the annihilation of an electron positron pair.
 ↑ Actual experiments don't use the polarizing filters depicted in the figure.
 ↑ See also Wikipedia:Photon_polarization#Energy_density_of_classical_electromagnetic_waves
 ↑ In other words we need not consider elliptical or circular polarization. This restriction is possible because the first measurement (typically on the "remote" photon) is made with a linear filter. This "forces" (whatever that means) the other photon to be also linearly polarized.
 ↑ One result from quantum mechanics is needed: Both theoretical calculation, as well as experiment, have established that the two entangled photons are always orthogonal (in the generalized sense, see w:Orthogonality#Orthogonal_states_in_quantum_mechanics.
 ↑ "...whenever they are measured" sounds odd. But quantum mechanics suggests that variables do not have numerical value until they are measured.
Bell's theorem/Introduction Bell's theorem/Probability Bell's theorem/Inequality sandbox