INDIAN INSTITUTE OF TECHNOLOGY MADRAS

DEPARTMENT OF PHYSICS

PH352 Quantum Physics Problem Set 1 11.1.2008

Some mathematical preliminaries: Linear vector spaces

A linear vector space (or LVS) V is a set of elements |i , |i , . . . called vectors,

with an operation called addition satisfying the following properties:
|i + |i = |i + |i V , |i , |i V
|i + (|i + |i) = (|i + |i) + |i
a unique null vector |i V such that |i + |i = |i , |i V
a unique vector |i V for every |i such that |i + (|i) = |i .
There is also an operation of multiplication by scalars a, b, . . . belonging to R
(the field of real numbers) or C (the field of complex numbers), such that
a |i V , a R or C and |i V
a (b |i) = (ab) |i
a (|i + |i) = a |i + a |i
(a + b) |i = a |i + b |i
1 |i = |i
0 |i = 0 . It is because of this property that we may as well use the usual
symbol 0 for both the null vector |i and for the usual scalar zero.
If the scalars a, b, . . . are restricted to the real numbers, the LVS is a real LVS; if
the scalars are complex numbers, we have a complex LVS. (The scalars may also
be drawn from fields other than the real or complex number fields, but we shall not
consider these generalizations.)

Every LVS has a dual that is also an LVS. The notation used for the elements
of this dual LVS is h| , h| . . . . The inner product h|i has the general properties
h|(|i + |i) = h|i + h|i
h|i = h|i
h|a i = ah|i
ha |i = a h|i
The norm of a vector is given by k k = h|i1/2 . The norm is positive definite
for every non-null vector. It vanishes iff |i = |i.

1. Check whether the following sets of elements form an LVS. If they do, find
the dimensionality of the LVS.
(a) The set of all n n matrices with complex entries.
(b) The set of all polynomials (of all orders) of a complex variable z.
(c) The set of all (Cartesian) tensors of rank 2 in three dimensions.

1
 (d) The set of all antisymmetric (Cartesian) tensors of rank 2 in three di-
mensions.
(e) The set of all 2 2 matrices whose trace is zero.
d2 y dy
(f) The set of all solutions of the differential equation 2
3 + 2y = 0 .
dx dx
(g) The set of all nn unitary matrices. (U is unitary iff U U = U U = I .)
(h) The set of all n n hermitian matrices (with multiplication by real
scalars).
2. In the three-dimensional LVS with basis vectors

1 0 0
|1 i = 0 , |2 i = 1 , |3 i = 0 ,
0 0 1

find a vector |i such that hi |i = 1 for i = 1, 2 and 3.

3. In a three-dimensional LVS, consider the three vectors

1 1 1
|1 i = 1 , |2 i 1 , |3 i = 0 .
1 0 1

(a) Show that they are linearly independent.

(b) Use the Gram-Schmidt procedure to construct an orthonormal basis
{|i i} from these three vectors.
4. The Cauchy-Schwarz inequality is of fundamental importance. It says
that
|h |i| k k k k ,
the equality holding iff |i and |i are linearly dependent. In terms of ordinary
vectors in Euclidean space, it amounts to saying that the cosine of the angle
between two vectors has a magnitude between 0 and 1, the limiting value of
unity occurring iff the vectors are collinear. Establish the Cauchy-Schwarz
inequality.
Hint: Consider the inner product h + a | + a i where |i, |i V, and
a is an arbitrary complex number. Choosing a appropriately leads to the
desired inequality.
5. Use the Cauchy-Schwarz inequality to establish the triangle or Minkowski
inequality
k + kk k + k k
for any two vectors |i and |i V.
6. (a) Let |i and |i be two linearly independent vectors in a real LVS. Find
the value of the (real) scalar that makes k k a minimum.
(b) Same problem, in a complex LVS. (Note that any complex number and
its complex conjugate are linearly independent!)
7. Show that the 2 2 unit matrix I and the three Pauli matrices, namely,
       
1 0 0 1 0 i 1 0
I= , 1 = , 2 = , 3 = ,
0 1 1 0 i 0 0 1
form a basis in the LVS of all 2 2 matrices. What happens if we restrict
ourselves to the set of all traceless 2 2 matrices?

2
 8. Which of the following infinite sequences (x1 , x2 , ) listed below belong to
`2 , the LVS of square-summable sequences?
(a) xn = (1)n (ln n)/n
(b) xn = n!/(2n)!
(c) xn = (2/n)n
(d) xn = (2n + 1)/(3n + 4)2
(e) xn = en /nn
(f) xn = 2n/2 .
9. Identify the functions that belong to L2 (, ) , the LVS of square-integrable
fucntions of a real variable:
(a) f (x) = (x2 + 1)1/4
(b) f (x) = ex cos x
2
(c) f (x) = e1/x
(d) f (x) = (sin x)/x
2
(e) f (x) = x3 ex
(f) f (x) = (tanh x)/x .
10. In an n-dimensional LVS, consider the vectors |k i (k = 1, 2, . . . , n) defined
by
|1 i = (1, 0, 0, , 0) , |2 i = (21/2 , 21/2 , 0, 0, , 0) ,
, |n i = (n1/2 , n1/2 , n1/2 , , n1/2 ) .
Construct a vector |i such that hk |i = 1 for every k (1 k n) .
11. The set of all n n matrices (with complex entries) forms an LVS (Problem
1(a) above). The inner product of two elements in this space may be defined
as
(A, B) = Tr (A B),
where A denotes the hermitian conjugate of A.
(a) If A is hermitian, show that
1
(A, A) (I, A)2 .
n

(b) If A is an arbitrary n n matrix, and U is an unitary n n matrix, show

that
1
(A, A) |(U , A)|2 .
n