Transcribed Text
3
Hydrogen and the Postulates [32 points]
an electron in a hydrogen potential. In this problem as express usual. We all
energies For this as problem multiples we of will the consider Bohr energy E0. All basis states are labeled by {n,€,m))
will ignore the electron's spin in this problem!
Postulate 2 (Observables): The electric dipole moment vector is an observable whose op 0
erator
is given by a = ef Consider the equal 1s2p superposition state given at time t =
by
Ja(0)) = 12(11,0,0)+12.19).
(a)
[6 points] Show that the expectation value of the dipole moment vector is oscillatory in time
for the state a(t)). That is, show (d2) = Acoswt. You will have to determine the frequency and
amplitude of oscillation w and A.
[Note: There are a few groundwork steps you will have to do! One first step would be to find the
timedependent wave function for this system 2(r,8,4.t). While you may use integral solvers or
tables to help you with this problem you must show your work here of the setup and take some steps
to simplify it before resorting to a solver. Simply jumping from the initial integral to the final answer
is not sufficient.]
Postulates 3, 4, and 5 (Measurement Possibilities, Probabilities, and Collapse) Next
consider the states
=
where a is a real, positive constant. Note that I have already normalized these states for you! At
time t = 0 the magnitude of angular momentum L2 of the state IA) is measured and the energy of
the state l7) is measured.
(b) [6 points] What are the possible results of this measurement and what are the probabilities
of getting each result? Describe and set up (but don't bother solving) how to find the probability
that a measurement of energy for the state hr) would yield the firstexcited state energy?
(c)
[6 points] For each of the possibilities you found in part (b), find the state of the system
immediately after making the measurement of L2 on the state IB). Describe and set up (but don't
bother solving) how to find the state of the system immediately after making the energy measurement
on the state Iy).
Postulate 6 (Time Evolution): Instead of measuring, suppose we start with the states
and hr(0)) and let them naturally evolve in time.
(d) [5 points] What is the state 8(t)) at a later time? Describe and set up (but don't bother
solving) how to find the state brtt)) at a later time.
Parity Recall that parity is an active transformation whose action is to spatially invert the system. the
The parity operator it is both Hermitian and unitary and under the unitary transformation
position and momentum operators transform as F F
and
p
H
p

As a consequence the
spherical coordinates transform as (r,8,y)
(r,T,,,+++T).
(e)
[5 points] Show that L, commutes with the parity operator.
[Hint: First write an equation showing how L. transforms under parity and then manipulate this
equation until you get a commutator on one side and zero on the other!]
(f)
[4 points] For each of the states (1,0,0), (2,1,0), and (0)), determine if the state is an
eigenstate of parity and, if applicable, determine the eigenvalue.
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