Electrons with Spin

Abstract

In this chapter, we describe a simple experiment about the spin property of an electron and use it to determine what is a physical observable in quantum mechanics. We give a general description of physical systems with two distinct states. We conclude this chapter with a brief description how spin can be used in magnetic resonance imaging, a technique that is used in hospitals to reveal what is going on inside your brain.

Exercises

• 1. The state of an electron spin is given by

  $$ |{\psi }\rangle = \frac{1}{\sqrt{3}} |{\uparrow }\rangle + \sqrt{\frac{2}{3}} |{\downarrow }\rangle \, . $$

  What is the probability of finding spin $\downarrow$ in a measurement outcome? What is the probability of finding measurement outcome “+”, with $|+\rangle = (|{\uparrow }\rangle + |{\downarrow }\rangle )/\sqrt{2}$?

• 2. Normalise the state $2|{\uparrow }\rangle + 4|{\downarrow }\rangle$.

• 3. An electron is prepared in the spin state $2|{\uparrow }\rangle -3i|{\downarrow }\rangle$. Normalise this state and calculate the probability of finding spin “up” and spin “$+$”, corresponding to $|{\uparrow }\rangle$ and $|{+}\rangle = (|{\uparrow }\rangle + |{\downarrow }\rangle )/\sqrt{2}$, respectively. What is the expectation value of the z-component of the spin?

• 4. Construct the matrix form of $S_y$ similar to Eq. (3.23).

• 5. Give the matrix representation of the observable $S_\phi = \cos \phi \; S_x + \sin \phi \; S_y$. Calculate the expectation value of $S_\phi$ given the spin state

  $$ |{\psi }\rangle = \frac{1}{\sqrt{3}} |{\uparrow }\rangle + \frac{\sqrt{2}\, e^{i\pi /6}}{\sqrt{3}} |{\downarrow }\rangle \, , $$

  with $|{\uparrow }\rangle$ and $|{\downarrow }\rangle$ the spin up and down states in the z-direction. What is the uncertainty $\Delta S_\phi$?

• 6. For the state in Exercise 5, calculate the probabilities of finding spin “up” and spin “down” in the z-direction.

• 7. Determine the angles $\theta$ and $\phi$ for which the state in Exercise 5 is the quantum state associated with the “up” direction.

• 8. Calculate the uncertainty $\Delta S_\theta$ for the operator $S_\theta = \cos \theta \; S_z + \sin \theta \; S_x$ given the spin state $|{\uparrow }\rangle$. Does your result conform to your expectation?

• 9. In Eq. (3.45) we have covered the Bloch sphere using two angles. However, we have covered two special points on the sphere too many times. Can you tell which points? Does it matter in this case? Hint: remember that a global phase is unobservable.

• 10. Construct a matrix form of the observable associated with the path in a Mach–Zehnder interferometer where we find the photon. You will need to take special care in choosing the measurement values.

• 11. We prepare an electron spin state in the direction $(\theta ,\phi )$, which can be written as

  $$ \frac{\hbar }{2} \begin{pmatrix} \sin \theta \cos \phi \\ \sin \theta \sin \phi \\ \cos \theta \end{pmatrix} = \frac{\hbar }{2} \begin{pmatrix} 0.433 \\ 0.750 \\ 0.500 \end{pmatrix} . $$

  What is the quantum state of the electron?

• 12. For the electron spin state of the previous question, relate the probabilities of finding outcomes $+$ and − in the x-direction to the projection onto the x-axis.

• 13. An electron with spin state

  $$ |\psi \rangle = \frac{3}{5} |{\uparrow }\rangle + \frac{4}{5} |{\downarrow }\rangle $$

  has its spin measured in the x-direction. What is the expectation value $\langle S_x\rangle$? What is the uncertainty $\Delta S_x$?

• 14. An electron with spin state

  $$ |\psi \rangle = \frac{3}{5} |{\uparrow }\rangle + \frac{4}{5} |{\downarrow }\rangle $$

  has its spin measured in the y-direction. What is the expectation value $\langle S_y\rangle$? What is the uncertainty $\Delta S_y$?

• 15. An electron with spin state

  $$ |\psi \rangle = \frac{3}{5} |{\uparrow }\rangle + \frac{4}{5} |{\downarrow }\rangle $$

  has its spin measured in the z-direction. What is the expectation value $\langle S_z\rangle$? What is the uncertainty $\Delta S_z$?

• 16. The state of an electron spin is given by

  $$ |\psi \rangle = \frac{2}{\sqrt{13}} |{\uparrow }\rangle + \frac{3i}{\sqrt{13}} |{\downarrow }\rangle \, . $$

  Calculate $\langle S_\theta \rangle$ and $\Delta S_\theta$.