Download 1. Consider an infinite dimensional vector space consisting of all

1. Consider an infinite dimensional vector space consisting of all positive real numbers with
vector addition defined as:
|x ! + |y ! = |xy !
(1)
so that, for example
|2.2 ! + |4 ! = |8.8 !
(2)
(a) What is the “additive identity” vector for this space? What is the “additive inverse”
vector for |x !?
(b) If we expand the domain to all real numbers, do we still have a vector space? Why or
why not?
(c) Our vector space comes with scalars in R which we will define to have the normal
addition and multiplication rules among themselves. Show, however, that the rules of
vector spaces forbid us to define scalar-vector multiplication in the usual way.
2. Consider a three-dimensional vector space. When a particular orthonormal basis, which we
shall designate by |1 !, |2 !, and |3 !, is used to define a matrix representation, the operators
A and B have matrix representations




a 0
0
b 0
0
A ←−−→ 0 −a 0 
B ←−−→ 0 0 −ib
(3)
|j !
|j !
0 0 −a
0 ib 0
with a and b both real.
(a) Obviously A has degenerate eigenvalues. Does B also have degenerate eigenvalues?
(b) Show that A and B commute.
(c) Find a new set of orthonormal kets which are simultaneous eigenkets of both A and
B. Specify the eigenvalues of A and B for each of the three eigenkets. Does your
specification of eigenvalues completely characterize each eigenket?
(d) Find the unitary operator that transforms the {|j !} basis to the eigenbasis of A and
B that you have found. Write the matrix representations of |1 !, |2 !, and |3 ! in the
eigenbasis you have found.
3. Suppose F is a Hermitian operator and U (α) = eiαF with α real. Given an arbitrary operator
A, show by power series expansion that, to first order in α, the unitary transformation of A
satisfies
U † (α) A U (α) = A − i α[F, A]
(4)
where [F, A] = F A − AF is the commutator of F and A. By “first order in α”, we mean for
you to use the power series expansion and ignore terms of order α2 and higher.
Under what condition is the operator A unchanged by the transformation? If A is Hermitian,
what does this say about how the eigenvectors of A are affected by the transformation?
This is called an infinitesimal unitary transformation. The operator A is invariant under the
transformation if it is not changed. If A is invariant under the infinitesimal transformation,
then it also invariant under any finite version of the transformation (i.e., without the firstorder in α approximation). A is also said to be conserved if this holds. F is said to generate
the transformation. If A is the Hamiltonian (energy operator) for a system, then we say the
transformation is a symmetry of the system.
2
4. Suppose {|j !} and {|j " !} are orthonormal bases for a N -dimensional inner product space V.
Show that a unitary basis transformation operator that transforms the |j ! basis into the |j " !
basis,
U |j ! = |j " !
(5)
may be written as
U=
N
%
j=1
&
' &
'
&
'
|j " !%j | = |1 " !%1 | + |2 " !%2 | + · · · + |N " !%N |
(6)
Note that this is not our conventional bilinear form because |j ! and |j " ! belong to different
bases.
Show that the operator is unitary by showing that U † U = I. Be careful here, we have never
shown that
& "
'†
|j !%j | = |j !%j " |
(7)
You will need to show this to calculate U † .
5. Given an orthonormal basis {|j !} for a vector space V and two vectors |a ! and |b ! defined
by
1
|a ! = √ |1 ! +
2
1
|b ! = √ |1 ! +
3
i
1
|2 ! + |3 !
2
2
i
√ |3 !
3
(8)
(9)
(a) Are the vectors normalized? If not, normalize them.
(b) Calculate the matrix representation of the projection operators onto the subspaces Va ,
the subspace spanned by |a !, and Vb , the subspace spanned by |b !.
(c) Gram-Schmidt orthogonalize |a ! and |b !. (It is suggested you take |b ! as your new |1 !
to minimize the algebra.)
6. Given a Hermitian operator Ω and α a real number with |α| < 1.
(a) It is possible to define by power series expansion the function ln(I + α Ω), where ln
indicates “natural logarithm.” Under what condition on Ω is this operator well defined
for all real α with |α| < 1 (i.e., the power series is convergent for all real α with |α| < 1)?
(b) Calculate the derivative of ln(I + α Ω) with respect to α using the power series for ln.
Rewrite the series in a closed form (not a power series). Again, assume α is real and
|α| < 1. Is the convergence condition on Ω for this series the same as in (a)?
You will need to use the following, which hold for x real and |x| < 1:
ln(1 + x) =
∞
%
n=1
(−1)n+1
xn
n
3
(1 + x)−1 =
∞
%
n=0
xn
(10)