Chapter 2

Show that $$ \lim _{N \rightarrow \infty}\left(1+\frac{x}{N}\right)^{N}=e^{x} $$ by comparing the Taylor series expansions for the two functions.

Use Dirac notation (the properties of kets, bras, and inner products) directly without explicitly using matrix representations to establish that the projection operator \(\hat{P}_{+}\)is Hermitian. Use the fact that \(\hat{P}_{+}^{2}=\hat{P}_{+}\)to establish that the eigenvalues of the projection operator are 1 and 0 .

Suppose in a two-dimensional basis that the operators \(\hat{A}\) and \(\hat{B}\) are represented by the \(2 \times 2\) matrices $$ \hat{A} \rightarrow\left(\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right) \quad \hat{B} \rightarrow\left(\begin{array}{ll} 5 & 6 \\ 7 & 8 \end{array}\right) $$ Show that \((\hat{A} \hat{B})^{\dagger}=\hat{B}^{\dagger} \hat{A}^{\dagger} .\)

A photon polarization state for a photon propagating in the \(z\) direction is given by $$ |\psi\rangle=\sqrt{\frac{2}{3}}|x\rangle+\frac{i}{\sqrt{3}}|y\rangle $$ (a) What is the probability that a photon in this state will pass through an ideal polarizer with its transmission axis oriented in the \(y\) direction? (b) What is the probability that a photon in this state will pass through an ideal polarizer with its transmission axis \(y^{\prime}\) making an angle \(\phi\) with the \(y\) axis? (c) A beam carrying \(N\) photons per second, each in the state \(|\psi\rangle\), is totally absorbed by a black disk with its normal to the surface in the \(z\) direction. How large is the torque exerted on the disk? In which direction does the disk rotate? Reminder: The photon states \(|R\rangle\) and \(|L\rangle\) each carry a unit \(\hbar\) of angular momentum parallel and antiparallel, respectively, to the direction of propagation of the photons. (d) How would the result for each of these questions differ if the polarization state were $$ \left|\psi^{\prime}\right\rangle=\sqrt{\frac{2}{3}}|x\rangle+\frac{1}{\sqrt{3}}|y\rangle $$ that is, the " \(i\) " in the state \(|\psi\rangle\) is absent?

A system of \(N\) ideal linear polarizers is arranged in sequence, as shown in Fig. 2.13. The transmission axis of the first polarizer makes an angle of \(\phi / N\) with the \(y\) axis. The transmission axis of every other polarizer makes an angle of \(\phi / N\) with respect to the axis of the preceding one. Thus, the transmission axis of the final polarizer makes an angle \(\phi\) with the \(y\) axis. A beam of \(y\)-polarized photons is incident on the first polarizer. (a) What is the probability that an incident photon is transmitted by the array? (b) Evaluate the probability of transmission in the limit of large \(N\). (c) Consider the special case with the angle \(\phi=90^{\circ}\). Explain why your result is not in conflict with the fact that \(\langle x \mid y\rangle=0 .^{16}\)

Construct projection operators out of bras and kets for \(x\)-polarized and \(y\) polarized photons. Give physical examples of devices that can serve as these projection operators. Use (a) the properties of bras and kets and (b) the properties of the physical devices to show that the projection operators satisfy \(\hat{P}_{x}^{2}=\hat{P}_{x}, \hat{P}_{y}^{2}=\hat{P}_{y}\), and \(\hat{P}_{x} \hat{P}_{y}=\hat{P}_{y} \hat{P}_{x}=0\).

What is the probability that a right-circularly polarized photon will pass through a linear polarizer with its transmission axis along the \(x^{\prime}\) axis, which makes an angle \(\phi\) with the \(x\) axis?

Linearly polarized light of wavelength \(5890 \AA\) is incident normally on a birefringent crystal that has its optic axis parallel to the face of the crystal, along the \(x\) axis. If the incident light is polarized at an angle of \(45^{\circ}\) to the \(x\) and \(y\) axes, what is the probability that the photons exiting a crystal of thickness \(100.0\) microns will be right-circularly polarized? The index of refraction for light of this wavelength polarized along \(y\) (perpendicular to the optic axis) is \(1.66\) and the index of refraction for light polarized along \(x\) (parallel to the optic axis) is \(1.49\).

A beam of linearly polarized light is incident on a quarter-wave plate with its direction of polarization oriented at \(30^{\circ}\) to the optic axis. Subsequently, the beam is absorbed by a black disk. Determine the rate at which angular momentum is transferred to the disk, assuming the beam carries \(N\) photons per second.

(a) Show that if the states \(\left|a_{n}\right\rangle\) form an orthonormal basis, so do the states \(\hat{U}\left|a_{n}\right\rangle\), provided \(\hat{U}\) is unitary. (b) Show that the eigenvalues of a unitary operator can be written as \(e^{i \theta}\).

The Hermitian operator \(\hat{A}\) corresponding to the observable \(A\) has two eigenstates \(\left|a_{1}\right\rangle\) and \(\left|a_{2}\right\rangle\) with eigenvalues \(a_{1}\) and \(a_{2}\), respectively. Assume \(a_{1} \neq a_{2}\). Show that \(\hat{A}\) can be written in the form $$ \hat{A}=a_{1}\left|a_{1}\right\rangle\left\langle a_{1}\left|+a_{2}\right| a_{2}\right\rangle\left\langle a_{2}\right| $$ and that $$ \langle\psi|\hat{A}| \psi\rangle=\langle A\rangle $$