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Unformatted text preview: 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 ) , ( ) ( = 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 ψ ψ = 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 ψ ψ = , 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 >≡ 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. 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 ψ . 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 ψ ψ ψ ψ PHY4604 R. D. Field Department of Physics Chapter2_2.doc University of Florida Theory of Stationary States (2) Probability Density: The probability density, in general, depends on time and is given by ∑ ∑ ∑ ∞ = − ∞ ≠ = ∗ ∗ ∞ = ∗ ∗ ∗ + = Ψ Ψ = 1 1 1 ) , ( ) , ( ) , ( m t i m n n n m n m n n n n n mn e c c c c t x t x t x ω ψ ψ ψ ψ ρ where h / ) ( n m mn E E − = ω and called the “transition” frequencies. Average Energy: The average energy of the arbitrary state ∑ ∑ ∞ = − ∞ = = Ψ = Ψ 1 / 1 ) ( ) , ( ) , ( n t iE n n n n n n e x c t x c t x h ψ . is ∑ ∑ ∞ = ∞ = ∗ = >= < 1 2 1   n n n n n n n E c E c c E and P n = c n  2 is the probability that in a single measurement of the energy of the arbitrary state Ψ one would find E n . Proof: ∑ ∑ ∑ ∑ ∞ = ∞ = − − ∞ ≠ = ∞ = = > < + > < >= Ψ Ψ >=< < 1 * 1 / ) ( 1 * 1 *     n n n n m t E E i j m n n i n n m n n n n n n op E c c e E c c E c c H E m n h ψ ψ ψ ψ Overlap Functions: The complex constants are the overlap of the eigenstate Ψ n with the arbitrary state Ψ since > Ψ Ψ =<  n n c and > Ψ Ψ =< ∗ n n c  ....
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This note was uploaded on 04/19/2008 for the course PHY 4604 taught by Professor Field during the Spring '07 term at University of Florida.
 Spring '07
 Field
 mechanics

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