Transcript Quantum Mechanics

量子力學導論
Textbook
Introduction to Quantum Mechanics / David J. Griffiths
Prentice Hall / 1995
Reference
Principles of Quantum Mechanics
- as applied to chemistry and chemical physics
Donald D. Fitts
Cambridge University Press / 1999
量子力學導論
Chap
Chap
Chap
Chap
1 - The Wave Function
2 - The Time-independent Schrödinger Equation
3 - Formalism in Hilbert Space
4 - 表象理論
Quantum Mechanics
Chap 1 - The Wave Function
► Schrödinger equation
Classical mechanics:
Newton’s second law :
wave function
time-depedent Schröinger equation :
Quantum Mechanics
► Statistical interpretation
Born’s statistical interpretation :
{ probability of finding the particle
between x and (x+dx) at time t }
► Probability
is probability density
The probability of infinite interval :



( x, t ) dx
2
Quantum Mechanics
► Normalization



( x, t ) dx  1
2
(i) If  ( x, t ) is a solution , then A ( x, t ) is also a solution.
Normalized the wave function to determine the factor A
(ii) If the integral is infinite for some wave functions,
no factor to make it been normalizable.
The non-normalizable wave function cannot represent
particles.
(iii) the condition of wave function which can be normalizable
Quantum Mechanics
► Operator and expectation value (average / mean)
expectation value of position x :


x   x ( x, t ) dx   * xdx

2

expectation value of momentum :
operator x represent position;
operator
■
represent momentum in x-direction.
all physics quantities can be written in terms of position
and momentum
Quantum Mechanics
► Heisenberg uncertainty principle
(proof ref. chap 3)
standard deviation
the variance of distribution, where
individual physics quantity
 2  (j ) 2  ( j  j ) 2
 ( j2  2 j j  j )
2
 j2  2 j j  j
 j2  j
2
2
Quantum Mechanics
so
standard deviation in position
standard deviation in momentum