Chapter2_all

# Chapter2_all - PHY4604 R D Field Theory of Stationary...

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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 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: 0 ) , ( ) ( ) , ( ) ( )) , ( ) , ( )( ( ) , ( ) ( 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

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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 | . P n = |c n | 2 is the probability that in a single measuremen of an arbitrary state Ψ would find it in the eigenstate Ψ n . Time Dependence: Suppose we know Ψ (x,t) at t = 0 . Then we know Ψ (x,t) at all later times! Proof: = = Ψ = = Ψ 1 0 ) ( ) ( ) 0 , ( n n n x c x t x ψ with > Ψ =< 0 | n n c ψ = = Ψ 1 / ) ( ) , ( n t iE n n n e x c t x h ψ
PHY4604 R. D. Field Department of Physics Chapter2_3.doc University of Florida The Infinite Square Well (1) Particle in a One-Dimensional Box: Consider the solution of ) ( ) ( ) ( ) ( 2 2 2 2 x E x x V dx x d m ψ ψ ψ = + h , where h / ) ( ) , ( iEt e x t x = Ψ ψ , for the case V(x) = if x 0 and V(x) =

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