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Chapter2_1

# Chapter2_1 - Proof 2 2 1 1 2 2 1 1 = Ψ ∂ ∂ − Ψ ∂...

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PHY4604 R. D. Field Department of Physics Chapter2_1.doc University of Florida Theory of Stationary States (1) Time Dependent Equation: Look for solutions of the equation (A) t t x i t x H op Ψ = Ψ ) , ( ) , ( h or 0 ) , ( ) ( = Ψ t x t i H op h Stationary States: The stationary states are solutions of the form ) ( ) ( ) , ( t x t x Φ = Ψ ψ with ) ( ) ( x E x H op = and ) ( ) ( t E t t i Φ = Φ h which implies that h / ) ( iEt e t = Φ . Eigenvalue Equation: The eigenstates of the system are determined from the eigenvalue equation ) ( ) ( x E x H n n n op = , with n = 1,2, 3, . .. Orthonormal Set: The eigenstates form an orthornormal set of wavefunctions such that ij j i j i j i dx x x dx t x t x δ = = Ψ Ψ >≡ Ψ Ψ < +∞ +∞ ) ( ) ( ) , ( ) , ( | . Superposition Principle: The Hamiltonian operator is a “linear operator” which means that ) , ( ) , ( ) , ( 2 2 1 1 t x c t x c t x Ψ + Ψ = Ψ , is also a solution of (A) , where c 1 and c 2 are arbitrary complex numbers.
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Unformatted text preview: Proof: ) , ( ) ( ) , ( ) ( )) , ( ) , ( )( ( ) , ( ) ( 2 2 1 1 2 2 1 1 = Ψ ∂ ∂ − + Ψ ∂ ∂ − = Ψ + Ψ ∂ ∂ − = Ψ ∂ ∂ − t x t ih H c t x t ih H c t x c t x c t ih H t x t ih H op op op op Most General Solution: The most general solution of (A) is ∑ ∑ ∞ = − ∞ = = Ψ = Ψ 1 / 1 ) ( ) , ( ) , ( n t iE n n n n n n e x c t x c t x h . Normalization: The arbitrary complex constants must satisy 1 1 = ∑ ∞ = ∗ n n n c c . Proof: ∑ ∑ ∑ ∑ ∞ = ∗ ∞ = − − ∞ ≠ = ∗ ∞ = = > < + > < >= Ψ Ψ =< 1 1 / ) ( 1 1 * | | | 1 n n n m t E E i n m n n m n m n n n n n c c e c c c c m n h...
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