a complex issue that we will come back to when we discuss Stern–Gerlach experiments in
Chapter 7 and entangled states in Chapter 30.
Problems
Homework Assignment1.14
01: Basic
Formalism I (Due: 01/28/2016)
Problem 1 (24%, 4% each part)
1. Define the following two state vectors as column matrices:
" #
1
|α 1 ⟩ =
0
" #
0
|α 2 ⟩ =
1
and
with their Hermitian conjugates given by
$
%
⟨α 1 | = 1 0
and
respectively. Show the following for i, j = 1, 2:
(i) The |α i ⟩’s are orthonormal.
(ii) Any column matrix
22
$
%
⟨α 2 | = 0 1
" #
a
b
Basic formalism
can be written as a linear combination of the |α i ⟩’s.
(iii) The outer products |α i ⟩ ⟨α j | form 2 × 2 matrices which can serve as operators.
(iv) The |α i ⟩’s satisfy completeness relation
!
|α i ⟩ ⟨α i | = 1
i
where 1 represents a unit 2 × 2 matrix.
(v) Write
"
#
a b
A=
c d
as a linear combination of the four matrices formed by |α i ⟩ ⟨α j |.
(vi) Determine the matrix elements of A such that |α 1 ⟩ and |α 2 ⟩ are simultaneously the
eigenstates of A satisfying the relations
A |α 1 ⟩ = + |α 1 ⟩
and A |α 2 ⟩ = − |α 2 ⟩ .
(The above properties signify that the |α i ⟩’s span a Hilbert space. These abstract
representations of the state vectors actually have a profound significance. They
represent
thestates
states
of particles
spin(e.g.,
½. electrons).
We will discuss this in detail in
represent the
of particles
with with
spin-half
Chapter 5.)
2. Show that if an operator A is a function of λ then
1
dA−1
= −A−1
dA
A−1 .
Problem 2 (40%)
Let A be a positive definite Hermitian operator (i.e., its matrix representation is positive
definite). Prove that for two arbitrary states u and v ,
u Av
2
≤ u Au v Av .
[Hint: All eigenvalues of A are positive, or equivalently, ψ A ψ ≥ 0 for any state ψ . One
has to use this property to prove the above.]
Problem 3 (36%) [Note: I want you to be familiar with matrix math, so please DON’T use
Mathematica in this problem.]
(a) [6%] By considering the commutator, show that the following two matrices:
⎡ 1 0 1 ⎤
⎡ 2 1 1 ⎤
⎢
⎥
A = ⎢ 0 0 0 ⎥ and B = ⎢⎢ 1 0 −1 ⎥⎥ ,
⎢⎣ 1 0 1 ⎥⎦
⎢⎣ 1 −1 2 ⎥⎦
can be simultaneously diagonalized.
(b) [20%] Find the 3 eigenvectors common to both. (This part is slightly tricky due to degeneracy.)
(c) [10%] Verify that under a unitary transformation to this basis, both matrices are diagonalized.
2