PHY4604 R. D. Field Department of Physics Chapter1_10.doc University of Florida Expectation Values and Differential Operators (1) One Space Dimension: To make things easier we will start with just one spatial dimension x so that dx t x dx t x 2 ) , ( ) , ( Ψ = ρ is the probability of finding the particle at time t between x and x+dx and ∫ ∫ Ψ Ψ = Ψ = 2 1 2 1 ) , ( ) , ( ) , ( ) , , ( * 2 2 1 x x x x dx t x t x dx t x t x x P is the probability of finding the particle at time t between x 1 and x 2 ( i.e. x 1 ≤ x ≤ x 2 ). The average value of x is called the “expectation value” of x and is given by ∫ ∫ ∞ ∞ − ∞ ∞ − Ψ Ψ = >= < dx t x x t x dx t x x x ) , ( ) , ( ) , ( * Dynamical Quantities: What about momentum and energy? The average momentum (“expectation value” of p x ) is ∫ ∞ ∞ − Ψ Ψ >= < dx t x p t x p x x ) , ( ) , ( * How do we proceed? Classically x = x(t) and p x = mdx/dt . Schrödinger’s Equation:
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