Mathematical Methods for Physicists: A concise introduction - Site Map

LINEAR VECTOR SPACES
or
U
~ ‡ ˆ U
~ 1 :
…5:25†
We often use Eq. (5.25) for the de®nition of the unitary operator.
Unitary operators have the remarkable property that transformation by a unitary
operator preserves the inner product of the vectors. This is easy to see: under
the operation U , a vector jvi is transformed into the vector jv 0 iˆU jvi. Thus, if
~ ~
two vectors jvi and jui are transformed by the same unitary operator U , then
~
u 0 v 0 ˆhU ujU viˆhujU ‡ U vi ˆ h uv ji;
~ ~ ~ ~
that is, the inner product is preserved. In particular, it leaves the norm of a vector
unchanged. Thus, a unitary transformation in a linear vector space is analogous
to a rotation in the physical space (which also preserves the lengths of vectors and
the inner products).
Corresponding to every unitary operator U , we can de®ne a Hermitian operator
H and vice versa by
~
~
U ˆ e i" H ~
; …5:26†
~
where " is a parameter. Obviously
U ‡ ˆ e …i"H † ‡
~ =e i"H 1 ~ ˆ U :
~ ~
A unitary operator possesses the following properties:
(1) The eigenvalues are unimodular; that is, if U
~
jvi ˆjvi, then jj ˆ1.
(2) Eigenvectors belonging to di€erent eigenvalues are orthogonal.
(3) The product of unitary operators is unitary.
The projection operators
A symbol of the type of juihvj is quite useful: it has all the properties of a linear
operator, multiplied from the right by a ket ji, it gives jui whose magnitude is
hvj i; and multiplied from the left by a bra hjit gives hvj whose magnitude is hjui.
The linearity of juihvj results from the linear properties of the inner product. We
also have
fjiv
u hjg ‡ ˆ jiu v h j:
The operator P j ˆ jij j hjis a very particular example of projection operator. To
~
see its e€ect on an arbitrary vector jui, let us expand jui:
222