Consider a wave function given by $\psi \left(x\right)=Asinkx$, where $k=2\pi /\lambda$ and $A$ is a real constant.
(a) For what values of $x$ is there the highest probability of finding the particle described by this wave function? Explain.
(b) For which values of $x$ is the probability zero? Explain.

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?

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.

A particle in a box is a billiard ball ($m = 0.20$ kg) and the box has a width of $1.3$ m, the size of a billiard table. (Assume that the billiard ball slides without friction rather than rolls; that is, ignore the rotational kinetic energy.) What is the difference in energy between the $n=2$ and $n=1$ levels? Are quantum­ mechanical effects important for the game of billiards?

A particle is described by a wave function $\psi \left(x\right)=A{e}^{-\alpha x^2}$, where $A$ and $\alpha$ are real, positive constants. If the value of $\alpha$ is increased, what effect does this have on (a) the particle’s uncer­tainty in position and (b) the particle’s uncertainty in momentum? Explain your answers.

(a) Find the lowest energy level for a particle in a box if the particle is a billiard ball ($m=0.20$ kg) and the box has a width of $1.3$ m, the size of a billiard table. (Assume that the billiard ball slides without friction rather than rolls; that is, ignore the rotational kinetic energy.)

(b) Since the energy in part (a) is all kinetic, to what speed does this correspond? How much time would it take at this speed for the ball to move from one side of the table to the other?

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 $t=\frac{2\pi }{\omega }$.