Problems
The symbols√
, , etc. are explained on page 1014.
1 Nearly all naturally occurring oxygen nuclei are the isotope(^16) O. The extremely neutron-rich isotope (^22) O has been produced in
accelerator experiments, but only with great difficulty, and little is
known about its properties. The only states that have been observed
and assigned reliable spins are the ground state, with spin 0, and
an excited state with spin 2 and an excitation energy of 3.2 MeV.
The excited state was detected by observing gamma rays for the
2 →0 transition. On the hypothesis that the spin-2 excited state
is a rotation, predict the gamma-ray energy that experimentalists
should expect from the 4→2 transition in the same rotational band.
√
2 For vectors in two dimensions, which of the following are
possible choices of a basis?
{ˆx} {xˆ,yˆ} {−ˆx,ˆx+ˆy} {xˆ,ˆy,xˆ+yˆ}
.Solution, p. 1052
3 (a) Consider the set of vectors in two dimensions. This set P
is a vector space, and can be visualized as a plane, with each vector
being like an arrow that extends from the origin to a particular
point. Now consider the linedefined by the equationy= xin Cartesian coordinates, and the rayrdefined byy=xwithx≥0. Sketchandr. If we considerandras subsets of the arrows in P, isa vector space? Isr?
(b) Consider the set C of angles 0≤θ < 2 π. Define addition on C by
adding the angles and then, if necessary, bringing the result back into
the required range. For example, ifx=πandy= 3π/2, thenx+y=
π/2. Thus if we visualize C as a circle, every point on the circle has
a single number to represent it, not multiple representations such
asπ/ 2 and 5 π/2. Suppose we want to make C into a vector space
over the real numbers, so that elements of C are the vectors, while
a scalarαcan beanyreal number, not just a number from 0 to 2π.
Then for example ifα= 2 is a scalar andv=πis a vector, then
αv = 0. Find an example to prove that C is not a vector space,
because it violates the distributive propertyα(v+w) =αv+αw.
.Solution, p. 1052
4 In the SI, we have three base units, the kilogram, the meter,
and the second. From these, we form expressions such as m/s to
represent units of velocity, and kg·m/s^2 for force. Show that these
expressions form a vector space with the rational numbers as the
scalars. What operation on the units should we take as the “ad-
dition” operation? What operation should scalar “multiplication”
be? .Solution, p. 1053
1008 Chapter 14 Additional Topics in Quantum Physics