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Unformatted text preview: PHY4604 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:
∞ ∑c < E >=< Ψ  H op  Ψ >=
+ ∞ ∞ ∑∑ m =1 n =1
n≠m *
cmcn E n < ψ m ψ
 n =1 n *
n cn E n < ψ n ψ
 > e −i( E n − E m )t / h = 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 = cn2 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 Chapter2_2.doc University of Florida ...
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This note was uploaded on 05/29/2011 for the course PHY 4064 taught by Professor Fields during the Spring '07 term at University of Florida.
 Spring '07
 FIELDS
 mechanics

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