Bell changed all that. To everyone's great surprise, in 1964 he constructed a thought experiment where quantum mechanics and hidden variable theories actually predict different results. This thought experiment suddenly made it an experimental question, so nature could tell us who was right!
Bell's clever idea rested on a variation of the EPR thought experiment where the Stern-Gerlach device that is measuring the positron spin is rotated. The electron's spin is still measured the same way, so we will find \(S_z = +\hbar/2\) or \(-\hbar/2\) as before. But the positron's spin is measured along some direction \(\hat n\) that makes an angle \(\theta\) with respect to the \(z\) axis, as shown in Figure 8.4. For simplicity, we will take \(\theta = 45^\circ\text{.}\)
Figure8.4.The Bell thought experiment, which is the same as the EPR experiment except that the positron Stern-Gerlach detector is rotated.
Subsection8.6.1QM Prediction for the Bell Experiment
If we rotate our positron detector to be \(45^\circ\) from the \(z\)-axis and measure this component of the spin, call it \(S_{45^\circ}\text{,}\) what possible values will we find? Just like with any spin component, we'll find \(+\hbar/2\) and \(-\hbar/2\text{.}\) And after the measurement, the spin will be in either the \(\ket{\nearrow}\) state or the \(\ket{\swarrow}\) state, just as an \(S_x\) measurement results in either the \(\ket{+x}\) or \(\ket{-x}\) states.
We can relate our familiar up and down spins to this new pair of states:
where \(\theta\) is measured from the \(z\)-axis as shown in Figure 8.4.
We'll create an electron-positron pair in the initial state given by (8.14). Let's explore what quantum mechanics predicts for the Bell Experiment. We're going to consider the case where an observer measures \(S_z^\text{ elec }\) to be spin down. According to quantum mechanics, a measurement of \(S_z^\text{ elec }\) = \(-\hbar/2\) collapses the state of our electron-positron pair to be \(\ket{\psi} = \ket{\downarrow\uparrow}\text{.}\) To put it another way, if we measure the state of the electron to be \(\ket{\downarrow}\text{,}\) quantum entanglement says that the state of the positron must be \(\ket{\uparrow}\text{.}\) The subsequent positron measurement will find \(S_{45^\circ}\) to be either \(+\hbar/2\) or \(-\hbar/2\) with probabilities that depend on the \(b_+\) and \(b_-\) variables. Since the positron state is \(\ket{\uparrow}\text{,}\) the probability of measuring \(S_{45^\circ}^\text{ pos }\) = \(+\hbar/2\) is \(\left|b_+\right|^2\) and the probability of measuring \(S_{45^\circ}^\text{ pos }\) = \(-\hbar/2\) is \(\left|b_-\right|^2\text{.}\) In your homework, you will calculate these probabilities.
Subsection8.6.2Hidden Variable Prediction for the Bell Experiment
Now here is where it gets interesting: Bell constructed a proof that any hidden variable theory 1 with the detectors oriented as in Figure 8.4 and with \(\theta=45^\circ\) will result in probabilities that are incompatible with the predictions of quantum mechanics. So, either the hidden variable theory or quantum mechanics must be wrong!
While we won't provide a proof of Bell's theorem in full generality here, we will show how to determine the predictions of hidden variable theory introduced in Example 8.3 of Section 8.5. According to this hidden variable theory, the electron and positron have fully determined spin values \(\vec S^\text{ elec } = -\vec S^\text{ pos }\text{,}\) but we just don't know them.
Figure8.5.The circle represents all possible \(\vec S^ pos\) directions. After a measurement of \(S_z^ elec =-\hbar/2\text{,}\) the possible directions of \(\vec S^ pos\) are limited to the upper hemisphere (shaded region).
Picture the positron spin variable \(\vec S^\text{ pos }\text{.}\) Since we don't know its value and we aren't in any way controlling how it is set when the electron-positron pair are created, it is reasonable to assume it is equally likely to be pointing in any direction. We can represent all possible directions of \(\vec S^\text{ pos }\) by the surface of a sphere, where the vector \(\vec S\) has its tail on the origin. A circular cross section of this sphere is shown in Figure 8.5. The dashed line in the figure is the boundary between the positive and negative values for \(S_{45^\circ}^\text{ pos }\text{.}\) That is, for spins \(\vec S^\text{ pos }\) above and to the right of the dashed line a measurement would result in \(S_{45^\circ}^\text{ pos }\) = \(+\hbar/2\text{.}\) The region below and to the left corresponds to \(S_{45^\circ}^\text{ pos }\) = \(-\hbar/2\text{.}\)
Now consider the case where we have measured the electron's \(z\)-component of spin and found \(S_z^\text{ elec }\) = \(-\hbar/2\text{.}\) This happens half the time, and when it happens the positron \(\vec S^\text{ pos }\) must be somewhere in the upper hemisphere, shown as the shaded region in Figure 8.5. This agrees with the QM prediction so far, where if we measure \(S_z^\text{ elec }\) = \(-\hbar/2\text{,}\) we know that we would also measure \(S_z^\text{ pos }\) = \(+\hbar/2\text{.}\) But, here, rather than the positron actually being in a spin up state, the hidden variable theory says that the original \(\vec S^\text{ pos }\) should determine the probabilities for measuring \(S_{45^\circ}^\text{ pos }\) as spin up or down. According to the figure, a fraction 3/4 of the shaded region results in a value \(S_{45^\circ}^\text{ pos }\) = \(+\hbar/2\text{.}\) Bell constructed a proof that for any hidden variable theory, this fraction of 3/4 is the maximum probability of measuring \(S_{45^\circ}^\text{ pos }\) = \(+\hbar/2\text{.}\)
So, let's recap. We create an electron-positron pair in the state:
We make a measurement of the \(z\)-component of the electron's spin, and we find that the electron is spin down. Quantum mechanics says that the positron is thus in the state \(\ket{\uparrow}\text{,}\) which means the probability of measuring \(S_{45^\circ}^\text{ pos }\) = \(+\hbar/2\) is \(\left|b_+\right|^2\text{,}\) or 85%. The hidden variable theory says that the positron spin is determined by its original \(\vec S^\text{ pos }\text{,}\) and we have a \(\le\)75% probability of measuring \(S_{45^\circ}^\text{ pos }\) = \(+\hbar/2\text{.}\)
… so which is right?
Technically, any local hidden variable theory, to satisfy the lawyers and mathematicians.
