Recall that $\left(\mid \psi {\mid }^2\right)dx$ is the probability of finding the par­ticle that has normalized wave function $\psi \left(x\right)$ in the interval $x$ to $x+dx$. Consider a particle in a box with rigid walls at $x=0$ and $x=L$. Let the particle be in the ground level and use ${\psi }_n$ as given in Eq. ($40.35$) ${\psi }_n\left(x\right)=\sqrt{\frac{2}{L}}sin\left[\right(\frac{n\pi x}{L}\left)\right]$ where $n=1,2,3,\dots$.
(a) For which values of $x$, if any, in the range from $0$ to $L$ is the probability of finding the particle zero?
(b) For which values of $x$ is the probability highest?
(c) In parts (a) and (b) are your answers consistent with Fig. $40.12$? Explain.

A free particle moving in one dimension has wave function $\psi (x,t)=A[e^{i(kx-\omega t)}-e^{i(2kx-4\omega t)}]$ where $k$ and $v$ are positive real constants.

(a) At $t=0$, what are the two smallest positive values of $x$ for which the probability function $\mid \psi (x,t){\mid }^2$ is a maximum?

(b) Repeat part (a) for time $t=\frac{2\pi }{\omega }$.

A free particle moving in one dimension has wave function ψ(x,t) = A[e^i(kx-ωt) -e^i(2kx-4ωt)] where k and v are positive real constants. (c) Calculate v_av as the distance the maxima have moved divided by the elapsed time.

Let ${\psi }_1$ and ${\psi }_2$ be two solutions of Eq. ($40.23$) [$-\frac{h^2}{2m}\frac{d^2\psi \left(x\right)}{d{x}^2}+U\left(x\right)\psi \left(x\right)=E\psi \left(x\right)$] with energies $E_1$ and $E_2$ respectively, where $E_1\ne E_2$. Is $\psi =A{\psi }_1+B{\psi }_2$, where $A$ and $B$ are nonzero constants, a solution to Eq. ($40.23$)? Explain your answer.

When a hydrogen atom undergoes a transition from the $n=2$ to the $n=1$ level, a photon with $\lambda =122$ nm is emitted.

(a) If the atom is modeled as an electron in a one-­dimensional box, what is the width of the box in order for the $n=2$ to $n=1$ transi­tion to correspond to emission of a photon of this energy?

(b) For a box with the width calculated in part (a), what is the ground­ state energy? How does this correspond to the ground ­state energy of a hydrogen atom?

(c) Do you think a one­-dimensional box is a good model for a hydrogen atom? Explain. (Hint: Compare the spacing between adjacent energy levels as a function of $n$.)

Show that the following combination of the three fundamental constants of nature that we used in Example 1–10 (that is G, c, and h) forms a quantity with the dimensions of time: tₚ = $\sqrt{Gh}$/c⁵. This quantity, tₚ, is called the Planck time and is thought to be the earliest time, after the creation of the Universe, at which the currently known laws of physics can be applied.

Photoelectrons are observed when a metal is illuminated by light with a wavelength less than 388 nm. What is the metal's work function?

Potassium and gold cathodes are used in a photoelectric-effect experiment. For each cathode, find: The threshold frequency.

BIO The wavelengths of light emitted by a firefly span the visible spectrum but have maximum intensity near 550 nm. A typical flash lasts for 100 ms and has a power output of 1.2 mW. How many photons does a firefly emit in one flash if we assume that all light is emitted at the peak intensity wavelength of 550 nm?