Assignment 2: Zettili Chapter 2 & 3 problems
Due: 13 March 2019, 15:00
For this assignment you are required to use the matrix representation of state vectors and operators. As we said in
class, we will not necessarily cover all the mathematical tools of QM in class. Therefore, work through section 2.5
of Zettili on your own. The concepts and mathematics presented there is not beyond your ability.
1. Consider two kets |αi and |βi. Let {|a1 i , |a2 i , . . .} be a complete set of basis kets. Taking into account that
|αi and |βi can be expanded as a linear combination of the basis kets, find the matrix representation of the
operator |αihβ| where the matrix entries should be expressed in terms of hai |αi and hai |βi. Start with the
definition of the matrix representation of an operator in a given basis and work from there. All steps must be
clear.
2. Consider a space of kets spanned by the eigenkets {|ai i} (with eigenvalues ai ) of a Hermitian operator A.
There is no degeneracy. Prove that
Y
(A − ai )
ai
is the null operator.
3. Consider a neutron which is confined to an infinite potential well of width a = 8 fm. At time t = 0 the
neutron is assumed to be in the state
r
! r
!
πx r 2
4
2πx
8
3πx
sin
+
sin
+
sin
Ψ(x, 0) =
7a
a
7a
a
7a
a
(a) If an energy measurement is carried out on the system, what are the values that will be found for the
energy and with what probability? Express your answer in MeV.
(b) If this measurement is repeated on many identical systems, what will be the average energy? Show
how you get to the answer (also in MeV).
(c) Using the uncertainty principle, estimate the order of magnitude of the neutron’s speed in the well.
Express your answer as a function of the speed of light, c.
4. In Zettili it is shown that if two operators A and B commute then it follows that they share the same set of
orthonormal eigenkets. Now, let A and B be observables and suppose that the simultaneous eigenkets of A
and B, denoted by |a, bi, form a complete orthonormal set of eigenkets, that is, a basis for the space of kets.
Can we always conclude that
[A, B] = 0?
If your answer is yes, prove the assertion. If your answer is no, give a counter example. You can assume that
there is no degeneracy.
Note: We can rephrase this as follows: Two operators commute if and only if they have a set of simultaneous
eigenkets that form a complete and orthonormal basis.
5. Consider a quantum system in state |ψi. The state operator is defined as Pψ = |ψihψ|. Show that
X
hAi = hψ|A|ψi =
hφm |Pψ A|φm i = Tr(Pψ A)
m
where in the last term A is the matrix representation of the operator for an observable and {|φm i} is a complete set of eigenstates of A. Tr() is the trace of the quantity in the brackets. Hint: You have to use the
completeness relation (eq. 2.172) twice which will give a double summation. Then identify the factor that
is the matrix representation of A and use the definition of the state operator. See also p.110.
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