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# notes04 - 5.73 Lecture#4 4-1 Lecture#4 Stationary Phase and...

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() 5.73 Lecture #4 4 - 1 Lecture #4: Stationary Phase and Gaussian Wavepackets Last time: tdSE motion, motion requires non-sharp E phase velocity began Gaussian Wavepacket goal: x , x, p = h k , p = h k by construction or inspection Ψ (x,t) is a complex function of real variables. Difficult to visualize. What are we trying to do here? techniques for solving series of increasingly complex problems illustrate philosophical points along the way to solving problems. free particle So far: infinite well very artificial δ -function * nothing particle-like * nothing molecule-like * no spectra Minimum Uncertainty (Gaussian) Wavepacket -- QM version of particle. We are going to construct a Ψ ( x , t ) for which | Ψ ( x,t )| 2 is a Gaussian in x and the FT of Ψ ( x,t ), gives Φ ( k,t ), for which | Φ ( k , t )| 2 is a Gaussian in k . center of wavepacket follows Newton’s Laws extra stuff: spreading interference tunneling Today: (improved repeat of material in pages 3–4 through 3–1 infer k by comparing g(k) to std. G(x; x 0 , x) gk α () = | ( ) | e ik for k near k 0 d α ≡− x 0 STATIONARY PHASE dk = kk 0 2 x t Ψ (, ) moving, spreading wavepacket how is it possible that the center of the wavepacket v G v φ moves at a different velocity than its center k- component revised 9/4/02 2:33PM

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4 5.73 Lecture #4 4 - 2 Here is a normalized Gaussian (see Gaussian Handout) Gx;x 0 , x ) = (2 π ) 1/2 1 x e −( x x 0 ) 2 [ 2( x) 2 ] ( normalized −∞ G ( x; x 0 , x ) dx = 1  ± center x = x 0 by construction std. dev. x [ x 2 x 2 ] 1/2
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notes04 - 5.73 Lecture#4 4-1 Lecture#4 Stationary Phase and...

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