1. 1. The Wave Functions The Schrodinger Equation
The Statistical Interpretation
Probability
Normalization
Momentum
The Uncertainty Principle

2. 1.1. The Schrodinger Equation
The dynamical state of a particle is completely specified by a wave function .
For a non-relativistic, spinless particle,
the evolution of (r,t) is governed by the Schrodinger equation
Laplacian

3. Quantization Rule
The Schrodinger equation can be obtained by applying the operator rules
to the classical relation for a conservative system
so that

4. 2. The Statistical Interpretation
Born:  = Probability amplitude
probability of finding particles between a & b at time t.
likely to find particle here
no chance of finding particle here

5. Philosophy
Realist (Einstein): There is an objective reality. Causality & locality hold. Statistical nature of quantum theory means that it is incomplete (there are hidden variables).
Orthodox (Copenhagen interpretation / Born ): Statistical nature of measurement is a fundamental property of nature (“strict” causality does not exist).
Agnostic: No need to worry about such metaphysical questions.
Answer to the question:
“Where is the particle just before measurement shows that it is at point C.
Realist : At C.
Orthodox: Nowhere.
Agnostic: Meaningless question.
John Bell
see p.4 & footnote

6. Collapse of Wave Function
Repeating a measurement right away always gives the same value.
Wave function collapses to an eigenfunction of the measurement operator.

7. 3. Probability 3.1. Discrete Variables
Let N(j) = number of people whose age is j.
In a certain room,
N(14) = 1, N(15) = 1, N(16) = 3, N(22) = 2, N(24) = 2, N(25) = 5,
N(j) = 0, otherwise
Total number of people:
Probability of finding a person of age j in room:
Probability of finding a person of age j or k :
Can always find a person of age between 0 and  :
Sum rule

8. Most probable age : 25
(highest probability)
Median age : 23
( 7 people with age below 23; 7 above)
Average age : 21
Expectation value
Average (expectation) value of a function f of j :

9. Most probable j = 5.
median j = 5.
mean j = 5.
Variance ( Standard deviation ) :

10.   probability density.
3.2. Continuous Variables
Probability of variable having a value between x and x+dx is (x) dx for dx  0.
  probability density.
Probability of variable having a value between a and b is

11. Example 1.1
A rock drops from rest a height h. What is the time average of the distance dropped?
Answer: Let x(t) be the distance dropped at time t, with x(T) = h.
All time intervals of the same length are the same. 
Constant acceleration :
with

12. Example 1.1 Alternative Solution
Let x be the random variable ( as Griffiths assumed by measuring a million photos )   
where

13. 4. Normalization
probability of finding particles between a & b at time t.  if
is finite.
(  is square-integrable
or normalizable )
Setting 
and
(  is normalized )

14. Schrodinger Eq. Preserves Normalization 
if  is normalizable
(  0 as |x|   ) 

15. Equation of Continuity
3D case : 
Equation of continuity.
Conservation of probability.
where
See Prob. 1.14

16. 5. Momentum Expectation value of x :
 x   average value of x measured on an ensemble of particles in the same state.
Reminder: repeated measurements (within a short time interval) on a particle gives only ONE value of x.
Integration by part : 
( Operator rule for 1st quantization )

17. In general, any dynamical function f (r, p) in classical mechanics can become a quantum operator by applying the “quantization” rule
( r - representation )
where upon
Eg.
Full discussion in Chapter 3.

18. 6. The Uncertainty Principle
de Broglie formula:
If  of a particle is a travelling wave of wavelength ,
the magnitude of the momentum of the particle is
= wave number
For a wave with a definite , p is definite but x is undefined.
For a wave with a small spread in , p is spreaded while x is better defined.
For a wave composited of all ’s, p is undefined while x is definite.
Uncertainty principle:
f = standard deviation of the values of f measured over an ensemble of identically prepared systems.