1. The Wave Functions

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. 3.2. Continuous Variables Probability of variable having a value between x and x+dx is (x) dxfor 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.