Skip to main content

Exercises 8.8 Problems

1.

An electron-positron pair is in the state

\begin{equation*} \ket{\psi} = \frac{1}{\sqrt{10}}\ket{\uparrow\uparrow} +\frac{1}{\sqrt{5}}\ket{\uparrow\downarrow} + \frac{1}{\sqrt{5}}\ket{\downarrow\uparrow} + \frac{1}{\sqrt{2}}\ket{\downarrow\downarrow}\text{.} \end{equation*}

  1. Calculate the probability that a measurement of the electron \(S_z\) spin component will give the value \(+\hbar/2\text{.}\)

  2. For the same state \(\ket{\psi}\text{,}\) calculate the probability that a measurement of the positron \(S_z\) component will give the value \(-\hbar/2\text{.}\)

2.

Show that the state \(\ket{\psi}\) in (8.5) takes the form shown in (8.6), with positron states \(\ket{\phi_1}\) and \(\ket{\phi_2}\) given by (8.7).

3.

An electron-positron pair is in the state

\begin{equation*} \ket{\psi} = \frac{1}{\sqrt{10}}\ket{\uparrow\uparrow} +\frac{1}{\sqrt{5}}\ket{\uparrow\downarrow} + \frac{1}{\sqrt{5}}\ket{\downarrow\uparrow} + \frac{1}{\sqrt{2}}\ket{\downarrow\downarrow}\text{.} \end{equation*}

  1. Convert this state to the form \(\ket{\psi} = c_+\ket{\uparrow}\ket{\phi_1} + c_-\ket{\downarrow}\ket{\phi_2}\text{.}\) Determine the coefficients \(c_+\) and \(c_-\) and the normalized positron states \(\ket{\phi_1}\) and \(\ket{\phi_2}\text{.}\)

  2. Check that your coefficients satisfy \(|c_+|^2+|c_-|^2=1\text{.}\)

  3. Is your answer consistent with Problem 8.8.1(a)?

4.

An electron-positron pair is in the state

\begin{equation*} \ket{\psi} = 0.6 \ket{\uparrow} \ket{\phi_1} + 0.8 \ket{\downarrow} \ket{\phi_2} \end{equation*}

with positron states

\begin{equation*} \ket{\phi_1} = \sqrt{\frac{1}{3}} \ket{\uparrow} - \sqrt{\frac{2}{3}} \ket{\downarrow} \qquad \ket{\phi_2} = \sqrt{\frac{2}{3}} \ket{\uparrow} + \sqrt{\frac{1}{3}} \ket{\downarrow}\text{.} \end{equation*}

  1. What is the probability that a measurement of the \(z\)-component of the positron's spin will find a result \(S_z^\text{ pos } = +\hbar/2\text{?}\)

  2. You now measure the \(z\)-component of the electron's spin and find a value \(S_z^\text{ elec } = +\hbar/2\text{.}\) Write down the new state \(\ket{\psi_\text{ new } }\) of the electron-positron system immediately after this measurement.

  3. What is the probability that a measurement of the \(z\)-component of the positron's spin will now find a result of \(S_z^\text{ pos } = +\hbar/2\text{?}\) Is it the same as your answer to part (a)?

  4. Is the original state \(\ket{\psi}\) an entangled state? How can you tell?

5.

Construct a separable state that has all four coefficients in (8.2) not equal to zero.

6.

Two Bell states were provided in the reading, in Eqs. (8.13) and (8.14). Find two more Bell states.

7.

Let's verify the details for the quantum mechanics prediction for the Bell experiment with the positron detector rotated \(45^\circ\) from the \(+z\) direction. We'll use a simulation of the Stern-Gerlach Experiment.

  1. Go to https://phet.colorado.edu/sims/stern-gerlach/stern-gerlach_en.html

  2. In the simulation rotate the angle to \(45^\circ\text{,}\) and change the spin orientation to “\(+z\)”. You now have the simulation set up to take atoms with \(S_z\) = \(+\hbar/2\) and measure the spin of these particles along a \(45^\circ\) angle.

  3. To get a feel for how this works, try firing a few atoms. You'll notice a counter at the bottom of the page keeping track of how many atoms have been measured with spin up or spin down as measured along a \(45^\circ\) angle.

  4. Turn on “AutoFire” and keep it running until the percentages are no longer changing. (Increase the speed of the auto fire so that you don't have to wait forever.) Based on the results of the simulation, what is the probability of measuring \(S_{45^\circ}\) = \(+\hbar/2\text{?}\)

  5. Now, using (8.15), calculate the probability of measuring a particle to have \(\ket{\nearrow}\) if it started in the state \(\ket{\uparrow}\text{.}\) Do your results agree with those of the simulation?

