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lecture9 - 5.61 Fall 2007 Lecture#9 page 1 VARIANCE...

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Unformatted text preview: 5.61 Fall 2007 Lecture #9 page 1 VARIANCE, ROOT-MEAN SQUARE, OPERATORS, EIGENFUNCTIONS, EIGENVALUES ≡ Deviation of i th measurement from average value < x > x i − x x i − x ≡ Average deviation from average value < x > But for particle in a box, x i − x = 0 ) 2 ≡ Square of deviation of i th measurement from average value < x > ( x i − x ) 2 ( x i − x ≡ σ 2 ≡ the Variance in x x Note ) 2 2 2 2 ( x i − x = x − x = σ x The Root Mean Square (rms) or Standard Deviation is then ⎤ ⎦ − ⎡ ⎣ σ 2 1 2 2 = x x x ⎢ ⎥ The uncertainty in the measurement of x , Δ x , is then defined as Δ x = σ x σ x for particle in a box a ∞ σ x 2 = ∫ ψ * x 2 ψ x dx − ∫ ψ * x x dx ( ) x ( ) ( ) x ψ ( ) 0 −∞ 2 ⎛ 2 ⎞ a ⎛ n π x ⎞ ⎡⎛ 2 ⎞ a ⎛ n π x ⎞ ⎤ = ⎝ ⎜ a ⎠ ⎟ ∫ 0 x 2 sin 2 ⎝ ⎜ a ⎠ ⎟ dx − ⎣ ⎢ ⎝ ⎜ a ⎠ ⎟ ∫ 0 x sin 2 ⎝ ⎜ a ⎠ ⎟ dx ⎦ ⎥ 5.61 Fall 2007 Lecture #9 page 2 Evaluate integral by parts ⎡ 2 2 ⎤ ⎡ 2 ⎤ ⇒ σ 2 = ⎢ ⎢ a 3 − a ) 2 ⎥ ⎥ −...
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lecture9 - 5.61 Fall 2007 Lecture#9 page 1 VARIANCE...

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