# If A, B Hermitian, then <v|AB|v>=<v|BA|v>*. Why?

1. Feb 23, 2014

This is certainly an elementary question, so I would be all the more grateful for the answer. Given: A and B are two Hermitian operators and v is a vector in C. If <v|AB|v>=x+iy (for x and y real), then <v|BA|v> = x-iy.
Why?

2. Feb 24, 2014

### Fredrik

Staff Emeritus
Bra-ket notation can be confusing, so I will translate to inner product notation. If I denote the inner product of two arbitrary vectors x and y by (x,y), and denote the ket |v> by v, then <v|AB|v> means (v,ABv). The definition of "inner product" says that $(x,y)^*=(y,x)$ for all x,y. This implies that $(v,BAv)^*=(BAv,v)$. Now you can use that A and B are hermitian, and the definition of the adjoint operator to evaluate the right-hand side.
$$(v,BAv)^*=(BAv,v)=(B^*A^*v,v)=(A^*v,Bv)=(v,ABv).$$

3. Feb 24, 2014

Many thanks, Fredrik. That clears that step up.