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# Dirac_1 - sides to be equal to the same constant C Hence 2...

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PHY4604 R. D. Field Department of Physics Dirac_1.doc University of Florida Schrödinger’s Equation (Non-Relativistic) The Classical Hamiltonian: Classically the energy is the sum of the kinetic energy plus the potential energy as follows ( in one dimension ): ) ( 2 2 x V m p E x + = , and hence corresponding Quantum Mechanical Hamiltonian operator is ) ( 2 2 2 2 x V x m H op + = h and t i H op = h . We now operate on the wave function Ψ (x,t) with both forms of H op yielding t t x i t x x V x t x m Ψ = Ψ + Ψ ) , ( ) , ( ) ( ) , ( 2 2 2 2 h h . Look for “stationary state” solutions of the form ) ( ) ( ) , ( t x t x Φ = Ψ ψ . Substituting Ψ (x,t) into the Schrodinger’s equation yields Φ Φ = + dt t d i t x x V dx x d m x ) ( ) ( 1 ) ( ) ( ) ( 2 ) ( 1 2 2 2 h h ψ ψ ψ The only way both sides can be equal for all values of x and t is for both
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Unformatted text preview: sides to be equal to the same constant C. Hence, ) ( ) ( ) ( ) ( 2 2 2 2 x C x x V dx x d m = + − h ) ( ) ( t iC dt t d Φ − = Φ h The time equation is a first order equation with one solution t C i e t h − = Φ ) ( and t C i e x t x h − = Ψ ) ( ) , ( . We see that C dx t t x t x i dx t x E t x E op = ∂ Ψ ∂ Ψ = Ψ Ψ >= < ∫ ∫ ∞ ∞ − ∗ ∞ ∞ − ∗ ) , ( ) , ( ) , ( ) , ( h 2 2 2 2 2 2 ) , ( ) , ( ) , ( ) , ( C dx t t x t x dx t x E t x E op = ∂ Ψ ∂ Ψ − = Ψ Ψ >= < ∫ ∫ ∞ ∞ − ∗ ∞ ∞ − ∗ h Hence 2 2 = > < − > < = Δ E E E and E C = . 1 st order differential equation!...
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