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Physical Chemistry - Variance, Root-Mean Square, Operators, Eigen Functions, Eigen Values_lecture9
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This course presents an introduction to quantum mechanics. It begins with an examination of the historical development of quantum theory, properties of particles and waves, wave mechanics and applications to simple systems — the particle in a box, the harmonic oscillator, the rigid rotor and the ...

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Document information

  • April 25, 2023
  • 6
  • 2007/2008
  • Lecture notes
  • Prof. robert guy griffin
  • All classes

Subjects

  • variance
  • root mean square
  • eigen functions

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  • MIT Courses
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5.61 Fall 2007 Lecture #9 page 1



VARIANCE, ROOT-MEAN SQUARE, OPERATORS,
EIGENFUNCTIONS, EIGENVALUES

xi − x ≡ Deviation of ith measurement from average value <x >


xi − x ≡ Average deviation from average value <x >


But for particle in a box, xi − x =0


(x − x )
2
i
≡ Square of deviation of ith measurement from average
value <x >


(x − x )
2
i
≡ σ x2 ≡ the Variance in x



(x − x )
2 2
Note i
= x2 − x = σ x2


The Root Mean Square (rms) or Standard Deviation is then

12
σ x = ⎡ x2 − x ⎤
2

⎢⎣ ⎦⎥

The uncertainty in the measurement of x, Δx , is then defined as

Δx = σ x

σ x for particle in a box


() () () ()
a ∞
σ x2 = ∫ 0
ψ * x x 2ψ x dx − ∫ ψ * x xψ x dx
−∞
2
⎛ 2⎞ a ⎛ nπ x ⎞ ⎡⎛ 2 ⎞ a 2 ⎛ nπ x ⎞
⎤
= ⎜ ⎟ ∫ x 2 sin 2 ⎜ dx − ⎢ ⎜ ⎟ ∫0 x sin dx ⎥
⎝ a⎠ 0 ⎝ a ⎟⎠ ⎣⎝ a ⎠
⎜⎝ a ⎟⎠
⎦

, 5.61 Fall 2007 Lecture #9 page 2




Evaluate integral by parts


⎡ 2 ⎤ ⎡ 2⎤
⎢ a a2 ⎥ − ⎢a ⎥
⇒ σ =
2
−
⎢ 3 2 nπ
( ) ⎥ ⎣4⎦
x 2

⎣ ⎦


( )
⎡ nπ ⎤
2
2
a ⎢
σ = 2
− 2⎥
( ) ⎢ 3 ⎥
x 2
4 nπ ⎣ ⎦
12
⎡
( ) ⎤
2
a ⎢ nπ
Δx = σ x = − 2⎥
2 nπ ⎢ 3
⎣
( ) ⎥
⎦

Note that deviation increases with a, and depends weakly on n.



Now suppose we want to test the Heisenberg Uncertainty Principle for the
particle in a box.

12
p and p 2 to get Δp = σ p = ⎡ p 2 − p ⎤
2
We need
⎢⎣ ⎥⎦


() ()
∞
But do we write p = ∫ −∞
ψ * x pψ x dx ?

what do we put in here??

We need the concept of an OPERATOR

ˆ x =g x
Af () ()
operator acts on function to get a new function

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