Atoms and Energy

Abstract

In this chapter, we look at the energy of electrons in an atom and determine how the quantum state of the atom changes over time. This leads us to introduce the energy operator, or Hamiltonian, and the Schrödinger equation that governs the behaviour of quantum systems. We show how light can interact with an atom, and as a demonstration of the quantum theory developed so far we explain how atomic clocks work.

Exercises

• 1. Normalise the state $3|g\rangle -i|e\rangle .$

• 2. An atomic state

  $$ |\psi \rangle = \frac{1}{\sqrt{3}} |g\rangle + \sqrt{\frac{2}{3}} |e\rangle $$

  evolves with an evolution matrix $\begin{pmatrix} e^{-i\omega t/2} &{} 0 \\ 0 &{} e^{i\omega t/2} \end{pmatrix}.$ Calculate the state $|\psi (t)\rangle$ at time t. Calculate the probability of finding $|+\rangle = (|g\rangle +|e\rangle )/\sqrt{2}$ at time t.

• 3. A two-level atom with ground state $|g\rangle$ and excited state $|e\rangle$ is prepared in the state

  $$|+\rangle = \frac{1}{\sqrt{2}}|g\rangle + \frac{1}{\sqrt{2}}|e\rangle $$

  at time $t=0.$ The energy difference between $|g\rangle$ and $|e\rangle$ is $\hbar \omega .$

  • (a) Calculate the probability of finding the atom in the state

    $$|R\rangle = \frac{1}{\sqrt{2}}|g\rangle + \frac{i}{\sqrt{2}}|e\rangle \, . $$

  • (b) The atom evolves according to the Schrödinger equation with the Hamiltonian

    $$ H = \frac{\hbar \omega }{2} \begin{pmatrix} 1 &{}\quad 0 \\ 0 &{}\quad -1 \end{pmatrix} . $$

    Calculate the probability of finding the atom in the state $|R\rangle$ at time t.

  • (c) Sketch this probability as a function of time.

  • (d) Sketch the trajectory of the quantum state over time in the Bloch sphere.

• 4. An electron with spin 1 / 2 is prepared in the state $|\uparrow \rangle$ in the z-direction. A magnetic field is pointing $45^\circ$ away from the z-axis and towards the x-axis.

  • (a) Using Eq. (3.63), construct the matrix form of the Hamiltonian H in the Schrödinger equation.

  • (b) Show that the states

    $$\begin{aligned} |+45\rangle&= \cos \left( \frac{\pi }{8}\right) |\uparrow \rangle + \sin \left( \frac{\pi }{8}\right) |\downarrow \rangle \\ |+135\rangle&= - \sin \left( \frac{\pi }{8}\right) |\uparrow \rangle + \cos \left( \frac{\pi }{8}\right) |\downarrow \rangle \end{aligned}$$

    are eigenstates of the matrix H. What are the corresponding eigenvalues?

  • (c) The eigenstates form the poles of the rotation axis determined by H. Sketch the time trajectory of the electron spin in the Bloch sphere.

  • (d) Does the electron spin ever reach the state $|\downarrow \rangle$?

• 5. At time $t=0$ an atom is prepared in the state

  $$ |\psi \rangle = \frac{|g\rangle - i\sqrt{2}|e\rangle }{\sqrt{3}}\, .$$

  • (a) Calculate the probability of finding the measurement outcome “+” and “–”, associated with the states $|\pm \rangle = (|g\rangle \pm |e\rangle )/\sqrt{2}.$

  The free evolution of the atom is determined by the Hamiltonian

  $$ H = -\frac{1}{2}\hbar \omega \left( |g\rangle \langle g| - |e\rangle \langle e|\right) . $$

  • (b) Calculate the state at time t.

  • (c) Sketch the rotation axis of the evolution due to $H'$ on the Bloch sphere.

• 6.

  • (a) A photon in a polarisation state $|\psi \rangle = \alpha |H\rangle + \beta |V\rangle$ with $|\alpha |^2+|\beta |^2=1$ is measured in the polarisation basis $\{|H\rangle ,|V\rangle \}.$ What are the possible measurement outcomes and their corresponding probabilities?

  • (b) If we measure an identically prepared photon in the circular polarisation basis $(|H\rangle \pm i|V\rangle )/\sqrt{2}$, what will be the measurement outcomes and their corresponding probabilities?

  • (c) The free evolution of the photon is governed by the Hamiltonian

    $$H = i\hbar \omega (|H\rangle \langle V| - |V\rangle \langle H|)\, ,$$

    with $\omega$ the angular frequency of the light. Find the state of the photon at time $t=T$ given that the state at $t=0$ is given by $|\psi \rangle = \alpha |H\rangle + \beta |V\rangle .$

  • (d) Calculate and sketch the probability of finding the outcomes of a measurement in the polarisation basis $\{|H\rangle ,|V\rangle \}$ in the time interval $0\le t \le T$ when $\alpha =\beta =1/\sqrt{2}.$