(a) A distance scale is shown below the wavefunctions and
probability densities illustrated in section 5.3. Compare this with
the order-of-magnitude estimate derived in section 5.4 for the radius
r at which the wavefunction begins tailing off. Was the estimate in
section 5.4 on the right order of magnitude?
(b) Although we normally say the moon orbits the earth, actually
they both orbit around their common center of mass, which is below
the earth's surface but not at its center. The same is true of the
hydrogen atom. Does the center of mass lie inside the proton or
The figure shows eight of the possible ways in which an electron
in a hydrogen atom could drop from a higher energy state to a state
of lower energy, releasing the difference in energy as a photon. Of
these eight transitions, only D, E, and F produce photons with
wavelengths in the visible spectrum.
(a) Which of the visible transitions would be closest to the violet
end of the spectrum, and which would be closest to the red end?
(b) In what part of the electromagnetic spectrum would the photons
from transitions A, B, and C lie? What about G and H? Explain.
(c) Is there an upper limit to the wavelengths that could be emitted
by a hydrogen atom going from one bound state to another bound
state? Is there a lower limit? Explain.
Before the quantum theory, experimentalists noted that in
many cases, they would find three lines in the spectrum of the same
atom that satisfied the following mysterious rule: 1/λ1 =
1/λ3. Explain why this would occur. Do not use reasoning that
only works for hydrogen - such combinations occur in the spectra
of all elements. [Hint: Restate the equation in terms of the energies
Find an equation for the wavelength of the photon emitted
when the electron in a hydrogen atom makes a transition from energy
level n1 to level n2. [You will need to have read optional section
(a) Verify that Planck's constant has the same units as angular
(b) Estimate the angular momentum of a spinning basketball, in
units of h[bar].
Assume that the kinetic energy of an electron in the n = 1
state of a hydrogen atom is on the same order of magnitude as the
absolute value of its total energy, and estimate a typical speed at
which it would be moving. (It cannot really have a single, definite
speed, because its kinetic and potential energy trade off at different
distances from the proton, but this is just a rough estimate of a
typical speed.) Based on this speed, were we justified in assuming
that the electron could be described nonrelativistically?
The wavefunction of the electron in the ground state of a hydrogen
where r is the distance from the proton, and a = h[bar]2/kme2 =
5.3 × 10-11 m is a constant that sets the size of the wave.
(a) Calculate symbolically, without plugging in numbers, the probability
that at any moment, the electron is inside the proton. Assume
the proton is a sphere with a radius of b = 0.5 fm. [Hint:
Does it matter if you plug in r = 0 or r = b in the equation for the
(b) Calculate the probability numerically.
(c) Based on the equation for the wavefunction, is it valid to think
of a hydrogen atom as having a finite size? Can a be interpreted
as the size of the atom, beyond which there is nothing? Or is there
any limit on how far the electron can be from the proton?
Use physical reasoning to explain how the equation for the
energy levels of hydrogen,
should be generalized to the case of a heavier atom with atomic
number Z that has had all its electrons stripped away except for
This question requires that you read optional section 5.4. A
muon is a subatomic particle that acts exactly like an electron except
that its mass is 207 times greater. Muons can be created by
cosmic rays, and it can happen that one of an atom's electrons is
displaced by a muon, forming a muonic atom. If this happens to
a hydrogen atom, the resulting system consists simply of a proton
plus a muon.
(a) How would the size of a muonic hydrogen atom in its ground
state compare with the size of the normal atom?
(b) If you were searching for muonic atoms in the sun or in the
earth's atmosphere by spectroscopy, in what part of the electromagnetic
spectrum would you expect to find the absorption lines?
Consider a classical model of the hydrogen atom in which the
electron orbits the proton in a circle at constant speed. In this
model, the electron and proton can have no intrinsic spin. Using
the result of problem 17 from book 4, ch. 6, show that in
this model, the atom's magnetic dipole moment Dm is related to
its angular momentum by Dm = (-e/2m)L, regardless of the details
of the orbital motion. Assume that the magnetic field is the
same as would be produced by a circular current loop, even though
there is really only a single charged particle. [Although the model
is quantum-mechanically incorrect, the result turns out to give the
correct quantum mechanical value for the contribution to the atom's
dipole moment coming from the electron's orbital motion. There are
other contributions, however, arising from the intrinsic spins of the
electron and proton.]