Chapter3_1 - PHY4604 R D Field Eigenfunctions and...

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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,
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