Probabilities And Wave Function-Quantum Physics and Mechanics-Lecture Slides, Slides of Quantum Mechanics

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Announcements

Homework # 2: Today Sep 7 by 7pm

Pb 1.4, 1.5, 1.7, 1.

Help sessions: T Th 3-6pm

Homework # 3: Friday Sep 9 by 7pm

Pb 1.9, 1.14, 2.1, 2.

Quiz 2a

“If the wave function is normalized at a time t,

it is then normalized at any time.”

A. True

B. False

Evolution of Y in time?

Expectation values

Density of probability: ( ) x

Average position x:

x x ( ) x dx





 (^) 

Probabilities

Average value for f(x):

f ( ) x f ( ) x ( ) x dx





 (^) 

Quantum Mec.

2 x x ( , ) x t dx





 (^)  Y

2 Y ( , ) x t

2 f f ( ) x ( , ) x t dx





 (^)  Y

2

f f x t dx

f x t f x t dx



 



 Y

 Y Y





Expectation values

The expectation value is the average of all the measurements of the quantity f on a ensemble of identically prepared particles

Differentiation between expectation value and most probable value

See pb 1.4 and pb 1.

Expectation values

Generalization

x x dx





 (^)  Y Y

“Operator” x

p i dx x





 Y   Y



(^) “Operator” p

Q Q x , i dx x





 Y   Y



Expectation values

Examples

(^2 2 )

2

1

2 2

T i dx dx m x m x

 

 

 ^    Y (^)   (^)  Y   Y Y  ^  

 

• Kinetic energy:

2 (^12)

2 2

p T mv m

 

• Angular momentum: L  r  p

and so on..

Effect of potential offset?

V V + V 0

Y?

Q?

See pb 1.

Uncertainty principle

Quiz 2b

Which statement is accurate for these electronic wave functions?

A. Both the position x and the momentum are fairly well defined

B. The position of the particle is fairly well defined but the momentum is poorly defined

C. The momentum of the particle is fairly well defined but the position is poorly defined

D. Both the position and the momentum are poorly defined.

p