Mathematical Methods for Physicists: A concise introduction - Site Map

PROBLEMS
5.10 Given the following three vectors from the vector space of real 2 2
matrices:
j1i ˆ 0 1
; j2i ˆ 1 1 2 1
; j3i ˆ ;
0 0
0 1
0 2
determine whether they are linearly dependent or independent.
5.11 If S ˆ fj1i; ji; 2 ...; jnig is a basis for a vector space V, show that every set
with more than n vectors is linearly dependent.
5.12 Show that any two bases for a ®nite-dimensional vector space have the same
number of vectors.
5.13 Consider the vector space E 3 with the Euclidean inner product. Apply the
Gram±Schmidt process to transform the basis
j1i ˆ…1; 1; 1†; j2i ˆ…0; 1; 1†; j3i ˆ…0; 0; 1†
into an orthonormal basis.
5.14 Consider the two linearly independent vectors of Example 5.10:
jUi ˆ…3 4i†j1i‡…5 6i†j2i;
jWi ˆ…1 i†j1i‡…2 3i†j2i;
where j1i and j2i are an orthonormal basis. Apply the Gram±Schmidt process
to transform the two vectors into an orthonormal basis.
5.15 Show that the eigenvalue of the square of an operator is the square of the
eigenvalue of the operator.
1 1
5.16 Show that if, for a given A , both operators A L and A R exist, then
~ ~ ~
A
~
1
L
ˆ A
~
1
R A
~ 1 :
5.17 Show that if a unitary operator U can be written in the form U ˆ 1 ‡ ie F ,
~ ~ ~
where e is a real in®nitesimally small number, then the operator F is
~
Hermitian.
5.18 Show that the di€erential operator
p ˆ p d
~ i dx
is linear and Hermitian in the space of all di€erentiable wave functions …x†
that, say, vanish at both ends of an interval (a, b).
5.19 The translation operator T…a† is de®ned to be such that T…a†…x† ˆ
…x ‡ a†. Show that:
(a) T…a† may be expressed in terms of the operator
p ˆ p d
~ i dx ;
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