Physical Chemistry - Variance, Root-Mean Square, Operators, Eigen Functions, Eigen Values_lecture9

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