Lecture9 - MonJan31,2011 Lecture9 StaConaryStates i 2 = V x...

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1/26/11 1 PHYS 360 Quantum Mechanics Mon Jan 31, 2011 Lecture 9: If all the eigenfuncCons are staConary states, how can QM handle Cme‐dependent dynamics? Recap from last lecture…. StaConary States: Ψ n ( x , t ) = ψ n ( x ) ϕ n ( t ) i 1 d ϕ n ϕ n dt = 2 2 m 1 ψ n d 2 ψ n dx 2 + V ( x ). i ∂Ψ t = 2 m 2 2 Ψ x 2 + V ( x , t ) Ψ THE Schrödinger EquaCon: 1. Require that V(x,t)=V(x). 2. Try separaCon of variables: 3. Rearrange terms: 2 2 m 1 ψ n d 2 ψ n dx 2 + V = E n 2 2 m d 2 ψ n dx 2 + V ψ n = E ψ n i 1 d ϕ n ϕ n dt = 2 2 m 1 ψ n d 2 ψ n dx 2 + V ( x ). Segregate x on one side, t on the other: i 1 ϕ n d ϕ n dt = E n d ϕ n dt = iE n ϕ n We now have 2 equaCons instead of one, but they are much simpler. d ϕ n dt = iE n ϕ n This equaCon is especially easy to solve, because V does not appear in it: ϕ n ( t ) = e iE n t / Note: φ n ( t ) 2 = ϕ n *( t ) ϕ n ( t ) = e iE n t / e + iE n t / = e 0 = 1 The square of this eigenfuncCon is independent of Cme.
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