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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 = ) , ( ) , ( h or ) , ( ) ( = t x t i H op h Stationary States: The stationary states are solutions of the form ) ( ) ( ) , ( t x t x = with ) ( ) ( x E x H op = and ) ( ) ( t E t t i = h which implies that h / ) ( iEt e t = . 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 = = > < + + ) ( ) ( ) , ( ) , (  . Superposition Principle: The Hamiltonian operator is a linear operator which means that ) , ( ) , ( ) , ( 2 2 1 1 t x c t x c t x + = , 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 = + = + = 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 = = = = 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: = = = = = > < + > < >= =< 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 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 = = = + = = 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 where h / ) ( n m mn E E = and called the transition frequencies. Average Energy: The average energy of the arbitrary state = = = = 1 / 1 ) ( ) , ( ) , ( n t iE n n n n n n e x c t x c t x h . 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 one would find E n . Proof: = = = = = > < + > < >= >=< < 1 * 1 / ) ( 1 * 1 *     n n n n m t E E i n m n n m 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 Overlap Functions: The complex constants are the overlap of the eigenstate n with the arbitrary state since > =<  n n c and > =< n n c  ....
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 Spring '07
 FIELDS
 mechanics

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