Particle in three-dimensional
space
The world has three
spatial dimensions
Generalize simple
problem to reflect the
three- dimensional
aspects of problem
Example: a free particle
in 3-D space
Appropriate model for a
gas molecule in an
unconfined space
The particle’s motions in
the x, y, and z
dimensions are
considered to be
independent of each
other
Physical Chemistry
Lecture 14
Solving multidimensional problems
Functions of multiple
independent variables
Functions can be


Very complex
Very simple
Independent
variables

Each is required,
independent of the
others, to describe
any function
Functions of multiple
variables may be


Inseparable
Separable
F ( x, y , z ) 
x2

xy 
Particle in three-dimensional
space
In free space, there is no
potential energy
All energy is kinetic
z
x

Sets of common independent variables
( x, y , z )

Inseparable :
F ( x, y ) 
f ( x) g ( y )
F ( x, y ) 
f ( x) g ( y )
Example :
F ( x, y )  e ( cx
2
 dy )
 e cx e  dy
f ( x)  g ( y )
2
Each term depends on an
independent variable (x, y, or
z)
Substitution by
correspondence gives a
Hamiltonian operator that has
three terms, each depending
on a different co-ordinate only
and
p x2
2m
H

 Hx
p y2

2m
 Hy
p z2
2m
 Hz
2  2
2
2 




2m  x 2 y 2 z 2 
Hˆ
 
Hˆ
 Hˆ x

Hˆ y
 Hˆ z
For a particle in a three-dimensional box, wave
function depends on three variables
Hˆ  ( x, y, z )  E  ( x, y, z )
Hˆ
x

 Hˆ y  Hˆ z  ( x, y, z )  E  ( x, y, z )
Distribution of the operators gives
Hˆ x  ( x, y, z )  Hˆ y  ( x, y, z )  Hˆ z  ( x, y, z )  E ( x, y, z )

g ( y)  C

Schroedinger’s equation

f ( x)  C
p p
2m
Three contributions to the
kinetic energy
Equality of functions of different
variables
Two functions of
two different
independent
variables
Equality means that
they must be equal
to a constant, some
number that is not a
function of either
variable
 T

Hamiltonian function is parsed
into a sum of three terms
(r , ,  )
Separable :
H
T

A sum of three terms
Each operator only operates on a specific co-ordinate’s
functions
1
Particle in three-dimensional
space
 (x, y, z)

Hˆ

Hˆ


x
Hˆ  ( x , y , z )
Hˆ x 
x
x
(x)
Hˆ

y
( y ) z (z)

y
Hˆ
 ( x, y, z )  x ( x)y ( y )z ( z )  e ik x x e
E
z
E  (x, y, z)
( x ) y ( y ) z ( z )

Hˆ z  x ( x ) 
y
 Ex  E y  Ez
k y2  2
k x2  2
k 2 2

 z
2m
2m
2m


Hˆ y  x ( x )  y ( y )  z ( z )
( y ) z (z)

E  x (x) y ( y ) z (z)
Assume the 3D wave function is separable into a
product of functions of the independent variables
Hamiltonian is sum of Hamiltonians

Particle in three-dimensional
space
Each depends on a single independent variable
y ( y )z ( z )
x ( x)y ( y )z ( z )
Hˆ x x ( x) 
1
Hˆ x x ( x ) 
x ( x)
x ( x )z ( z )
Hˆ y y ( y ) 
x ( x)y ( y )z ( z )
1
Hˆ y y ( y ) 
y ( y )
1
Hˆ x x ( x )  E 
x ( x)
x ( x)y ( y )
x ( x )y ( y )z ( z )
Hˆ z z ( z )  E
1 ˆ
H z z ( z )  E
z ( z )
1
Hˆ y y ( y ) 
y ( y )
1
Hˆ z z ( z )
z ( z )
F ( x )  G ( y, z )

Other separable models
Particle in a box of n dimensions
Modes of motion of a molecule



Translation of center of mass
Electronic state
Whole-body rotation of the molecule
Vibrational modes of the molecule
  trans elect rot vib
E  Etrans
 Eelect
 Erot
 Evib
Each depends on a single independent variable
F ( x)  G ( y , z )
1
Hˆ x x ( x)  E x
x ( x )
 Hˆ x x ( x)  E x x ( x )
Hˆ y y ( y )z ( z )  Hˆ z y ( y )z ( z ) 

Hˆ y y ( y )  E y y ( y )
E  Ex y ( y )z ( z )
Hˆ z z ( z )  E z z ( z )
 Ex
 Ey
 Ez
Assumption gives 3 equations for the three parts of
the wave function

2
k k
2m
Each wave function depends on its own quantum number

Particle in three-dimensional
space
E
 e ik r
Total energy is a sum of the energies of the
independent motions

Assume the 3D wave function is separable into a
product of functions of the independent variables
Hamiltonian is sum of Hamiltonians
e
Assumption gives a wave function that is a
product
Particle in three-dimensional
space
y ( y )z ( z ) Hˆ x x ( x)  x ( x )z ( z ) Hˆ y y ( y )  x ( x )y ( y ) Hˆ z z ( z )  Ex ( x )y ( y )z ( z )

ik y y ik z z
Each equation is like the 1-D equation previously solved
Total energy is a sum of the energies of the
independent motions
Particle in a 3-D box
Each mode is exactly like the particle in
a 1-D box
Solutions and energies of these modes
are known
Overall solution
nx n y nz ( x, y, z ) 
En x n y n z

23
 n x   n yy   nzz 
 sin 
sin  x  sin

abc  a   b   c 
2
h 2  nx2 n y nz2 
 2 2
2
8m  a
b
c 
2
Probability plots for a particle
in a 2-D box
Upper graph


nx = 1
ny = 1
Lower graph


Summary
Can sometimes separate wave functions into parts
that represent independent energies of the
molecule
Separation of variables simplifies the complex
problem into simpler lower-dimensional problems
Have to identify independent variables
nx = 1
ny = 2


f
Easy for simple problems
More complex for real problems
Example of a particle in three dimensions
Note the symmetry of the
graphs and how it changes
depending on the
relationship of the
eigenvalues


Translation separable into movement along three
orthogonal directions
Reduces to three 1-D problems
Multiple dimensions introduces the effects of
symmetry on the problem

High symmetry results in degeneracy of energy levels
g
Symmetry and degeneracy
For the particle in a 2-D box,
the energies depend on the
size of the box in each
direction
When a = b,


The box is square
The states (n,m) and (m,n)
have the same energy
Symmetry of the potential
energy increases the number
of states at a particular
energy


Degeneracy increases
because of symmetry
Very important relation used
to determine symmetry
properties of systems
Quantum model problems
System
Model
Gas molecule
Particle in a
Box
Bond vibration
Harmonic
oscillator
Potential
Energy
Differential
Equation
Solutions
Either 0 or 
Bounded
wave
equations
Sines and cosines
(k/2)(r-req)2
Hermite’s
equation
Hermite polynomials
Spherical
harmonic
(angular
momentum)
Spherical harmonic
functions
Either 0 or 
Molecular rotation
Rigid rotor
Hydrogen atom
Central-force
problem
-Ze2/r
Legendre’s
and
Laguerre’s
equations
Legendre
polynomials,
Laguerre
polynomials,
spherical harmonic
functions
Complex systems
Multi-mode
systems
Complex
Complicated
equations
Complex products of
functions
3