(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 stick of length ℓ₀, at rest in reference frame S, makes an angle θ with the x axis. In reference frame S', which moves to the right with velocity $\overrightarrow{v}$ = vî with respect to S, determine (a) the length l of the stick, and (b) the angle θ it makes with the x' axis.

A pi meson of mass m_π decays at rest into a muon (mass m_μ) and a neutrino of negligible or zero mass. Show that the kinetic energy of the muon is K_μ = (m_π - m_μ)² c² / (2m_π).

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.

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.

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 proton is in a box of width $L$. What must the width of the box be for the ground­-level energy to be $5.0$ MeV, a typi­cal value for the energy with which the particles in a nucleus are bound? Compare your result to the size of a nucleus — that is, on the order of ${10}^{-14}$ m.

An electron in a one­-dimensional box has ground ­state energy $2.00$ eV. What is the wavelength of the photon absorbed when the electron makes a transition to the second excited state?

(a) Find the excitation energy from the ground level to the third excited level for an electron confined to a box of width $0.360$ nm.

(b) The electron makes a transition from the $n=1$ to $n=4$ level by absorbing a photon. Calculate the wave­length of this photon.