Lecture9Notes

# Lecture9Notes - Chem 120A Hermitian Operators Spring 2006...

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Unformatted text preview: Chem 120A Hermitian Operators 02/06/06 Spring 2006 Lecture 9 READING: QCS Chapter 3; FL3 Chapter 8-1 to 8-6; Supplemental reading posted in ”Lecture Notes” on blackboard.berkeley.edu We start by remembering Rule 3(the big rule) of our quantum mechanical postulates from last lecture: All physical quantities of classical mechanics (energy, position, momentum, etc) are represented by Hermitian operators which have the following property h i | O | j i * = h j | O | i i where O is a Hermitian operator which acts on states | i i and | j i The eigenvalue problem is an integral part of determining observables (measurable quantities) for quantum mechanical systems. An eigenvalue equation can be written in the form: (operator) × (eigenvector) = (eigenvalue) × (same eigenvector), where the eigenvalue is a constant factor. So that when an operator A acts on a state | α i A | α i = λ α | α i , likewise h α | A = h α | λ * α where α = 1 ... N If our state is normalized then h α | α i = 1 which just says that the probability of finding the state α in itself is 1. Then with the operator, A, acting on the ket h α | A → | α i = h α | λ α | α i (1) = λ α h α | α i (2) = λ α (3) Likewise when A acts on the bra h α | A ← | α i = λ * α So that when A is a Hermitian operator λ α = λ * α In other words λ α is a real number. This brings us to our next rule:is a real number....
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Lecture9Notes - Chem 120A Hermitian Operators Spring 2006...

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