A 4.653-μm emission line of atomic hydrogen corresponds to transition between the states 𝑛𝑓 = 5 and 𝑛𝑖. Find 𝑛𝑖.
Answer: 7
4.3
For an electron in a hydrogen atom in the n = 2 state, compute:
(a) the angular momentum
(b) the kinetic energy
(c) the potential energy
(d) the total energy
Answer: (a) 1.316 x 10-15 eV•s (b) 3.4 eV (c) -6.8 eV (d) -3.4 eV
4.4
In the double-slit interference pattern for helium atoms, the beam of atoms had a kinetic energy of 0.020 eV.
(a) What is the de Broglie wavelength of a helium atom with this kinetic energy?
(b) Estimate the de Broglie wavelength of the atoms from the fringe spacing in the figure and compare your estimate with the value obtained in part (a). The distance from the double slit to the scanning slit is 64 cm and the slit separation is 8 µm.
Answer: (a) 1.02 Å (b) 1 Å
4.5
The neutrons produced in a reactor are known as thermal neutrons, because their kinetic energies have been reduced (by collisions) until K = (3/2)kT where T is room temperature (293 K) and k is the Boltzmann constant.
(a) What is the kinetic energy of such neutrons?
(b) What is their de Broglie wavelength? Because this wavelength is of the same order as the lattice spacing of the atoms of a solid, neutron diffraction (like X-ray and electron diffraction) is a useful means of studying solid lattices.
Answer: (a) 0.038 eV (b) 0.146 nm
4.6
Through what potential difference must electrons be accelerated to resolve the following particles?
(a) a virus of diameter 15 nm
(b) an atom of diameter 0.096 nm
(c) a proton of diameter 1.2 fm
Answer: (a) 0.0067 V (b) 163 V (c) 1 x 1012 V
4.7
The speed of an electron is measured to within an uncertainty of 2.8 × 104 m/s. What is the size of the smallest region of space in which the electron can be confined?
Answer: 2.1 nm
4.8
The radii of atomic nuclei are of the order of 5.0 x 10-15 m.
(a) Estimate the minimum uncertainty in the momentum of a proton if it is confined within a nucleus.
(b) Take this uncertainty in momentum to be an estimate of the magnitude of the momentum. Use the relativistic relationship between energy and momentum, equation E2=(mc2)2 + (pc)2, to obtain an estimate of the kinetic energy of a proton confined within a nucleus.
(c) For a proton to remain bound within a nucleus, what must the magnitude of the (negative) potential energy for a proton be within the nucleus? Compare this to the potential energy for an electron in a hydrogen atom, which has a magnitude of a few tens of eV. (This shows why the interaction that binds the nucleus together is called the “strong nuclear force.”)
Answer: (a) 19.7 MeV/c (b) 200 keV (c) -200 keV
4.9
A pi meson (pion) and a proton can briefly join together to form a Δ particle. A measurement of the energy of the 𝜋p system (see figure) shows a peak at 1236 MeV, corresponding to the rest energy of the Δ particle, with an experimental spread of 120 MeV. What is the lifetime of the Δ?
Answer: 2.7 x 10-24 s
4.10
A proton or a neutron can sometimes “violate” conservation of energy by emitting and then reabsorbing a pi meson, which has a mass of 135 MeV/c2. This is possible as long as the pi meson is reabsorbed within a short enough time Δt consistent with the uncertainty principle.
(a) Consider p → p + 𝜋. By what amount ΔE is energy conservation violated? (Ignore any kinetic energies.)
(b) For how long a time Δt can the pi meson exist?
(c) Assuming that the pi meson travels at very nearly the speed of light, how far from the proton can it go?
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