This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Physics 225/315 January 21, 2008 Operators Matrix Representation An operator transforms a state. A  ψ =  φ . Expand the states in the z basis.  ψ =  + +  ψ +   ψ  φ =  + +  φ +   φ Then A  ψ =  φ bcomes A (  + +  ψ +   ψ ) =  + +  φ +   φ Take the inner product with +  and also with +  A  + +  ψ + +  A   ψ = +  φ A  + +  ψ + A   ψ = φ Write this as a matrix equation. ˆ +  A  + +  A  A  + A  !ˆ +  ψ ψ ! = ˆ +  φ φ ! 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 ψ  A  ψ = ψ  + +  ψ a + + ψ   ψ a = P + a + + P a where P + and P are the probabilities for finding each of the basis states....
View
Full Document
 Spring '10
 ROTHBERG
 Physics, Linear Algebra, Cos, Eigenvalue, eigenvector and eigenspace, Hilbert space

Click to edit the document details