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Lecture4 - WedJan19,2011 Lecture4 ,howdowe...

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PHYS 360 Quantum Mechanics Wed Jan 19, 2011 Lecture 4: Since QM relies on probabiliDes, how do we quanDfy uncertainDes?
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Recap of last lecture….
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Vibra&ng String Schrödinger (QM) Wave equaDon SeparaDon of variables Eigenvalue equaDon EigenfuncDon example: Length L, ends clamped: Infinite square well, width “a”: “Eigenstate” Wave funcDon 2 y ( x , t ) x 2 = 1 c 2 2 y ( x , t ) t 2 i ∂Ψ ( x , t ) t = 2 2 m 2 Ψ ( x , t ) x 2 + V Ψ ( x , t ) 2 2 m d 2 ψ n ( x ) dx 2 + V ψ n ( x ) = E n ψ n ( x ) d 2 f ( x ) dx 2 = k 2 f ( x ) y n ( x , t ) = f n ( x ) g n ( t ) Ψ n ( x , t ) = ψ n ( x ) ϕ n ( t ) ψ n ( x ) = A n sin( n π x a ) f n ( x ) = a n sin( n π x L ) y n ( x , t ) = f n ( x )exp( i ω n t ) Ψ n ( x , t ) = ψ n ( x )exp( i ω n t ) y ( x , t ) = a n sin( n π x L ) n = 1 exp( i ω n t ) Ψ ( x , t ) = A n sin( n π x a )exp( i ω n t ) n = 1
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0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 n=2 n=3 Ψ ( x , t ) = Ψ 2 ( x , t ) + Ψ 3 ( x , t ) = sin( 2 π x a )exp( i ω 2 t ) + sin( 3 π x a )exp( i ω 3 t ) 0.0 0.5 1.0 Not staDonary! Re( Ψ ( x , t )) Ψ ( x , t ) 2
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Ψ ( x , t ) 2 dx = { } a b Probability of finding the parDcle between a and b , at Dme t .
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