Homework FOUR Ch 4: due October 1, 2007 PHYS 560
1. (based on Shankar Example 4.2.4) This is a long, calculationally intensive (fun!) prob-
lem. If using Mathematica, Maple, etc for evaluating some of the integrals, of course
please note this. It is in your interests to examine the integrals carefully to consider how
one might do them given only some integral tables. Also, the section on Γ functions
at the end of the book will be very useful. I have changed the notation from Shankar
slightly so that we use σas the parameter of width in the function. Although the valid
interval is not stated, note that wave function appears to be valid from −∞ to +∞(that
is, it is normalizable). Given the wave equation below, calculate many things. Note, this
is NOT the same wavefunction in book (because of that extra x!)
ψ(x) = Cxe
−(x−a)2
2σ2
.
(a) Find expectation value of hˆ
Xiand hˆ
X2i.
(b) Find the uncertainty (or the root mean square deviation) of ˆ
X, ∆ ˆ
X.
(c) Find the wave function expressed in momentum space, ˜
ψ(p).
(d) Find the the expectation values hˆ
Piand hˆ
P2i.
(e) Find the uncertainty in ˆ
P, ∆ ˆ
P2.
(f) Using work from above, Find the ∆ ˆ
X∆ˆ
P.
2. An ensemble of particles have been prepared in the state |ψ0i= 3|lz= 1i − i|lz=−1i,
where |lz= 1,0,−1iare the possible eigenstates of the operator ˆ
Lz. (See the text book
problem in the chapter for the operators ˆ
Lz,ˆ
Lx,ˆ
Ly. These eigenstates are orthonormal.
Although it is not necessary to turn it in, convince yourself that these operators do not
commute (and find the commutation rules), and convince yourself that, (as they are all
represented in the same basis set), that basis set for the representation is that of the
eigenkets of ˆ
Lz.
(a) The state |ψ0ias given is not normalized. Normalize it. That is now |ψ0i.
(b) Calculate the eigenkets and eigenvalues for the ˆ
Lxoperator. Calculate the eigenkets
and eigenvalues for ˆ
L2
z(These are used in next problem). Write down (because it
should be trivial) the eigenkets and eigenvalues for ˆ
Lz.
(c) For this ensemble, find hˆ
Lziand hˆ
Lxi.
(d) For the ensemble of particles in |ψ0i, an experimenter measures the observable lx
associated with ˆ
Lx. Calculate the probability that the observer measures lx= 1,
and lx= 0, and lx=−1? What is the probability that the observer measures
lx=.2? (Trick question).
(e) Experimenter takes ensemble of particles in |ψ0iand measures Lx, then takes those
particles with lx= 0 and performs a second measurement Lz. What is the prob-
ability in that second measurement, that the experimenter measures any particles
with lz= 0?
revision printed: September 24, 2007 Professor Thompson
