Dirac_3 - Hence 2 2 2 2 2 2 x C x mc dx x d c = − h 2 2 2...

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PHY4604 R. D. Field Department of Physics Dirac_3.doc University of Florida Klein-Gordon Equation Relativistic Schrödinger’s: A relativistic free particle with mass m satisfies the following equation ( one dimension ): 2 2 2 2 ) ( ) ( mc cp E x + = , and the corresponding Quantum Mechanical Hamiltonian operator ( squared ) is 2 2 2 2 2 2 2 ) ( mc x c H op + = h and 2 2 2 2 t H op = h . We now operate on the wave function Ψ (x,t) with both forms of H op squared yielding 2 2 2 2 2 2 2 2 2 ) , ( ) , ( ) ( ) , ( t t x t x mc x t x c Ψ = Ψ + Ψ h h . Look for “stationary state” solutions of the form ) ( ) ( ) , ( t x t x Φ = Ψ ψ . Substituting Ψ (x,t) into the Klein-Gordon equation yields Φ Φ = + 2 2 2 2 2 2 2 2 2 ) ( ) ( 1 ) ( ) ( ) ( ) ( 1 dt t d t x mc dx x d c x h h The only way both sides can be equal for all values of x and t is for both sides to be equal to the same constant C.
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Unformatted text preview: Hence, ) ( ) ( ) ( ) ( 2 2 2 2 2 2 x C x mc dx x d c = + − h ) ( ) ( 2 2 2 t C dt t d Φ − = Φ h The time equations is a second order equation with two solutions t C i e t h m = Φ ± ) ( and t C i e x t x h m ) ( ) , ( = 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 C dx t t x t x dx t x E t x E op = ∂ Ψ ∂ Ψ − = Ψ Ψ >= < ∫ ∫ ∞ ∞ − ± ∗ ± ∞ ∞ − ± ∗ ± 2 2 2 2 2 ) , ( ) , ( ) , ( ) , ( h Hence 2 2 = > < − > < = Δ E E E and E C = . 2 nd order differential equation!...
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