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Lecture 2: The Born Interpretation of the
Wavefunction
This lecture covers the following parts of Atkins
11.4 The Born interpretation of the wavefunction
Lectures on-line
Born Interpretation (PDF)
Born Interpretation (PowerPoint)
Tutorials on-line
The postulates of quantum mechanics (This is the writeup for Dry-lab-II)(
This lecture coveres parts of postulate 2)
Probability
The wave function and probability
The Development of Classical Mechanics
Experimental Background for Quantum mecahnics
Early Development of Quantum mechanics
Audiovisuals on-line
review of the Schrödinger equation and the Born postulate (PDF)
review of the Schrödinger equation and the Born postulate (HTML)
review of Schrödinger equation and Born postulate (PowerPoint **,
1MB)
Slides from the text book (From the CD included in Atkins ,**)
Quantum Mechanical Hamiltonian
From Last Lecture
ˆ is constructed by the
The quantum mechanical Hamiltonian H
following transformations:


1
ˆ
HClass  H 
pˆ x2  pˆ 2y  pˆ z2  V( xˆ , yˆ , zˆ )
2m
x  xˆ ;
y   yˆ ;
z  zˆ ;

i x

py   pˆ y  
i y
px   pˆ x  

pz   pˆ z  
i z
2
2
2



ˆ 
H
[ 2  2  2 ]  V(x, y,z)
2m x
y
z
2
Solve Schrödinger
equation
ˆ  = E
H
to obtain

Quantum Mechanical Hamiltonian
Postulate 2
To every physical observable
class
there corresponds a linear
QM
Hermitian operator. In order
class (x,y,z, px ,py,pz )
to obtain this operator,
we write out the
ˆ QM ( xˆ , yˆ , zˆ , pˆ x , pˆ y,pˆ z )

classical - mechanical expression for
the observable in terms of
Cartesian coordinates and
corresponding linear momentum
components and perform the
following transformations :
xˆ   x ;
yˆ   y ;
zˆ   z ;

i x

py   pˆ y  
i y
px   pˆ x  

pz   pˆ z  
i z
Interpretation of the wavefunction in 1- D
ˆ  = E
Schrödinger Equation
H
1. The wavefunction contains all the dynamic information
about the system it describes
2. The square modulus of the wavefunction at x is
proporional to the probability of finding the particle at x
If the wavefunction of a particle has the value
(x) at a point x,than the probability of finding the
particle between x and x + x is :
P  (x) (x)dx
*
Max Born
Interpretation of the wavefunction in 1- D
(x)  probability amplitude
positive, negative, complex
| (x) |2  (x)(x)*
 always positive
c = a + ib
a, b real
cc  (a + ib)(a + ib)
 (a + ib)(a - ib)
*
P  (x) (x)dx
*
= a 2 - iab + iab + b 2
*
Interpretation of the wavefunction in 1- D
(x)  probability amplitude
positive, negative,
complex
2
| (x) |  (x)(x)*
 always positive
P  (x) (x)dx
*
Interpretation of the wavefunction in 3- D
If the wavefunction of a particle has the value
(x, y, z) at a point (x,y, z) than the probability of finding the
particle between x and x + x; y and y + y; z and z + z is :
dz
dy
dx
P(x, y, z)  (x, y, z) (x, y, z)dxdydz
*
Probability
Probability density=
Probability per volume Volume element
unit
QuickTime™ and a
Video decompressor
are needed to see this picture.
Spherical Coordinates
We are going to make use of the spherical polar coordinate system
Longitude
Latitude
Distance
Spherical Coordinates
We are going to make use of the spherical polar coordinate system
Z (x,y,z) 
(r,  

r

X
Y
We have the following
relation
x= r sincos
y= r sinsin
z= r cos
Spherical Coordinates
The volume between (r, ,  ) and
(r + r,  + ,   )
r
r

rsin
rsin

r
  r sinr
2
Volume element
Spherical Coordinates
Z (x,y,z) 
(r,  

r

X
Y
The Laplacian is in C artesian coordinates given by
2 d2 d2
d
2 = [ 2 + 2+ 2 ]
dx
dy dz
In po lar sphe rical coordinates we ge t "after some rather comp licated calculation s "
d2 2 d
1 d2 1
d
1
d2
2
 =[ 2 + r
+ 2
+ 2 cot
+ 2 2
]

2
d
r
d

d

dr
r d r
r sin 
Normalization
If yis a solution to the time-independent Schrödinger equation
 2 y(x)

 y (x)V(x)  Ey(x) (1)
2
2m x
2
then Ny will also be a solution
Proof:
2 2
 (Ny(x))

