- #1
binbagsss
- 1,254
- 11
Homework Statement
Hi,
Just watching Susskind's quantum mechanics lecture notes, I have a couple of questions from his third lecture:
Homework Equations
[/B]
1) At 25:20 he says that
## <A|\hat{H}|A>=<A|\hat{H}|A>^*## [1]
##<=>##
##<B|\hat{H}|A>=<A|\hat{H}|B>^*=## [2]
where ##A## and ##B## are vectors, ##H## is a hermitian operator and ##^*## means to complex conjugate.
The proof of the bottom implying the top is trivial, setting ##B=A##.
I have attempted the proof that the top implies the bottom :
The Attempt at a Solution
Let ##|B>=\hat{H}|A>##
Then ##<B|=<A|\hat{H^*}##
Consider ##<B|\hat{H}|A>=<A|\hat{H^*}\hat{H}|A>=<A|\hat{C}|A>##
Where ##\hat{H^*}\hat{H}=\hat{C}##, since ##H=H^*## is a hermitian operator, they commute and so by the theorem that operators that commute are hermitian, ##\hat{C}## is hermitian (or is this theorem made redundant by the fact that ##\hat{C}## is real and all real operators are hermitian?) , so [1] which holds for a hermitian operator is true and therefore
##<A|\hat{C}|A>=<A|\hat{C}|A>^*## by [1]
##=<A|\hat{H^*}\hat{H}|A>=<A|\hat{H^*}|B>^*##
##=<A|\hat{H}|B>^*## by ##H=H^*##.
- Is this last line valid? I'm not sure that is the strict definition of hermitian as different notes seem to suggest different things, some define hermitian as simply ##H=H^*## others more via the inner product and [2] holding.
Many thanks in advance.