Chapter6_3

# Chapter6_3 - x = mdx/dt Plane Traveling Probability Wave...

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PHY3063 R. D. Field Department of Physics Chapter6_3.doc University of Florida Expectation Values and Differential Operators (1) One Space Dimension: To make things easier we will start with just one space 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 ) , ( ) , ( ) , ( * Similarly the expectation value of x 2 is Ψ Ψ = >= < dx t x x t x dx t x x x ) , ( ) , ( ) , ( 2 * 2 2 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
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Unformatted text preview: x = mdx/dt. Plane Traveling Probability Wave Amplitude: The key on how to proceed comes from looking at a plane wave h / ) ( ) ( ) , ( Et x p i t kx i x e e t x − − = = Ψ ω , where I used h / x p k = and h / E = . Notice that ) , ( ) , ( t x p i x t x x Ψ = ∂ Ψ ∂ h and x t x i t x p x ∂ Ψ ∂ − = Ψ ) , ( ) , ( h ) , ( ) , ( t x E i t t x Ψ − = ∂ Ψ ∂ h and t t x i t x E ∂ Ψ ∂ = Ψ ) , ( ) , ( h Associate Dynamical Variables with Differential Operators: Postulate that x i p x ∂ ∂ − ↔ h and t i E ∂ ∂ ↔ h The key to the development of Quantum Mechanics! Need to express this in terms of x and t!...
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