 Ny(x)V(x)  ENy (x) (2)
2
2m
x
or
 (Ny(x))

[
]  Ny(x)V(x)  ENy(x)
2m x
x
2
Normalization
 (Ny(x))

[
]  Ny(x)V(x)  ENy(x)
2m x
x
2

y(x)

[N
]  Ny(x)V(x)  ENy(x)
2m x
x
2
 2 y (x)

N
 Ny(x)V(x)  ENy(x)
2
2m
x
2
Thus Eq.(2) is equivalent to Eq.(1)
Normalization
However we shall shortly show that of all the solutions Ny
the only acceptable is the one that satisfy :
x b
y( x)y( x)*
 Ny (Ny ) dx  1
*
x a
a
b
For a wavefunction defined in
Normalized
interval a to b
Thus :
N
1
x b
1/2

  y ( y ) dx  1
x a

*
Normalization
The normalization of N is required to satisfy the Born
interpretation of the function N
y ( x)
Born interpretation:
x xx
P(x)  Ny(Ny ) x
P( x ) 

x x
*
P(x)  Ny(Ny ) x
*
a
x x+x
*
N y ( N y ) dx
b
Probability of finding
particle between x and x
Normalization and Born Interpretation
By dividing the interval between a and b into sections
y( x)y( x) *
P(xi )  N 2 (xi )(xi )* dxi
P(x1 )
P(xi )
P(xk )
xi
We have that the probability
of finding the particle in its
Possible range between a and
b is:
P(xn )
x1
a
a
xi
xk
xn
b
b
i n
Pab   P(xi )
i 1
Normalization and Born Interpretation
y( x)y( x) *
i n
Pab   P(xi )
i 1
Pab  N  (xi )(xi ) xi
2
xi
x1
a
a
in
*
i1
x
i
xk
x
n
Pab  N  (xi )(xi ) dx i
2
b
b
b
a
*
Normalization and Born Interpretation
y( x)y( x) *
The probability of finding the
Particle between a and b
b
Pab  N  (xi )(xi ) dx i
2
*
a
xi
Is 100 %
x1
a
x
i
xk
x
n
Thus by convention
b
b
N  (xi )(x i ) dxi  1
2
a
b
*
a
Probabilities are out of a total of 1
Generalization to 3D
QuickTime™ and a
Video decompressor
are needed to see this picture.
Properties of  and Born Interpretation
The Born intepretation imposes a number of restrictions on 
 Not continuous
2

so
and 2 not defined
x
 x
Schrödinger equation
H = E
not defined
 2 (x)

 V(x)(x)  E(x)
2
2m x
2
Normalization and Born Interpretation

Not continuous
x
so
2
 x
2
not defined
Schrödinger equation
H = E
not defined
 2 (x)

 V(x)(x)  E(x)
2
2m x
2
Normalization and Born Interpretation
P(x)  Ny(Ny ) x
*
not single valued
Not permissable
b
N  (xi )(x i ) dxi  
2
*
a
Not permissable
Postulate 1.
QM Postulate 1.
The state of a quantum system
is described by a function (x, y, z, t)
or (r, t), of the configuration space
variables and time. The function
is referred to as the state function or the
wavefunction. It contains all the
information that can be determined
about the system. Furthermore , we
require that (x, y, z, t) be single valued,
continuous, differentiable to all orders,
and quadratically integrable
Comments to Postulate 1
QM Postulate 1.
The wavefunction itself has no
physical interpretation since it
can be a complex function
However, the function
*
2
(r,t)(r,t) | (r,t) | , does
have an interpretation as a
probability density and, in
principle, is a measurable quantity
What you should know from this lecture
Born postulate in 1D
If the wavefunction of a particle has the value
(x) at a point x,than the probability of finding the
particle between x and x + x is :
P  (x) (x)dx
*
Born postulate in 3D
If the wavefunction of a particle has the value
(x, y, z) at a point (x,y, z) than the probability of finding the
particle between x and x + x; y and y + y; z and z + z is :
P(x, y, z)  (x, y, z) (x, y, z)dxdydz
*
What you should know from this lecture
You are not required to derive or remember the
expression for the Laplacian or the volume element
in spherical coordinates.
However you should know the definition of the three
variables r,, and their relations to x,y, z
You should know how to normalize a
function
You should understand why the integral
of the square modules of a wavefunction
integrated over all space in which the wavefunction
is defined must give 1 according to Borns postulate
You should be familiar with
QM postulate 1 and the conditions
it (The Born Postulate)
puts on a proper wavefunction