Uncertainty Relations

Abstract

In this chapter, we show how uncertainty relations arise naturally from quantum theory for position and momentum, time and energy, and so on. We explore the meaning of these relations and apply them to the quantum mechanical description of the pendulum. Finally, we show how we can use entanglement to enhance the precision of measurements.

Exercises

• 1. Show that the variance of A vanishes when $|{\psi }\rangle$ is an eigenstate of A.

• 2. Calculate the maximum time precision for a 1.0 kg pendulum of length 1.0 m and amplitude 10  cm. Is the precision of a grandfather clock restricted by the laws of quantum mechanics?

• 3. Show that the precision of a clock cannot be improved by increasing the size of the face plate.

• 4. We can create entangle ment by using a quantum erasure protocol. Consider two identical atoms with low lying energy states $|{\mathsf {0}}\rangle$ and $|{\mathsf {1}}\rangle$, and an excited state $|{e}\rangle$ that is coupled to $|{\mathsf {1}}\rangle$. We can generate a photon if the atoms is in state $|{\mathsf {1}}\rangle$ by exciting the atom to $|{e}\rangle$ with a laser, followed by the spontaneous emission of a photon.

  • (a) both atoms are prepared in the state $(|{\mathsf {0}}\rangle +|{\mathsf {1}}\rangle )/\sqrt{2}$. Write down the state of the two atoms and any emitted photons after we excite both atoms and let them spontaneously emit a photon.

  • (b) The modes that may contain photons are mixed on a beam splitter. What is the state of the atoms if we detect exactly one photon in the output modes of the beam splitter? You may assume our detectors are perfect.

  • (c) What is the probability that we find exactly one photon after the beam splitter?

• 5. Prove Eqs. (9.38) and (9.41).

• 6. Show that the momentum and the total energy can be measured simultaneously only when the potential is constant everywhere. What does a constant potential mean in terms of the dynamics of a particle?

• 7. A coherent state of a pendulum is defined by Eq. (9.56), and has a complex amplitude $\alpha$. Calculate the expectation value for the position and momentum of the pendulum. When we write $\alpha$ in polar coordinates, such that $\alpha = r e^{i\phi }$, give a physical interpretation of $\phi$.

• 8 Prove that the ladder operators obey the commutation relation $[\hat{a},\hat{a}^\dagger ]=1$.