Physics 2D Lecture Slides Mar 3 Vivek Sharma UCSD Physics Read - - PowerPoint PPT Presentation

Physics 2D Lecture Slides Mar 3

Vivek Sharma UCSD Physics

• Read Scientific American Special Issue on Modern Phys:

– “The Edge of Physics” : online @ www.sciam.com

• Hardcopy Available at Bookstar (Costa Verde in UTC area)
• Introducing Prof. Kim Griest :

– “Why study Physics ?”…how to make money & learn Cool stuff – Dark matter & Dark Energy in the Universe

Measurement Expectation: Statistics Lesson

• Ensemble & probable outcome of a single measurement or the

average outcome of a large # of measurements

1 1 2 2 3 3 1 1 2 3 1 *

( ) .... ... ( ) For a general Fn f(x) ( ( ) ( ) ( ) ( ) ) ( )

n i i i i i i n i i i

xP x dx n x n x n x n x n x x n n n n N P x dx n f x f x N x f x x dx P x dx ψ ψ

∞ = −∞ ∞ − ∞ −∞ ∞ −∞ ∞ =

+ + + < >= = = + + + < >= =

∑ ∫ ∫ ∫ ∑ ∫

2 i 2 2

Sharpness of A Distr Scatter around average

(x ) = = ( ) ( ) = small Sharp distr. Uncertainty X = : x N x x σ σ σ σ − − → ∆

∑

Particle in the Box, n=1, <x> & ∆x ?

• 2

2 2 2 2

2 (x)= sin L 2 <x>= sin L 2 = sin , change variable = L 2 <x>= sin , L 2L <x>= d 2 sin L 1 use sin cos2 (1 cos2

• 2

) 2

L

x L x dx x d L x x dx x x L L L

π π π

π ψ π π π θ θ θ π θ π θ θ θ θ π θ θ

∞ ∞

                =               ⇒  − ⇒  

∫ ∫ ∫ ∫ ∫

L 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

Similarly <x >= x s use ud L <x>= (same result as from graphing ( )) 2 2 in ( ) 3 2 and X= <x 0.18 3 2 4 X= 20% of L, Particle not sharply confi v=uv- ned vdu L L x dx L L L L L x L x π ψ π π π π    ⇒ = − ∆ >   =   − < > = − − =  ∆ 

∫ ∫ ∫

in Box

Expectation Values & Operators: More Formally

• Observable: Any particle property that can be measured

– X,P, KE, E or some combination of them,e,g: x2 – How to calculate the probable value of these quantities for a QM state ?

• Operator: Associates an operator with each observable

– Using these Operators, one calculates the average value of that Observable – The Operator acts on the Wavefunction (Operand) & extracts info about the Observable in a straightforward way gets Expectation value for that

• bservable

* * 2

ˆ ( , ) [ ] ˆ [ ] is the operator & is the Expectation va ( , ) is the observable, [X] = x , lue [P] = [P] [K] = 2 Exam i p : m les x t d Q x t Q Q Q d dx x Q

+∞ −∞

< >= Ψ < Ψ >

∫

• 2

2 2 [E] =

• 2m

i t x ∂ = ∂ ∂ ∂

Operators Information Extractors

2 + + * *

• 2

2

ˆ [p] or p = Momentum Operator i gives the value of average mometum in the following way: ˆ [K] or K = - <p> = (x) gi [ ] ( ) = (x) i Similerly 2m : d p x dx dx dx d dx d dx ψ ψ ψ ψ

∞ ∞ ∞ ∞

     

∫ ∫

• +

+ 2 2 * * 2

• +

*

• +

* *

• ( )

<K> = (x)[ ] ( ) (x) 2m Similerly <U> = (x ves the value of )[ ( )] ( ) : plug in the U(x) fn for that case an average K d <E> = (x)[ ( )] ( ) (x) E d x K x dx dx dx U x x dx K U x x dx ψ ψ ψ ψ ψ ψ ψ ψ ψ

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

  = −     + =

∫ ∫ ∫ ∫

• +

2 2 2

( ) ( ) 2m The Energy Operator [E] = i informs you of the averag Hamiltonian Operator [H] = [K] e energy +[U] d x U x dx dx t ψ

∞

  − +     ∂ ∂

∫

• Plug & play form