Mathematical Tools for Physics

7—Operators and Matrices 193

question: Do I start over with a new basis, or can I use the work that I’ve already done to transform everything
into the new basis?
For linear transformations, this becomes the problem of computing the components of an operator in a new
basis in terms of its components in the old basis.
First: Make sure that I can do this for vector components, something that I ought to be able to do. The
equation ( 5 ) defines the components with respect to a basis,anybasis. If I have a second proposed basis, then
by the definition of the word basis, every vector in that second basis can be written as a linear combination of the
vectors in the first basis. I’ll call the vectors in the first basis,~eiand those in the second basis~e′i, for example in
the plane you could have

~e 1 =ˆx, ~e 2 =ˆy, and ~e′ 1 = 2ˆx+ 0. 5 ˆy, ~e′ 2 = 0. 5 ˆx+ 2ˆy (31)

Each vector~e′iis a linear combination* of the original basis vectors:

~e′i=S(~ei) =

∑

j

Sji~ej (32)

This follows the standard notation of Eq. ( 6 ); you have to put the indices in this order in order to make the
notation come out right in the end. One vector expressed in two different bases is still one vector, so

~v=

∑

i

v′i~e′i=

∑

i

vi~ei

and I’m using the fairly standard notation ofv′ifor theithcomponent of the vector~vwith respect to the second
basis. Now insert the relation between the bases from the preceding equation ( 32 ).

~v=

∑

i

v′i

∑

j

Sji~ej=

∑

j

vj~ej

• There are two possible conventions here. You can write~e′iin terms of the~ei, calling the coefficientsSji,
  or you can do the reverse and callthosecomponentsSji. Naturally, both conventions are in common use. The
  reverse convention will interchange the roles of the matricesSandS−^1 in what follows.