9: Orthonormality, completeness, and projection

Calvin W. Johnson November 5, 2013

Unitary matrices, that is, matrices which satisfy

U−1 = U†, (1) are ubiquitous and important in quantum mechanics, in particular because they have some useful properties we’ll dissect here.

1 Orthornomality

From Eq. (1), we get U†U = 1. (2) What this really tells us is that the columns of the matrix U form a set of orthnormal vectors. We can interpret the columns of any matrix as a set of column vectors, e.g.,

(U) = (~v1,~v2,~v3, . . . ,~vN ) , (3) whereas for the inverse we get row vectors,

 †  ~v1  ~v†   2  −1
†
 †  U = U ==  ~v3  , (4)  .   .   .  † ~vN so that, following the usual rules for matrix multiplication,

 †  ~v1  ~v†   2  †  †  U U =  ~v3  (~v1,~v2,~v3, . . . ,~vN ) =  .   .   .  † ~vN

1  † † † †  ~v1 · ~v1 ~v1 · ~v2 ~v1 · ~v3 . . . ~v1 · ~vN  ~v† · ~v ~v† · ~v ~v† · ~v . . . ~v† · ~v   2 2 1 2 2 3 2 N   † † † †   ~v3 · ~v1 ~v3 · ~v2 ~v3 · ~v3 . . . ~v3 · ~vN  (5)    ..   .  † † † † ~vN · ~v1 ~vN · ~v2 ~vN · ~v3 . . . ~vN · ~vN or, using Dirac’s bra-ket notation,

  hv1|  hv2|     hv3|  =   (|v1i, |v2i, |v3i,..., |vN i) =  .   .  hvN |   hv1|v1i hv1|v2i hv1|v3i ... hv1|vN i  hv2|v1i hv2|v2i hv2|v3i ... hv2|vN i     hv3|v1i hv3|v2i hv3|v3i ... hv3|vN i    (6)  ..   .  hvN |v1i hvN |v2i hvN |v3i ... hvN |vN i But since we have †
−1
U U ij = U U ij = δij, (7) we get, for the columns of U (interpreted as vectors)

hvi|vji = δij (8) that is, they form an orthonormal set.

2 Completeness

From U†U = 1 we derived orthonormality. But we also expect UU† = 1. (9) What do we get from this? Something powerful, it turns out. Let’s try it:   hv1|  hv2|    †  hv3|  U U = (|v1i, |v2i, |v3i,..., |vN i)   = 1 (10)  .   .  hvN | Following carefully the rules for matrix-matrix multiplication, we get

|v1ihv1| + |v2ihv2| + |v3ihv3| + + ... + |vN ihvN | = 1 (11)

2 or, in more compact form, N X |viihvi| = 1. (12) i=1 This is known as a completeness relation and is very useful. While you may not be familiar with the completeness relation, it is best to think of it as the ‘flip side’ of orthonormality, as we can see from the definition of unitary matrices.

You try: While you have had practice in taking dot products such as ~v∗ · ~w = hv|wi, you probably don’t have practice in computing |wihv| (also sometimes called an outer product). However if you follow the rules of ma- trix multiplication, it is straightforward. Here is an example; be sure you can reproduce it!  1   2 3  2 3
= (13) −1 −2 −3

3 Projection

Without you realizing it, we have been using the completeness relation all along. Consider the expansion of a vector into components in a basis: X |wi = wi|vii (14) i where, if the set of basis vectors {|vii} are orthonormal,

wi = hvi|wi (15)

We can insert this into (14), using the fact that because the wi are scalars, we can put them anywhere: X X |wi = |viiwi = |viihvi|wi (16) i i But we can derive this another way: use the completeness relation, which is simply a fancy but useful way to write 1: ! X X |wi = 1 · |wi = |viihvi| |wi = |viihvi|wi (17) i i In other words, we were able to use the completeness relation to project a vector onto its components in a particular basis. For example, you know that for vectors |αi and |βi, we can take the inner product between them by using their components in a basis {|vii} :

X ∗ hα|βi = ai bi, (18) i

3 where ai = hvi|αi and bi = hvi|βi. But we can arrive at this directly by using the completeness relation: ! X hα|βi = hα|1|βi = hα| |viihvi| |βi = i X X ∗ X ∗ hα|viihvi|βi = hvi|αi hvi|βi = ai bi (19) i i i where we used the very important relation hv|wi = hw|vi∗. The next homework set (#8) will give you practice in applying these rela- tions.

You try : We can write the trace of an operator or matrix as X hvi|Tˆ|vii. (20) i Use the completness relation to prove that Tr AB = Tr BA.

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