Is $\cos x$ an eigenfunction of the operator $\hat{A}$ if $\hat{A} f(x)=x f(x)$ ?

A ground-state H atom absorbs a photon and makes a transition to the $n = 4$ energy level. It then emits a photon of energy $4.085\cdot 10^{-19}~\text{J}.$ What is the final energy and $n$ value of the atom?

(a) A glow-worm of mass 5.0 g emits red light (650 nm) with a power of 0.10 W entirely in the backward direction. To what speed will it have accelerated after 10 y if released into free space and assumed to live? (b) A photon-powered spacecraft of mass 10.0 kg emits radiation of wavelength 225 nm with a power of 1.50 kW entirely in the backward direction. To what speed will it have accelerated after 10.0 y if released into free space?

Determine in each of the following cases if the function in the first column is an eigenfunction of the operator in the second column. If so, what is the eigenvalue? a. $e^{-i(7 x+y)} \quad \quad \frac{\partial^2}{\partial x^2}$ b. $\sqrt{3 x^2+2 y^2} \quad \quad (1 / 3 x)\left(3 x^2+2 y^2\right) \frac{\partial}{\partial x}$ c. $\sin \theta \cos \theta \quad \quad \frac{1 d}{\sin \theta d \theta}\left(\sin \theta \frac{d}{d \theta}\right)$

A linear operator satisfies the condition $\hat{A}(f(x)+g(x))=\hat{A} f(x)+\hat{A} g(x)$. Are $\hat{A}$ or $\hat{B}$ linear operators if $\hat{A} f(x)=(f(x))^2$ and $\hat{B} f(x)=d f(x) / d x$ ?

Can the function $\sin k x$ be normalized over the interval $-\infty<x<\infty$ ? Explain your answer.

Show by carrying out the integration that $\sin (m \pi x / a)$ and $\cos (m \pi x / a)$, where $m$ is an integer, are orthogonal over the interval $0 \leq x \leq a$. Would you get the same result if you used the interval $0 \leq x \leq 3 a / 4$ ? Explain your result.

What is the usefulness of a set of complete functions?

Determine in each of the following cases if the function in the first column is an eigenfunction of the operator in the second column. If so, what is the eigenvalue? a. $\cos \theta \quad \frac{1}{\sin \theta} \frac{d}{d \theta}\left(\sin \theta \frac{d}{d \theta}\right)$ b. $e^{-\left(2 i x^2\right)} \quad \frac{d^2}{d x^2}+16 x^2$ c. $\cos x \sin x \quad \frac{d^2}{d x^2}-2$

Does the superposition $\psi(x, t)=A \sin (k x-\omega t)+$ $2 A \sin (k x+\omega t)$ generate a standing wave? Answer this question by using trigonometric identities to combine the two terms.

Why can we conclude that the wave function $\psi(x, t)=\psi(x) e^{-i(E / \hbar) t}$ represents a standing wave?

Which of the following wave functions are eigenfunctions of the operator $d^2 / d x^2$ ? If they are eigenfunctions, what is the eigenvalue? a. $a\left(e^{-3 x}+e^{-3 i x}\right)$ d. $\cos \frac{a x}{\pi}$ b. $\sin \frac{2 \pi x}{a}$ e. $e^{-i x^2}$ c. $e^{-2 i x}$

Distinguish between the following terms applied to a set of functions: orthogonal, normalized, and orthonormal.

Why does a quantum mechanical system with discrete energy levels behave as if it has a continuous energy spectrum if the energy difference between energy levels $\Delta E$ satisfies the relationship $\Delta E \ll k_B T$ ?