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

  1. Feb 23, 2014 #1
    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. jcsd
  3. Feb 24, 2014 #2

    Fredrik

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    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.
    $$(v,BAv)^*=(BAv,v)=(B^*A^*v,v)=(A^*v,Bv)=(v,ABv).$$
     
  4. Feb 24, 2014 #3
    Many thanks, Fredrik. That clears that step up.
     
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