Problem 1:
A negative K meson with mass m = 1000 electron masses is captured into a circular Bohr orbit around a lead nucleus (Z = 82). Assume it starts with principal quantum number n = 10 and then cascades down through n = 9, 8, 7, …, etc. What is the energy of the photon emitted in the n = 10 to n = 9 transitions?
Problem 2:
In an atom, a valence electron experiences a long range Coulomb force and the
potential well representing the interaction supports an infinite spectrum of
bound states. In contrast, the interaction between the outermost electron in a
negative ion and the neutral atomic core, which is weak and short ranged,
results in only a finite number of bound states.
For the case of s states (l = 0), the negative ion may be
approximated by a model in which the interaction between the outermost electron
of mass m and the core is represented by an attractive one-dimensional
central potential of the form
V(r) = -V0, 0 < r < a;
V(r) = 0 r ³ a.
(a) Solve the time independent Schroedinger equation and determine an
expression, in the form of a transcendental equation, relating the eigenvalues
of this system to the quantities V0, a, and m. Solve this equation graphically.
(b) Show, graphically or otherwise, that there will exist no bound states
unless
R = (2mV0a2/h2)1/2
³ p/2.
(c) Determine how many bound states exist if R = p.
(d) The condition for the existence of bound states depends on the product
of the depth and the square of the width of the potential well. Explain this in
terms of the uncertainty principle.
Problem 3:
Let r, f, z be the
cylindrical coordinates of a spinless particle. Assume the potential
energy of this particle depends only on r and not on f
and z.
(a) Write in cylindrical coordinates the differential operator associated
with the Hamiltonian. Show that H commutes with Lz and Pz.
Show from this that the wave function associated with the stationary states of
the particle can be written as
ynmk(r, f,
z) = fnm(r)exp(imf)exp(ikz),
where the values that can be taken on by the indices m and k are to be
specified.
(b) Write in cylindrical coordinates the eigenvalue equation of the
Hamiltonian of the particle. Derive from it the differential equation
which yields fnm(r).
Problem 4:
Consider the following non-pure state for a hydrogenic
atom:
|y> = a1|y100>
+ a2|y200> + a3|y210>
+ a4|y32-1> + a5|y432>
a1 = (3/10)1/2, a2 = (1/10)1/2,
a3 = (2/10)1/2, a4 = (1/10)1/2,
a5 = (3/10)1/2.
i) Show that |y> is normalized. What
property of the hydrogen wave functions must you exploit to show that?
ii) Calculate the probability of observing a 1s state.
iii) Calculate the probability of observing a state with n > 2.
iv) What are the possible values you would get if you measured the quantity
associated with Lz? Give the probability of measuring each of these
values.
v) Calculate the expectation value of Lz.
vi) What are the possible values you would get if you measured the quantity
associated with L2? Give the probability of measuring each of these
values.
vii) Calculate the expectation value of L2.
Problem 5:
Find the eigenvalues and eigenfunctions for a particle in a box with sides a, b, c by solving the time independent Schroedinger equation (-h2/(2m)) Ñ2f(r) + U(r)f(r) = Ef(r). Do not just write down your answer but derive it.
Problem 6:
A hydrogen atom is placed in a uniform electric field, E = -E.
Place the proton at the origin of your coordinate system. An electron in the
hydrogen atom then has potential energy U(r) = -kqe2/r
– qeEz. U(0,0,z) becomes increasingly positive for negative z . For
positive z the potential energy U(0,0,z) contains a “hill” and then decreases
with increasing z.
Sketch the potential energy of the electron, U(0,0,z), as function of z and
calculate the energy at the maximum at positive z. Equate this energy to the
energy of the unperturbed (zero field) hydrogen energy level and thereby
determine the value of the field required to field-ionize a hydrogen atom with
principle quantum number n (neglect tunneling).
Problem 7:
Two atoms of masses m1 and m2 are bound together in a diatomic molecule. The separation of their nuclei is r. What are the rotational kinetic energy levels of the molecule? How does the energy of the first excited state of 13C16O compare to that of 12C16O?