Mathematical Methods for Physicists: A concise introduction - Site Map

LINEAR VECTOR SPACES
Change of basis
The choice of base vectors (basis) is largely arbitrary and di€erent representations
are physically equally acceptable. How do we change from one orthonormal set of
base vectors j' 1 i; j' 2 i; ...; j" n i to another such set j 1 i; j 2 i; ...; j n i? In other
words, how do we generate the orthonomal set j 1 i; j 2 i; ...; j n i from the old
set j' 1 i; j' 2 i; ...; j' n ? This task can be accomplished by a unitary transformation:
j i iˆU
~
j' i i …i ˆ 1; 2; ...; n†: …5:29†
Then given a vector jXi ˆ Pn
iˆ1 a i j' i i, it will be transformed into jX 0 i:
jX 0 iˆU
~
jXi ˆU
~
X n
iˆ1
a i j' i i ˆ Xn
iˆ1
U a i j' i i ˆ Xn
a i j i i:
~
iˆ1
We can see that the operator U
~
equation
possesses an inverse U
~ 1 which is de®ned by the
j' i iˆU
~ 1 j i i
…i ˆ 1; 2; ...; n†:
The operator U is unitary; for, if jXi ˆ Pn
iˆ1 a i j' i i and jYi ˆ Pn
iˆ1 b i j' i i, then
~
hXjYi ˆ Xn X n
a*b i j ' i j' j ˆ a*b i i ; hUXjUYi ˆ Xn X n
a*b i j i j j ˆ a*b i i :
i;jˆ1
iˆ1
i;jˆ1
iˆ1
Hence
U 1 ˆ U ‡ :
The inner product of two vectors is independent of the choice of basis which
spans the vector space, since unitary transformations leave all inner products
invariant. In quantum mechanics inner products give physically observable quantities,
such as expectation values, probabilities, etc.
It is also clear that the matrix representation of an operator is di€erent in a
di€erent basis. To ®nd the e€ect of a change of basis on the matrix representation
of an operator, let us consider the transformation of the vector jXi into jYi by the
operator A
~
:
jYi ˆA
~
j…Xi:
…5:30†
Referred to the basis j' 1 i; j' 2 i; ...; j'i; jXi and jYi are given by
jXi ˆPn
iˆ1 a i j' i i and jYi ˆPn
iˆ1 b ij' i i, and the equation jYi ˆA jXi becomes
~
X n
iˆ1
X n
b i j' i i ˆ A
~
jˆ1
a j
' j :
224