Download A 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)
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
Solving Schrödinger Equation – 1st Example.
A Free (V = 0) Particle Moving in x (A Particle Not in a Box)
H  E 
2 d 2
H 
2m dx 2
 2 d 2

 E
2
2m dx
pˆ x 
Solutions
k 2 2
Ek 
2m
 k  Aeikx  Be  ikx
eikx  p x   k
  A
2
2
e  ikx  p x   k
  B
2
2
 d
i dx
  Aeikx
 d
  px
i dx
 d
  k
i dx
px  kh
  Aeikx
 d
  k
i dx
px  kh
The Uncertainty Principle
When momentum is known precisely, the position cannot be predicted
precisely, and vice versa.
  Ae
ikx
p x   kh
  A
2
2
When the position is known precisely,
Location becomes
precise at the expense
of uncertainty in
the momentum
Free Translation (V = 0) Confined within Boundaries:
A 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)
Rapidly
changing

Higher E
node
quantized
zero-point
energy
n  1,2,3,...
Energy, Wave function & Probability density
node
not constant over x
Quantum (confinement) effect
Particle in a Box: Classical vs. Quantum
From Wikipedia (particle in a box)
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 = 10 nm
Case II:
T = 300 K, m = 1 kg, L = 1 m
Postulate 2 of Quantum Mechanics (measurement)
• Once (r, t) is known, all observable properties of the system can be
obtained by applying the corresponding operators (they exist!) to the
wave function (r, t).
• Observed in measurements are only the eigenvalues {an } which satisfy
the eigenvalue equation.

A   a
eigenvalue
eigenfunction
(Operator)(function) = (constant number)(the same function)
(Operator corresponding to observable) = (value of observable)
A Free Particle Moving along x, Two Cases
A free particle not confined
  Aeikx
pˆ x 
A free article confined in a box of size L
 d
i dx
 d
  p x
i dx
the same function
 d
Aeikx  khAeikx  kh
i dx
constant
p x   kh
number
eikx  p x   k
+
e  ikx  p x   k
-
 k  Aeikx  Be  ikx
It’s not a momentum operator’s eigenfuncition.
The momentum is either
or
. (px 
)
The position x is partially known. (x  L)
It’s an eigenfunction of the momentum operator px
Only a constant momentum px (eigenvalue) is measured.
We know the momentum exactly. (px = 0)
The position x is completely unknown. (x = )
x  px  h  0
Heisenberg’s
uncertainty principle
Alice, Bob, and Uncertainty Principle…
Postulate 4 of Quantum Mechanics (average)
• For a system in a state described by a normalized wave function , the
average value of the observable corresponding to


 A    A d

is given by
= <|A|>

• For a special case when the wavefunction corresponds to an eigenstate,
Position, Momentum and Energy of PIB
momentum
p
Two independent quantum numbers