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Physics4DEquationSheet

# Physics4DEquationSheet - PHYSICS 4D EQUATION SHEET u = u v...

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PHYSICS 4D EQUATION SHEET ( ) ( ) ( ) ' ' ' 2 2 ' ' ' ' 2 ' 2 ' 2 ' 2 2 2 2 2 2 2 2 2 1 1 ( ) 1 / 1 / 1 / ( ) ( ) ( ) x x x y y x z z x u u v t t L L v c x x vt y y z z vx t t c u v u u v c u u u v c u u u v c p mu dp F dt K mc mc E mc K mc E pc mc γ γ γ γ γ γ γ γ γ γ = ∆ = ∆ = = = = = = = = = = = = = = + = + G G G G G G G 2 2 2 max max max 1 1 2 i i i res e s m c E u c E nhf E hf K m v eV K hf φ = = = = = = ( ) 2 2 2 2 2 2 2 2 2 2 2 sin 1 cos 1 2 13.6 n =1,2,3, ... / / / 2 2 ( ) ( , ) ( , ) ( , ) ( , ) e f i n e e n o n n o p g x b ab a d n h m c E E hf n r m ke m vr n ke E a n z E eV n n r a Z h p f E h v k d v dk p x E t P x dx x t dx x t dx p x t dx x t θ λ λ θ λ ω ω +∞ −∞ = = = = = = − = − = = = = = ∆ ≥ ∆ ∆ ≥ = Ψ Ψ = Ψ Ψ = = = = = ( ) i kx t Ae ω

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( ) 2 2 2 2 2 2 2 2 2 2 2 ( ,0) ( ) ( , ) ( ) ( , ) ( , ) ( ) ( , ) 2 ( , ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 ( ) sin ( 1/ 2) ( , ) ikx i kx t i t n n x a k e dk x t a k e dk x t x t U x x t i m x t x t x t x e d x U x x E x m dx n E mL n x x L L E n x x x t d ω ω ψ φ ψ ψ ψ ψ π π ψ ω +∞ −∞ +∞ −∞ Ψ = Ψ = ∂ Ψ ∂Ψ + Ψ = Ψ = = + = = = = + < >= Ψ = = = = = 2 2 2 * * * * 2 2 2 2 ( ) ( ) ( ) ( ) ( ) 1 ( , ) ( , ) ( ) ( , ) 2 ( ) ( ) ( ) ( ) 2 ( , , ) ( ) ( , ) ( l reflected incident transmitted incident m nl l x f f x dx x x x R T R T r t r t U r r t i m x r U r r E r m r R r Y L l l ψ ψ ψ ψ θ φ θ φ +∞ −∞ +∞ −∞ < >= Ψ = < > − < > ΨΨ = ΨΨ ΨΨ =
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