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 helium atom is in a finite potential well. The atom’s energy is 1.0 eV below U₀. What is the atom’s penetration distance into the classically forbidden region?

A particle confined in a rigid one-dimensional box of length 10 fm has an energy level E_n = 32.9 MeV and an adjacent energy level E_n+1 = 51.4 MeV. What is the mass of the particle? Can you identify it?

At an archeological site, a sample from timbers containing $500$ g of carbon provides $2690$ decays/min. What is the age of the sample?

An electron is moving as a free particle in the $-x$-direction with momentum that has magnitude $4.50\times {10}^{-24}$ kg*m/s. Let $k_2=3k_1=3k$. At $t=0$, the probability distribution func­tion $\mid \Psi (x,t){\mid }^2$ has a maximum at $x=0$.

(a) What is the smallest positive value of $x$ for which the probability distribution function has a maximum at time $t=\frac{2\pi }{\omega }$, where $\omega =h{k}^2/2m$?

(b) From your result in part (a), what is the average speed with which the probability distribution is moving in the $+x$­-direction?

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.