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This is basically more of a math question than a physics-question, but I'm sure you can answer it. My question is about linear operators. If I write an operator H as (<al and lb> being vectors):
<alHlb>
What is then the relationship between H action the ket and H action on the bra. Is this for example true:
<alHlb> = H <alb>
Which rules determine how the operator acts?
Can it be thought of as a scalar such that instead:
<alHlb> = H* <alb>
My question came naturally as I was applying the position-state <xl to the left side of the schrödinger-equation (H being the hamiltonian and lψ(t)> the state vector:
<xlHlψ(t)>
does this equal to:
H<xlψ(t)>
I reckon it must because I've seen it work the other way around, but why is it precisely so. Doesn't it have to do with the fact that H is hermitian?
<alHlb>
What is then the relationship between H action the ket and H action on the bra. Is this for example true:
<alHlb> = H <alb>
Which rules determine how the operator acts?
Can it be thought of as a scalar such that instead:
<alHlb> = H* <alb>
My question came naturally as I was applying the position-state <xl to the left side of the schrödinger-equation (H being the hamiltonian and lψ(t)> the state vector:
<xlHlψ(t)>
does this equal to:
H<xlψ(t)>
I reckon it must because I've seen it work the other way around, but why is it precisely so. Doesn't it have to do with the fact that H is hermitian?