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 1.0-mm-diameter sphere bounces back and forth between two walls at x = 0 mm and x = 100 mm. The collisions are perfectly elastic, and the sphere repeats this motion over and over with no loss of speed. At a random instant of time, what is the probability that the center of the sphere is at exactly x = 50.0 mm?

A 1.0-mm-diameter sphere bounces back and forth between two walls at x = 0 mm and x = 100 mm. The collisions are perfectly elastic, and the sphere repeats this motion over and over with no loss of speed. At a random instant of time, what is the probability that the center of the sphere is between x = 49.0 mm and x = 51.0 mm?

An experiment finds electrons to be uniformly distributed over the interval 0 cm ≤ x ≤ 2 cm, with no electrons falling outside this interval. If 10⁶ electrons are detected, how many will be detected in the interval 0.79 to 0.81 cm?

An experiment finds electrons to be uniformly distributed over the interval 0 cm ≤ x ≤ 2 cm, with no electrons falling outside this interval. What is the probability density at x = 0.80 cm?

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 $\psi (x,t)=A[e^{i(kx-\omega t)}-e^{i(2kx-4\omega t)}]$ 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 $t=\frac{2\pi }{\omega }$.