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Unformatted text preview: PHY4604 R. D. Field Department of Physics Chapter3_1.doc University of Florida Eigenfunctions and Eigenvalues Eigenvalue Equation: Consider the special case of an hermitian operator A op = A op ↑ acting on a wave function | ψ> such that A op |ψ> = a |ψ> , where a is a constant. The constant a is called the “eigenvalue” and the wave function | ψ > is called the “eigenfunction” or “eigenket” . Eigenkets have Unit Norm: We require that the eigenkets have finite norm and we normalize the wave functions such that < ψ | ψ > = 1 .. Eigenvalues Value of Hermitian Operators: The eigenvalues values of hermitian operators are real. Proof: We know that < ψ |A op | ψ >=a and taking the complex conjugate of both sides gives a* = < Ψ |A op | Ψ > * = < Ψ |A op ↑ | Ψ > = < Ψ |A op | Ψ > = a, Thus, a = a* which implies that a is real. Expectation Value of A op : The expectation value ( i.e. average value) of A op is given by <A> = < ψ |A op | ψ > = a< ψ | ψ > = a . Also, <A 2 > = < ψ |A 2 op | ψ > = a 2 < ψ | ψ > = a 2 . Deviation from the Mean: The square of the deviation of A from the mean, <A> , is given by σ 2 = ( ∆ A) 2 ≡ <(A - <A>) 2 > = <A 2 > - (<A>) 2 = a 2 –a 2 = 0, and σ = ∆ A = 0. This means that the dynamical quantity A op can only have the definite value a ( i.e. precise value). Ordinarily, when you measure an observable O on an ensemble of identically prepared systems, all in the same state | Ψ >, you do not get the same result every time. For “determinate states” you do get the same result every time, you get the eigenvalue. Eigenfunctions of Hermitian Operators with Different Eigenvalues: Now suppose that A op | ψ 1 > = a 1 | ψ 1 > and A op | ψ 2 > = a 2 | ψ 2 > with a 1 ≠ a 2 , then < ψ 2 | ψ 1 > = 0 . Eigenfunctions of hermitian operators corresponding to different eigenvalues are orthogonal. Proof: < ψ 2 |A op | ψ 1 > = a 1 < ψ 2 | ψ 1 > and taking the complex conjugate of both sides yields (< ψ 2 | A op | ψ 1 >) * = a 1 * < ψ 2 | ψ 1 > * = a 1 < ψ 1 | ψ 2 > . But (< ψ 2 | A op | ψ 1 >) * = < ψ 1 | A op ↑ | ψ 2 > = < ψ 1 | A op | ψ 2 > = a 2 < ψ 1 | ψ 2 > Thus, (a 2-a 1 )< ψ 1 | ψ 2 > = 0 which implies that < ψ 1 | ψ 2 > = 0 , where I used the fact that a 1 and a 2 are real since A op is hermitian. Called “determinant states”! PHY4604 R. D. Field Department of Physics Chapter3_2.doc University of Florida Obsevables and Vector Spaces Eigenvalue Equation: Consider the case of an hermitian operator A op acting on a set of wave functions, | ψ n >, such that A op | ψ n > = a n | ψ n > where a n is a constant. Note a n is real and < ψ i | ψ j > = δ ij ( i.e. orthonormal set)....
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This note was uploaded on 04/19/2008 for the course PHY 4604 taught by Professor Field during the Spring '07 term at University of Florida.
- Spring '07