35. Special Relativity

For crystal diffraction experiments (discussed in Section $39.1$), wavelengths on the order of $0.20$ nm are often appropriate. Find the energy in electron volts for a particle with this wavelength if the particle is a photon.

Draw a series of pictures, similar to Figure 41.21, for the ground states of Ca, Ni, As, and Kr.

How many lines of atoms would you expect to see on the collector plate of a Stern-Gerlach apparatus if the experiment is done with (a) lithium and (b) beryllium? Explain.

Consider the electron wave function where x is in cm. Draw a graph of ψ(x) over the interval −2 cm ≤ x ≤ 2 cm. Provide numerical scales on both axes.

What is the time for one complete revolution for a very high-energy proton in the 27-km LHC circumference?

A sodium atom emits a photon with wavelength 818 nm shortly after being struck by an electron. What minimum speed did the electron have before the collision?

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. Draw an energy-level diagram showing all energy levels from 1 through n+1. Label each level and write the energy beside it.

The energy-level scheme for the hypothetical one-electron element Searsium is shown in Fig. $E39.25$. The potential energy is taken to be zero for an electron at an infinite distance from the nucleus. An $18$-eV photon is absorbed by a Searsium atom in its ground level. As the atom returns to its ground level, what possible energies can the emitted photons have? Assume that there can be transitions between all pairs of levels.

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 beam of alpha particles is incident on a target of lead. A particular alpha particle comes in 'head-on' to a particular lead nucleus and stops $6.50\times {10}^{-14}$ m away from the center of the nucleus. (This point is well outside the nucleus.) Assume that the lead nucleus, which has $82$ protons, remains at rest. The mass of the alpha particle is $6.64\times {10}^{-27}$ kg.

(a) Calculate the electrostatic potential energy at the instant that the alpha particle stops. Express your result in joules and in MeV.

(b) What initial kinetic energy (in joules and in MeV) did the alpha particle have?

(c) What was the initial speed of the alpha particle?

INT An electron is confined in a harmonic potential well that has a spring constant of 2.0 N/m. What wavelength photon is emitted if the electron undergoes a 3→1 quantum jump?

A $4.78$-MeV alpha particle from a $226$Ra decay makes a head-on collision with a uranium nucleus. A uranium nucleus has $92$ protons.

(a) What is the distance of closest approach of the alpha particle to the center of the nucleus? Assume that the uranium nucleus remains at rest and that the distance of closest approach is much greater than the radius of the uranium nucleus.

(b) What is the force on the alpha particle at the instant when it is at the distance of closest approach?

CALC A particle of mass m has the wave function ψ(x) = Ax exp (−x²/a²) when it is in an allowed energy level with E = 0. At what value or values of x is the particle most likely to be found?

A rocket traveling at 0.50c sets out for the nearest star, Alpha Centauri, which is 4.3 ly away from earth. It will return to earth immediately after reaching Alpha Centauri. What distance will the rocket travel and how long will the journey last according to (a) stay-at-home earthlings and (b) the rocket crew? (c) Which answers are the correct ones, those in part a or those in part b?