8.

Einstein died in 1955, before Bell had derived his theorem and well before Aspect did the experiments that ruled out hidden variable theories. It is often stated that refusing to accept quantum mechanics is one of Einstein's mistakes. What do you think of that statement?

9.

Suppose we modified Bell's experiment and aligned the positron detector to be at a \(60^\circ\) angle with respect to the \(z\)-axis.

  1. In quantum mechanics, if the electron is measured to be spin down (\(S_z=-\hbar/2\)), what is the probability that the positron will be found to have \(S_{60^\circ} =+\hbar/2\text{?}\) There are two ways to approach this. You can use Eqs. (8.15) and (8.16). You can also use the same simulation as in Problem # Exercise 8.8.7.

  2. Now we turn to the hidden variable theory of Section 8.6. In this case, if the electron is measured to be spin down what is the maximum probability that the positron will be found to have \(S_{60^\circ}=+\hbar/2\text{?}\)

  3. Note the difference between the predictions of quantum mechanics versus the hidden variable theory in this case.

10.

Explain why hidden variable theories do not require “spooky action at a distance.”

11.

Show that the state

\begin{equation*} \ket{\psi} = \frac{1}{2}\ket{\uparrow\uparrow} -\frac{1}{2}\ket{\uparrow\downarrow} +\frac{1}{2}\ket{\downarrow\uparrow} -\frac{1}{2}\ket{\downarrow\downarrow} \end{equation*}

is separable and can be written as \(\ket{\psi} = \ket{\phi_\text{ electron } } \ket{\phi_\text{ positron } }\text{.}\) Find \(\ket{\phi_\text{ electron } }\) and \(\ket{\phi_\text{ positron } }\text{.}\)

12.

An electron-positron pair is in the state

\begin{equation*} \ket{\psi} = 0.5\ket{\uparrow\uparrow} + 0.1\ket{\uparrow\downarrow} + 0.7\ket{\downarrow\uparrow} + 0.5\ket{\downarrow\downarrow}\text{.} \end{equation*}

  1. Convert this state to the form \(\ket{\psi} = c_+\ket{\uparrow}\ket{\phi_1} + c_-\ket{\downarrow}\ket{\phi_2}\text{.}\) Determine the coefficients \(c_+\) and \(c_-\) and the normalized positron states \(\ket{\phi_1}\) and \(\ket{\phi_2}\text{.}\)

  2. Based on your answer to part (a), is \(\ket{\psi}\) entangled or not? Describe how you can tell.

13.

An electron-positron pair is in the state

\begin{equation*} \ket{\psi} = \frac{1}{\sqrt{10}}\ket{\uparrow\uparrow} +\frac{1}{\sqrt{5}}\ket{\uparrow\downarrow} + \frac{1}{\sqrt{5}}\ket{\downarrow\uparrow} + \frac{1}{\sqrt{2}}\ket{\downarrow\downarrow} \end{equation*}

which is the same state as in Problem 8.8.3.

  1. Determine the probability of a positron \(S_z\) measurement finding the value \(+\hbar/2\text{.}\)

  2. Starting from state \(\ket{\psi}\text{,}\) an electron \(S_z\) measurement has found the value \(+\hbar/2\text{.}\) Using your results from Problem 8.8.3, calculate the probability of the positron being spin up.

  3. Starting from state \(\ket{\psi}\text{,}\) an electron \(S_z\) measurement has found the value \(-\hbar/2\text{.}\) Using your results from Problem 8.8.3, calculate the probability of the positron being spin up.

14.

For each of the following states, determine whether they are separable or entangled.

  1. \(\displaystyle \ket{\uparrow\uparrow}\)

  2. \(\displaystyle \ket{\downarrow\downarrow}\)

  3. \(\displaystyle \displaystyle\frac{1}{\sqrt{2}}\ket{\uparrow\uparrow} + \frac{1}{\sqrt{2}} \ket{\downarrow\downarrow}\)

15.

Suppose we modified Bell's experiment and aligned the positron detector to be in the \(+x\) direction. That is, we will measure the \(S_x\) value for the positron.

  1. In quantum mechanics, if the electron is measured to be spin down (\(S_z=-\hbar/2\)), what are the probabilities that the positron will be found to have \(S_x=\pm\hbar/2\text{?}\)

  2. Now we turn to the hidden variable theory of Section 8.6. In this case, if the electron is measured to be spin down what are the probabilities that the positron will be found to have \(S_x=\pm\hbar/2\text{?}\)

  3. Will this modification of Bell's experiment allow us to distinguish between quantum mechanics and our hidden variable theory?