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# notes02 - 5.73 Lecture#2 2-1 free particle V(x)=V0 Last...

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2 - 1 5.73 Lecture #2 Last Time: A,B are complex constants, determined by “boundary conditions” general solution free particle V(x)=V 0 ψ = Ae ikx + Be –ikx and and probability distribution only get wiggly stuff when 2 or more different values of k are superimposed. In this special case we had +k and –k. TODAY 1. infinite box 2. δ (x) well 3. δ (x) barrier P x A B Re A B kx A B kx () == + + + ψψ * | | | | ( * )cos Im( * )sin 22 const. wiggly 12 4 43 44 4444444 3 k p e kE V m for E V ikx = =− h h from , eigenfunction of p, and the real number, p, is the eigenvalue / 0 2 0 2

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2 - 2 5.73 Lecture #2 What do we know about ψ (x) for physically realistic V(x)? ψ ψψ ()? *( ) ( ) ? /? ±∞ = −∞ xx d x dd x for all x? Continuity of and Computationally convenient potentials have steps and flat regions. infinite step finite step infinitely high but infinitely thin step,“ δ -function” ψ d ψ dx , d 2 ψ dx 2 d ψ dx continuous not continuous for infinite step, and not for δ -function is continuous for finite step More warm up exercises 1. Infinite box L 0 x V(x) ψ ψ ψ ( ) cos sin () ,, x Ae Be C kx D kx C Lk L n n ikx ikx =+ = + =⇒ = =⇒ =π = 00 0 0 1 2 why not n = 0? [C=A+B, D=iA – iB]
2 - 3 5.73 Lecture #2 kE V mn L V En mL n h mL n 2 0 2 22 2 0 2 2 2 2 2 2 0 28 =− () = π = π = π = h h Insert kL = n boundary condition. recall here. n = 0 would be empty box # of bound levels E 1 normalization (P=1 for 1 particle in well) 1 2 0 12 | | sin ( ) ( ) ( / ) sin( ) / Dd xn x xLn x L n ψ |D| = (2/L) 1/2 D = e i α arbitrary phase factor { cartoons of ψ n (x): what happens to { ψ n } and {E n } if we move well: left or right in x? up or down in E? Infinite well was easy: 2 boundary conditions plus normalization requirement.

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notes02 - 5.73 Lecture#2 2-1 free particle V(x)=V0 Last...

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