Chapter6_19 - PHY3063 R. D. Field Theory of Stationary...

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Unformatted text preview: PHY3063 R. D. Field Theory of Stationary States (2) Probability Density: The probability density, in general, depends on time and is given by ρ ( x, t ) = Ψ ∗ ( x, t)Ψ ( x, t) = ∞ ∑ n =1 where ∗ ∗ c n c nψ nψ n + ∞ ∞ ∑∑c m =1 n =1 n≠m ∗ m ∗ c nψ mψ n e − i ω mn t ω mn = ( E m − E n ) / h and called the “transition” frequencies. Average Energy: The average energy of the arbitrary state ∞ ∞ n =1 n =1 Ψ ( x, t ) = ∑ c n Ψn ( x, t ) = ∑ c nψ n ( x )e − iE n t / h . is ∞ ∞ n =1 n =1 ∗ < E >= ∑ c n c n E n = ∑ | c n | 2 E n 2 and Pn = |cn| is the probability that in a single measurement of the energy of the arbitrary state Ψ one would find En. Proof: < E >=< Ψ | H op | Ψ >= + ∞ ∞ ∑∑ m =1 n =1 n≠m * cm cn E n < ψ i ψ | j ∞ ∑c n =1 * n cn E n < ψ n > e −i( E n − E m )t / h = ψ | n ∞ > ∑c n =1 * n cn E n Overlap Functions: The complex constants are the overlap of the eigenstate Ψn with the arbitrary state Ψ since ∗ c n =< Ψn | Ψ > and c n =< Ψ | Ψn > . Pn = |cn|2 is the probability that in a single measuremen of an arbitrary state Ψ would find it in the eigenstate Ψn. Time Dependence: Suppose we know Ψ(x,t) at t = 0. Then we know Ψ(x,t) at all later times! Proof: ∞ Ψ ( x, t = 0) = Ψ0 ( x ) = ∑ c nψ n ( x ) with c n =< ψ n | Ψ0 > n =1 ∞ Ψ ( x, t ) = ∑ c nψ n ( x )e − iE n t / h n =1 Department of Physics Chapter6_19.doc University of Florida ...
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This note was uploaded on 05/29/2011 for the course PHY 3063 taught by Professor Field during the Spring '07 term at University of Florida.

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