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A wave traveling in the \(z\) direction is described by the wave function \(\Psi(z, t)=A_{1} \mathbf{x} \sin \left(k z-\omega t+\phi_{1}\right)+\) \(A_{2} \mathbf{y} \sin \left(k z-\omega t+\phi_{2}\right),\) where \(\mathbf{x}\) and \(\mathbf{y}\) are vectors of unit length along the \(x\) and \(y\) axes, respectively. Because the amplitude is perpendicular to the propagation direction, \(\Psi(z, t)\) represents a transverse wave. a. What requirements must \(A_{1}\) and \(A_{2}\) satisfy for a plane polarized wave in the \(x\) -z plane? The amplitude of a plane polarized wave is non-zero only in one plane. b. What requirements must \(A_{1}\) and \(A_{2}\) satisfy for a plane polarized wave in the \(y\) -z plane? c. What requirements must \(A_{1}\) and \(A_{2}\) and \(\phi_{1}\) and \(\phi_{2}\) satisfy for a plane polarized wave in a plane oriented at \(45^{\circ}\) to the \(x-z\) plane? d. What requirements must \(A_{1}\) and \(A_{2}\) and \(\phi_{1}\) and \(\phi_{2}\) satisfy for a circularly polarized wave? The phases of the two components of a circularly polarized wave differ by \(\pi / 2\)

Because \(\int_{-d}^{d} \cos (n \pi x / d) \cos (m \pi x / d) d x=0\) \(m \neq n,\) the functions \(\cos (n \pi x / d)\) for \(n=1,2,3, \dots\) form an orthogonal set in the interval \((-d, d) .\) What constant must these functions be multiplied by to form an orthonormal set?

If two operators act on a wave function as indicated by \(\hat{A} \hat{B} f(x),\) it is important to carry out the operations in succession with the first operation being that nearest to the function. Mathematically, \(\hat{A} \hat{B} f(x)=\hat{A}(\hat{B} f(x))\) and \(\hat{A}^{2} f(x)=\hat{A}(\hat{A} f(x)) .\) Evaluate the following successive operations \(\hat{A} \hat{B} f(x) .\) The operators \(\hat{A}\) and \(\hat{B}\) are listed in the first two columns and \(f(x)\) is listed in the third column. a. \(\frac{d}{d y} \quad y \quad y e^{-2 y^{3}}\) b. \(y \quad \frac{d}{d y} \quad y e^{-2 y^{3}}\) c. \(y \frac{\partial}{\partial x} \quad x \frac{\partial}{\partial y} \quad e^{-2(x+y)}\) d. \(x \frac{\partial}{\partial y} \quad y \frac{\partial}{\partial x} \quad e^{-2(x+y)}\) Are your answers to parts (a) and (b) identical? Are your answers to parts (c) and (d) identical? As we will learn in Chapter \(17,\) switching the order of the operators can change the outcome of the operation \(\hat{A} \hat{B} f(x)\)

Assume that a system has a very large number of energy levels given by the formula \(\varepsilon=\varepsilon_{0} l^{2}\) with \(\varepsilon_{0}=\) \(1.75 \times 10^{-22} \mathrm{J},\) where \(l\) takes on the integral values 1,2 \(3, \ldots .\) Assume further that the degeneracy of a level is given by \(g_{l}=2 l .\) Calculate the ratios \(n_{4} / n_{1}\) and \(n_{8} / n_{1}\) for \(T=125 \mathrm{K}\) and \(T=750 . \mathrm{K}\)

If two operators act on a wave function as indicated by \(\hat{A} \hat{B} f(x),\) it is important to carry out the operations in succession with the first operation being that nearest to the function. Mathematically, \(\hat{A} \hat{B} f(x)=\hat{A}(\hat{B} f(x))\) and \(\hat{A}^{2} f(x)=\) \(\hat{A}(\hat{A} f(x)) .\) Evaluate the following successive operations \(\hat{A} \hat{B} f(x) .\) The operators \(\hat{A}\) and \(\hat{B}\) are listed in the first and second columns and \(f(x)\) is listed in the third column. Compare your answers to parts \((a)\) and \((b),\) and to \((c)\) and \((d)\) a. \(\frac{d}{d x}\) b. \(x\) c. \(\frac{\partial^{2}}{\partial y^{2}}\) d. \(y^{2}\) $$\begin{array}{ll} x & x^{2}+e^{a x^{2}} \\ \frac{d}{d x} & x^{2}+e^{a x^{2}} \\ y^{2} & (\cos 3 y) \sin ^{2} x \\ \frac{\partial^{2}}{\partial y^{2}} & (\cos 3 y) \sin ^{2} x \end{array}$$

Find the result of operating with \(\left(1 / r^{2}\right)(d / d r)\left(r^{2} d / d r\right)+2 / r\) on the function \(A e^{-b r} .\) What must the values of \(A\) and \(b\) be to make this function an eigenfunction of the operator?

Normalize the set of functions \(\phi_{n}(\theta)=e^{i n \theta}\) \(0 \leq \theta \leq 2 \pi .\) To do so, you need to multiply the functions by a normalization constant \(N\) so that the integral \(N N^{*} \int_{0}^{2 \pi} \phi_{m}^{*}(\theta) \phi_{n}(\theta) d \theta=1\) for \(m=n\)

In normalizing wave functions, the integration is over all space in which the wave function is defined. a. Normalize the wave function \(x(a-x) y(b-y)\) over the range \(0 \leq x \leq a, 0 \leq y \leq b .\) The element of area in two-dimensional Cartesian coordinates is \(d x d y ; a\) and \(b\) are constants. b. Normalize the wave function \(e^{-(2 r / b)} \sin \theta \sin \phi\) over the interval \(0 \leq r<\infty, 0 \leq \theta \leq \pi, 0 \leq \phi \leq 2 \pi .\) The volume element in three-dimensional spherical coordinates is \(r^{2} \sin \theta d r d \theta d \phi,\) and \(b\) is a constant.

Show that the following pairs of wave functions are orthogonal over the indicated range. a. \(e^{-\alpha x^{2}}\) and \(x\left(x^{2}-1\right) e^{-\alpha x^{2}},-\infty \leq x<\infty\) where \(\alpha\) is a constant that is greater than zero b. \(\left(6 r / a_{0}-r^{2} / a_{0}^{2}\right) e^{-r / 3 a_{0}}\) and \(\left(r / a_{0}\right) e^{-r / 2 a_{0}} \cos \theta\) over the interval \(0 \leq r<\infty, 0 \leq \theta \leq \pi, 0 \leq \phi \leq 2 \pi\)