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Lecture19

# Lecture19 - FriFeb25,2010 Lecture19...

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2/25/11 1 PHYS 360 Quantum Mechanics Fri Feb 25, 2010 Lecture 19: Where do wave funcFons live? (In Hilbert space, of course!) 1 HW 5 due on Monday Feb 28: Ch 2, #23, 30, 34, 35 2 Let’s review some of the systems we have tackled so far: 1. Infinite square well 2. Harmonic oscillator 3. Free parFcle 4. Delta funcFon well 5. Finite square well V(x) ‐V o ‐a a E+V o 3 Recall the basic steps of QM problem‐ solving: 1. You are given a potenFal V(x). 2. You are also given an iniFal wave funcFon ψ(x,0). 3. Solve the Fme‐independent Schrödinger equaFon for the eigenfuncFons (ψ n ) and eigenvalues (E n ). 4. Solve for the coeﬃcients c n . 5. Construct the Fme‐dependent wave funcFon. (QED) Ψ ( x , t ) = c n n = 1 ψ n ( x ) e iE n t / c n = ψ n x ( ) * Ψ ( x ,0) dx H ψ n = E n ψ n Ψ ( x ,0) = c n ψ n ( x ) n = 1 4 There is a key assumpFon behind these steps: The eigenfunc+ons are orthogonal. For example, recall the infinite square well: ψ m x ( ) * ψ n x ( ) dx = 2 a sin m π x a 0 a sin n π x a dx = 0 ( m n ) ψ m x ( ) * ψ n x ( ) dx = 0 if m n 5 Where else do we see members of a set that are mutually orthogonal? Vectors . When we break a vector into x, y, and z components, we are using a basis set whose elements are mutually orthogonal: a = a x ˆ i + a y ˆ j + a z ˆ k ( ) b = b x ˆ i + b y ˆ j + b z ˆ k ( ) a i b = a x b x + a y b y + a z b z because ˆ i i ˆ i = 1 and ˆ i i ˆ j = 0, etc.

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Lecture19 - FriFeb25,2010 Lecture19...

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