Lecture Outline 3 - CHEM 332 Physical Chemistry II Lecture...

CHEM 332
Physical Chemistry II
Lecture Outline
11 February 2014
Particle in a One Dimensional Box and a Three Dimensional
Box
VIII. Particle in a One Dimensional Box with a Finite Wall
Write the Classical Hamiltonian
where V = ∞ Region I
0 Region II
U Region III
Write the Hamiltonian Operator
Solve the Schrodinger Wave Equation
Region I
 = 0
Region II
After Applying the Boundary Condition  = 0 at x = 0

Region III
Case for (E - U) < 0:
Boundary Condition 
II
= 
III
at x = L
Boundary Condition d 
II
/dx = d 
III
/dx at x = L
Solving the Above Two Equations Simultaneously for the Energy E Gives:
Lowe Provides Graphs of the Left-Hand Side and Right-Side of This
Equation for Specific Values of m, L and U . The Energy is Given by
the Intersection of These Graphs.

Notice:
(i) The Solution for E is Quantized. (Bound System)
(ii) The Energies for the Lowest Lying States are
Slightly Lower than Would be Expected if the
"Wall" at L were Infinite. (The Particle Behaves as
though It is In a Larger Box.)
Lowe Provides Graphs of