Requirements for a well behaved wave function | AP Physics 2 - Grade 9 PDF Download

 Page 1


REQUIREMENTS FOR AN ACCEPTABLE (WELL-BEHAVED) WAVEFUNCTION 
 
1. The wave function ? must be continuous.  All its partial derivatives must also be 
continuous (partial derivatives are etc. 
y
 , 
?
?
?
? ? ?
x
).  This makes the wave function 
“smooth”. 
 
2. The wave function ? must be quadratically integrable.  This means that the integral 
t ? ? d  
*
must exist. 
?
 
3. Since  is the probability density, it must be single valued. 
?
t ? ? d  
*
4. The wave functions must form an orthonormal set.  This means that  
• the wave functions must be normalized.  
 
?
8
8 _
*
  t ? ? d
i i
 = 1 
• the wave functions must be orthogonal. 
 
?
8
8 _
*
  t ? ? d
j i
 = 0 
OR         =   d
ij
  where d
ij
   = 1 when i = j and  d
ij
  = 0 when i ? j 
?
8
8 _
*
  t ? ? d
j i
 
d
ij
  is called Kronecker delta 
 
5. The wave function must be finite everywhere. 
 
6. The wave function must satisfy the boundary conditions of the quantum mechanical 
system it represents.  
 
Page 2


REQUIREMENTS FOR AN ACCEPTABLE (WELL-BEHAVED) WAVEFUNCTION 
 
1. The wave function ? must be continuous.  All its partial derivatives must also be 
continuous (partial derivatives are etc. 
y
 , 
?
?
?
? ? ?
x
).  This makes the wave function 
“smooth”. 
 
2. The wave function ? must be quadratically integrable.  This means that the integral 
t ? ? d  
*
must exist. 
?
 
3. Since  is the probability density, it must be single valued. 
?
t ? ? d  
*
4. The wave functions must form an orthonormal set.  This means that  
• the wave functions must be normalized.  
 
?
8
8 _
*
  t ? ? d
i i
 = 1 
• the wave functions must be orthogonal. 
 
?
8
8 _
*
  t ? ? d
j i
 = 0 
OR         =   d
ij
  where d
ij
   = 1 when i = j and  d
ij
  = 0 when i ? j 
?
8
8 _
*
  t ? ? d
j i
 
d
ij
  is called Kronecker delta 
 
5. The wave function must be finite everywhere. 
 
6. The wave function must satisfy the boundary conditions of the quantum mechanical 
system it represents.  
 
POSTULATES OF QUANTUM MECHANICS 
 
1. The state of a quantum-mechanical system is completely specified by its wave function 
?(r).  ?
*
(r) ?(r) dxdydz is the probability that the particle lies in the volume element 
dxdydz, located at r. (Note: we are considering the time independent wave function for all 
our work). 
 
2. To every observable in classical mechanics, there corresponds a linear operator in quantum 
mechanics. 
 
3. In any measurement of the observable associated with the operator
^
A , the only values that 
will ever be observed are the eigenvalues a, which satisfy the eigenvalue equation 
 
(r) a    (r) A
ˆ
? = ? 
 
4. If a system is in a state described by a normalized wave function ?, then the average value 
(or the expectation value) of the observable corresponding to operator A is given by  
 
t d  A
ˆ
     
*
? ? = > <
?
8
8 -
a 
 
 
5. The wave function of a system evolves in time according to the time dependent 
Schrodinger equation 
 
dt
d
  i     t) (x, H
ˆ
?
= ? h