100% satisfaction guarantee Immediately available after payment Both online and in PDF No strings attached
logo-home
Previously searched by you
Previously searched by you
logo
Physical Chemistry - Variance, Root-Mean Square, Operators, Eigen Functions, Eigen Values_lecture9 $2.60   Add to cart
Quickly navigate to
Preview
  2.  
  3. MIT Courses
  4.  
  5. MIT Courses
  6.  
  7. MIT Courses

Class notes

Physical Chemistry - Variance, Root-Mean Square, Operators, Eigen Functions, Eigen Values_lecture9
 2 views  0 purchase
  • Course
  • MIT Courses
  • Institution
  • MIT Courses

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 ...

[Show more]

Preview 2 out of 6  pages

Document information

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

Subjects

  • variance
  • root mean square
  • eigen functions

Written for

  • MIT Courses
All documents for this subject (18)

Seller

avatar-seller
tandhiwahyono

Reviews received

Content preview

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

The benefits of buying summaries with Stuvia:

Guaranteed quality through customer reviews

Guaranteed quality through customer reviews

Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.

Quick and easy check-out

Quick and easy check-out

You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.

Focus on what matters

Focus on what matters

Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!

Frequently asked questions

What do I get when I buy this document?

You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.

Satisfaction guarantee: how does it work?

Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.

Who am I buying these notes from?

Stuvia is a marketplace, so you are not buying this document from us, but from seller tandhiwahyono. Stuvia facilitates payment to the seller.

Will I be stuck with a subscription?

No, you only buy these notes for $2.60. You're not tied to anything after your purchase.

Can Stuvia be trusted?

4.6 stars on Google & Trustpilot (+1000 reviews)

79373 documents were sold in the last 30 days

Founded in 2010, the go-to place to buy study notes for 14 years now

Start selling
$2.60
  • (0)
  Add to cart