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Unformatted text preview: Physics 225/315 January 21, 2008 Operators Matrix Representation An operator transforms a state. A | = | . Expand the states in the z basis. | = | + + | + |- -| | = | + + | + |- -| Then A | = | bcomes A ( | + + | + |- -| ) = | + + | + |- -| Take the inner product with + | and also with-| + | A | + + | + + | A |- -| = + | -| A | + + | +-| A |- -| =-| Write this as a matrix equation. + | A | + + | A |--| A | +-| A |- ! + | -| ! = + | -| ! Average value The operator A is associated with the measurement of an observable. Expand operator A , as a spectral decomposition in terms of projection operators on basis states. A = | + a + + | + |- a--| where a + and a- are the measured values of the observable in each of the two basis states. For components of spin a + = h 2 and a- =- h 2 Evaluate | A | = | + + | a + + |- -| a- = P + a + + P- a- where P + and P- are the probabilities for finding each of the basis states....
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