0. Math Review

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 $n$.)

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 proton and an antiproton annihilate, producing two photons. Find the energy, frequency, and wavelength of each photon if the $p$ and $\(\overline{p}\)$ collide head-on, each with an initial kinetic energy of $620$ MeV.