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Lecture11 - FriFeb4,2011 Lecture11...

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1/26/11 1 PHYS 360 Quantum Mechanics Fri Feb 4, 2011 Lecture 11: The infinite square well: What can we do with a complete set of wave funcJons? 2 2 m d 2 ψ n dx 2 + V ψ n = E n ψ n The Jme‐independent Schrödinger equaJon: Inside infinite square well V=0, so: 2 2 m d 2 ψ n dx 2 = E n ψ n d 2 ψ n dx 2 = k n 2 ψ n , k n 2 = 2 mE n 2 Which of these are valid eigenfuncJons? 1. A sin( k n x ) 2. B cos( k n x ) 3. A sin( k n x ) + B cos( k n x ) 4. Dexp i k n x + θ ( ) 5. all of the above N.b. e i θ = cos θ + i sin θ The equaJon for the Jme‐dependence of the wavefuncJon: i 1 ϕ n d ϕ n dt = E n d ϕ n dt = iE n ϕ n Which of these are valid eigenfuncJons? 1. exp iE n t / [ ] 2. cos E n t / ( ) 3. i sin E n t / ( ) 4. all of the above Is it possible for a purely real wavefuncJon to solve the Jme‐dependent Schrödinger equaJon? 1. Yes 2. No i ∂Ψ t = 2 m 2 2 Ψ x 2 + V ( x , t ) Ψ THE Schrödinger EquaJon: Recap…. The Infinite Square Well V x ( ) = 0, if 0 x a, , otherwise 2 2 m d 2 ψ n dx 2 + V ψ n = E n ψ n We will solve this equaJon first outside the well, then inside, then match up the soluJons.
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1/26/11 2 ψ n x ( ) = 2 a sin n π a x
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