Download Particle in a Box

Quantum Chemistry: Our Agenda
• Postulates in quantum mechanics (Ch. 3)
• Schrödinger equation (Ch. 2)
• Simple examples of V(r)
 Particle in a box (Ch. 4-5)
 Harmonic oscillator (vibration) (Ch. 7-8)
 Particle on a ring or a sphere (rotation) (Ch. 7-8)
 Hydrogen atom (one-electron atom) (Ch. 9)
• Extension to chemical systems
 Many-electron atoms (Ch. 10-11)
 Diatomic molecules (Ch. 12-13)
 Polyatomic molecules (Ch. 14)
Computational chemistry (Ch. 16)
Simple Systems
Particle in a box (infinite or finite) (translation)
Harmonic oscillator (vibration)
V 
Particle on a ring or a sphere (rotation)
k : force constant
1 2
kx
2
Lecture 3. Simple System 1. Particle in a Box
References
• Engel, Ch. 4-5
• Molecular Quantum Mechanics, Atkins & Friedman (4th ed. 2005), Ch. 2
• Introductory Quantum Mechanics, R. L. Liboff (4th ed, 2004), Ch. 4
• A Brief Review of Elementary Quantum Chemistry
http://vergil.chemistry.gatech.edu/notes/quantrev/quantrev.html
• Wikipedia (http://en.wikipedia.org): Search for
 Particle in a box
Free Translational Motion (V = 0)
H  E 
2 d 2
H 
2m dx 2
 2 d 2

 E
2
2m dx
 k  Aeikx  Be  ikx
k 2 2
Ek 
2m
Solutions
eikx  p x   k
  A
2
2
e  ikx  p x   k
  B
2
2
Free Translation (V = 0) with Boundaries:
Particle in a Box (Infinite Square Wall Potential)
 2 d 2

 V ( x)  E
2
2m dx
V  0 for 0  x  L
 0 and x 
V   for x 
L
m
The same solution as the free particle
but with different boundary condition.
 k  C sin kx  D cos kx
A particle of mass m is confined
between two walls but free inside.
k 2 2
Ek 
2m
Applying boundary conditions
 k  C sin kx  D cos kx
 ( x  0)  0
 ( x  L)  0
D0
C sin kL  0
kL  n
 n ( x)  C sin( nx / L) n  1,2,3,... n cannot be zero.
(quantum number)
Normalization

L
0
 n 2 dx  C 2 
L
0
C2L
2
sin (nx / L)dx 
1
2
1/ 2
2
 n ( x)   
 L
sin( nx / L)
n  1,2,3,...
1/ 2
2
C  
 L
n2h2
En 
8mL2
Final Solution (Energy & Wave function)
n2h2
En 
8mL2
1/ 2
2
 n ( x)   
 L
sin( nx / L)
n  1,2,3,...
Rapidly
changing

Higher E
node
quantized
zero-point
energy
Energy, Wave function & Probability density
node
not constant over x
Quantum (confinement) effect
Classical Limit: Bohr’s Correspondence Principle
n 
by increasing E (~ kT) or m or L
What is the maximum value for n ?
Case I:
T = 300 K, m = me, L = 1 nm
Case II:
T = 300 K, m = 1 kg, L = 1 m
Position, Momentum and Energy of PIB
Two independent quantum numbers