17.4 The Orbitals of the Hydrogen-Like Atom 741
PROBLEMS
Section 17.3: The Radial Factor in the Hydrogen Atom
Wave Function. The Energy Levels of the Hydrogen Atom
17.13Calculate〈r〉, the expectation value ofrfor the 2sand 2p
states of a hydrogen-like atom. Comment on your answer.
17.14a.Find the value of the distancebsuch that there is a 95%
chance that an electron in a hydrogen atom in the 1s
state is no farther from the nucleus than the distanceb.
b.Find the ratio of the 1swave function atrbto the
same function atr0.
c.Repeat parts a and b for 90% probability instead
of 95%.
17.15For a hydrogen atom in a 1sstate, find the probability
that the electron is no farther from the nucleus than
(a)a, (b) 2a, (c) 3awhereais the Bohr radius.
17.16Calculate〈z〉andσzfor the electron in a hydrogen atom
in the 1sstate. Explain the meaning of the values. What
can you say about〈x〉andσx? What can you say about
〈y〉andσy?
17.17Find the most probable value of the electron’s distance
from the nucleus for a hydrogen atom in the 1s, the 2s,
and the 2pstates.
17.18Using formulas in Appendix F, verify the formula given
in Table 17.2 forR 32.
17.19a.Construct the formula that representsR 40 , using
formulas in Appendix F.
b.Construct a graph of the radial distribution function
for the 4s(400) state.
17.20a.Construct a graph of the radial distribution function
for the 3dstates of a hydrogen atom.
b.Find the values ofrat which the radial distribution
function vanishes.
c.Find the values ofrat which the radial distribution
function has relative maxima.17.21Find the shortest wavelength of light emitted by hydrogen
atoms in transitions between bound states. In what region
of the spectrum (visible, infrared, ultraviolet, etc.)
does it lie?17.22a.Calculate the percent difference between the energy of
an ordinary hydrogen atom and a deuterium atom in
the ground state.
b.Calculate the percent difference between the energy of
an ordinary hydrogen atom and a deuterium atom in
the 2sstate.
c.Calculate the percent difference between the energy of
a^4 He+ion and a^3 He+ion in the 2sstate.17.23Calculate the expectation values ofpxand ofp^2 xfor the
electron in a hydrogen atom in the 1sstate. Why does
〈p^2 x〉not equal〈px〉^2?17.24A positronium atom is a hydrogen-like atom
consisting of an electron and a positron (an antielectron
with charge+eand mass equal to the electron mass).
Find the energy of a positronium atom in the 1sstate.
Describe the classical motion of the two particles about
the center of mass. Find the value of the Bohr radius for
positronium.17.25Calculate the expectation value of the kinetic energy for a
hydrogen-like atom in the 1sstate forZ2,Z3, and
Z4.17.26Explain verbally why〈L^2 〉and〈Lz〉are independent ofZ
for all stationary states of the hydrogen-like atom. Since
the average distance from the nucleus depends onZ, what
does this mean about the average speed of the electron
around the nucleus for the 211 state?17.4 The Orbitals of the Hydrogen-Like Atom
Wave functions of single electrons are calledorbitals. We can choose between real
and complex orbitals. Each complex hydrogen atom orbital contains complexΦm
functions:ψnlmRnl(r)Θlm(θ)Φm(φ) (17.4-1)The complex orbitals are eigenfunctions of̂H,̂L^2 , and̂Lz. The real orbitals contain
eitherΦmxorΦmy. They are eigenfunctions of̂Hand̂L^2 but are not eigenfunctions