Free particle in one dimension
For a value of E there are
two solutions
Physical Chemistry


Lecture 13
Solving Schroedinger’s Equation
for Simple Systems


General solution for the
energy problem is a linear
combination of particular
solutions
No limits on values of k


Simple constant-energy model
systems
Free particle in one
dimension
Restricted translation

Particle in a onedimensional box
Vibrational motion

Particle in a harmonic
potential
Hˆ
 Tˆ
2 d 2
 
2m dx 2
Hˆ   E

 2 d 2
2m dx 2
d 2
dx 2
 
 E
2mE

2
d 2
dx 2
 
2mE

2
 k2 
 ( x) 
A eikx
 ( x) 
A e ikx
 ( x)   ( x)   ( x)

Ek
k 2 2
2m
Eigenfunctions of other operators
pˆ x
  i
d
dx
Eigenvalue equation
pˆ x  
 i
px 
 
d ikx
e
dx

  
k e  
 i ikeikx
p x eikx
p x eikx
ikx
p x e ikx
Orthogonality and
completeness
The free particle in one
dimension
The free particle only
has kinetic energy
No potential energy
Schroedinger’s
equation is a secondorder differential
equation
For a specific value of
E there are two
solutions
All values possible
No quantization of energy
Consider the particular
states of the free
particle
The particular states are
eigenfunctions of the
linear momentum
operator
The momentum is
precisely known for this
state, k
Corresponds to a state
with the particle
traveling in the positive
direction with a precise
speed, v = k/m
Unrestricted
translation

Particular solutions
Can be described in two
different, but related,
ways
Use exponential functions
Represent planes waves in
space, one moving in the
positive and one in the
negative direction
Eigenfunctions of a
*
hermitian operator
 a b dV  0 if a  b
all
corresponding to
space
different eigenvalues
are orthogonal.

ik ' x ikx
The set of all
 e e dx  0 if k  k '
eigenfunctions of a

hermitian operator is
complete. Any function
f ( x) 
ck k ( x)
of the coordinates can
be expressed as a
k
linear combination of
the members of the set.
ikx


Example: Expansion in
the members of the set
of eigenfunctions of the
linear momentum

c e
k
k
1
Eigenvalue equation for the
particle in a 1-D box
Uncertainty
Value of momentum is precisely
known for the free particle, px
= 0 in a state of constant
energy
Momentum eigenfunction
cannot be normalized over all
space





Solution is sines and cosines
 1 
  dx
 2L 
En



n 2 2
2ma 2
n2h2
8 2 ma 2

Function must be single-valued and continuous at all points
 (a)  0
Function should be normalized

  ( x) ( x)dx
*
H
 
2

2
 d
2m dx 2
Inside the box
 E
Provides a boundary condition on (x) at the box
edges
Inside the box, the equation is analogous to
the free particle’s equation
a

  ( x) ( x)dx
*
 1
0
Eigenvalue equation for the
particle in a 1-D box
Application of boundary conditions

Outside the box
   E
The only solution outside the box is (x) =
0


 ( x)  0
Schroedinger’s equation
 2 d 2

2m dx 2
nx
)
a
Application of boundary conditions
Eigenvalue equation for the
particle in a 1-D box

c ( x)  B cos(
Eigenvalue equation for the
particle in a 1-D box
In the box, V(x) = 0
Outside the box,
V(x) = 
Inside the box, the
Hamiltonian is well
known
nx
)
a
n ( x)  s ( x)  c ( x)

2
p x x 
True for all systems
Whether one can detect effects
of uncertainty depends on the
size of  and the accuracy of
measurements
A sin(
s ( x) 
A model for 1D
translation of a gas
molecule
Two regions

  E
L
Can be normalized over a finite
region
Probability of the particle being
at any place within one region
is independent of position
No localization of particle
Cannot be certain of position at
all
The particle in a one-dimensional
box

2 d 2
2m dx 2
Differential equation
L
 dx
Example of Heisenberg’s
Principle of Uncertainty

dx
P( x) dx 

Function must be single-valued and continuous at all points
 (0)  0

B  0
 (a )  0

n must be a nozero integer
Function must be normalized
  ( x) ( x)dx
 nx 
 | A |2  sin 2 
dx  1
 a 
0

2
a
a
*
0
| A |2

a

| A| 
2
a
2
Particle in a one-dimensional
box
Solve for trajectories for
constant energy
Fundamental frequency, 0
Oscillatory motion
Maximum displacements are
classical turning points
Energy diagram


Only integral values of the
quantum number
Limited energy levels
Zero-point energy


Classical harmonic oscillator in
one dimension

p(t )   2mE sin  0t
E = V(xmax)
0 
Lowest-energy state has a
finite energy
Particle cannot be isolated
at one point
xmax
Probability for the particle in a
one-dimensional box


Position
State
As the quantum number
increases, the
probability becomes
more uniform with
position
 Classical limit


y 2     0
y 
x
  
1/ 4
 2 

 mk 



2
E
 0
1-D harmonic-oscillator wave
functions and energies
Wavefunctions
Particle (mass m)
attached to a spring
of force constant, k
Hooke’s law
d 2
dy 2
Hermite’s associated differential equation
The harmonic oscillator in one
dimension

Quantum harmonic oscillator
in one dimension
k 2
 2  2

x   E
2m x 2
2
Convenient to make dimensionless equation
Lowest-energy state has
a finite energy
Particle cannot be
isolated at one point
Potential energy
depends on position
relative to
equilibrium
2E
m 02
 

Zero-point energy

k
m
Schroedinger’s equation
H  E
Probability of finding the
particle depends on

2E
cos  0t
m 02
x(t ) 
 ( x ) 
V
H


k
(r  req ) 2
2
2
p
2m


k 2
x
2
k 2
x
2
 x2 
A H ( x /  ) exp  2 
 2 
  0, 1, 2, 3, 
Energy eigenvalues
E

 


1
  0
2

 


1
h 0
2
3
Energy levels
Probability Functions
The square of the wave
function gives the
probability density at
each position
Finite possibility the
particle is outside of
the classical turning
points
The 1-D harmonic
oscillator has equally
spaced energy
states
Energy spacing
depends on the
fundamental
frequency
Energy levels are
nondegenerate


One state per level

Harmonic-oscillator wave
functions
Harmonic-oscillator wave functions are
related to the Hermite polynomials
Hermite polynomials are well-known
sets of functions

Unclassical behavior
As the quantum
number increases, the
probability distribution
becomes more uniform
H(y)
Symmetry
0
1
Even
1
2y
Odd
2
4y2 – 2
Even
3
8y3 – 12y
Odd
Classical limit
Summary
Eigenvalue equations give a means to find
eigenfunctions and eigenvalues
Model systems can be solved explicitly



Free particle in one dimension
Particle in a one-dimensional box
One-dimensional harmonic oscillator
Boundary conditions determine quantum
conditions
Limited possible energy levels
 Often in correspondence to the set of integers
Uncertainty is integral part of the wave function


Probability depends on the system’s parameters
Unclassical phenomena
Wave functions
Hermite polynomials
multiplied by a
Gaussian function
Note alternation in
symmetry about
x=0


Even
Odd
4