Chapter6_14

# Chapter6_14 - mean, &amp;amp;amp;lt;H&amp;amp;amp;gt; , is...

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PHY3063 R. D. Field Department of Physics Chapter6_14.doc University of Florida Eigenfunctions and Eigenvalues Eigenvalue Equation: Consider the special case of an hermitian operator H op acting on a wavefunction Ψ (x) such that H op Ψ (x) = a Ψ (x), where a is a constant (Note a is real since H op = H op ). The constant a is called the “eigenvalue” and the wavefunction Ψ (x) is called the “eigenfunction” . Expectation Value of H op : The expectation value ( i.e. average value) of H op is given by a dx t x t x a dx t x H t x H op = Ψ Ψ = Ψ Ψ >= < ) , ( ) , ( ) , ( ) , ( * * , since 1 ) ( ) ( * = Ψ Ψ dx x x . Also, = Ψ Ψ = Ψ Ψ >= < 2 * 2 2 * 2 ) , ( ) , ( ) , ( ) , ( a dx t x t x a dx t x H t x H op , Deviation of H from the Mean: The square of the deviation of H from the
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Unformatted text preview: mean, &amp;lt;H&amp;gt; , is given by ( H) 2 &amp;lt;(H - &amp;lt;H&amp;gt;) 2 &amp;gt; = &amp;lt;H 2 &amp;gt; - (&amp;lt;H&amp;gt;) 2 , where H is the root-mean-square deviation from the mean. We see that if H op (x) = a (x), where a is a constant, then &amp;lt;H&amp;gt; = a and H = 0 . This means that the dynamical quantity H op can only have the definite ( i.e. precise value) a . In general if H op n (x) = a n n (x), then n corresponds to an eigenstate of the system with a definite eigenvalue of the dynamical variable H op equal to a n and n is the quantum number that specifies the state. Called the dispersion Normalized Wavefunctions...
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