35. Special Relativity

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.

An electron is in a box of width $3.0\times {10}^{-10}$ m. What are the de Broglie wavelength and the magnitude of the momentum of the electron if it is in (a) the $n=1$ level; (b) the $n=2$ level; (c) the $n=3$ level? In each case how does the wavelength compare to the width of the box?

An electron is in a box of width 3.0*10^-10 m. What are the de Broglie wavelength and the magnitude of the momentum of the electron if it is in (a) the n = 1 level; (b) the n = 2 level; (c) the n = 3 level? In each case how does the wavelength compare to the width of the box?

(a) An electron with initial kinetic energy $32$ eV encoun­ters a square barrier with height $41$ eV and width $0.25$ nm. What is the probability that the electron will tunnel through the barrier?

(b) A proton with the same kinetic energy encounters the same barrier. What is the probability that the proton will tunnel through the barrier?

An electron with initial kinetic energy $6.0$ eV encounters a barrier with height $11.0$ eV. What is the probability of tunneling if the width of the barrier is (a) $0.80$ nm and (b) $0.40$ nm?

An electron with initial kinetic energy $5.0$ eV encoun­ters a barrier with height $U_0$ and width $0.60$ nm. What is the transmission coefficient if (a) $U_0=7.0$ eV; (b) $U_0=9.0$ eV; (c) $U_0=13.0$ eV?

While undergoing a transition from the $n=1$ to the $n=2$ energy level, a harmonic oscillator absorbs a photon of wavelength $6.50$ $\mu$m. What is the wavelength of the absorbed photon when this oscillator undergoes a transition (a) from the $n=2$ to the $n=3$ energy level and (b) from the $n=1$ to the $n=3$ energy level?

(c) What is the value of $\sqrt{\left(k^{\prime }/m\right)}$, the angular oscillation frequency of the corresponding Newtonian oscillator?