Download the Schrodinger wave equation

Schrodinger wave equation
► The position of a particle is distributed through space like the amplitude of
a wave.
► In quantum mechanics, a wavefunction describes the motion and location
of a particle.


► A wavefunction is just a mathematical function which may be large in one
region, small in others and zero elsewhere.
sin x or e  x
Concepts of Wave function
► If the a wavefunction is large at a particular point (i.e., the amplitude of the
wave is large), then the particle has a high probability of being found at that
point. If the wavefunction is zero at a point, then the particle will not be found
there.
► The more rapidly a wavefunction changes from place to place (i.e., the
greater the curvature of the wave), the higher kinetic energy of the particle it
describes.
Schrodinger wave equation
 2 d 2

2m dx 2
2 2


2m
where
 V( x )
 E
for a 1-D system
 V
 E
for a 3-D system
2
2
2
  2 2 2
x y
z
for Cartesian coordinate
2
2 1 2
  2
 2
r
rr r
for spherical coordinate
2
2
System with Spherical Symmetry
2  2
2
1 2 


 2    V  E

2
rr r
2m  r

where
1 2
1 

 

sin

sin 2   2 sin  

2
 2   2  2  1  1

2
1

 

  V  E

 r 2 rr r 2  sin 2   2  sin   sin   
2m 



Eigenstate
► in general, the Schrodinger eqn is
H  E 
► H is the Hamiltonian operator, i.e., the energy
operator.
E
H

► Schrodinger eqn is an eigenvalue equation
(operator)(function) = (constant) x (same function)
eigenvalue
eigenfunction
Operator
►(operator) (function) = (constant) x (same function)
̂f   f
The factor  is called the eigenvalue of the operator ̂
► The function f (which must be the same on each side in an eigenvalue
equation) is called an eigenfunction and is different for each eigenvalue.
► An eigenvalue is a measurable property of a system, an observable. Each
observable has a corresponding operator.
(operator for observable) (wavefunction) = (value of observable) x (wavefunction)
Examples of Operators
► A basic postulate of quantum mechanics is the form of the operator for
linear momentum.
pˆ  i
d
dx
► If we want to find the linear momentum of a particle in the x direction, we
use the following eigenvalue equation.
pˆ   p
► The eigenvalue, p, is the momentum.
Correspondence
► The kinetic energy operator is then created from the momentum
operator.
2
p2
mv 2

mv 

Ek 

2m
2
2m
2
2
2
ˆ
p

d
Eˆ k  H 

2m
2m dx 2
► If we want to find the kinetic energy of a particle in the x direction, we use
the following eigenvalue equation.
H  E 
 2 d 2

 E
2m dx 2
the eigenvalue, E, is the
kinetic energy
What is the wavefunction?
► So what’s the big deal? This should be straight forward??
► Just carry out some operation on a wavefunction, divide the result by the
same wavefunction and you get the observable you want.
► Each system has its own wavefunction (actually many wavefunctions) that
need to be found before making use of an eigenvalue equation.
► The Schrodinger equation and Born’s interpretation of wavefunctions will
guide us to the correct form of the wavefunction for a particular system.
Born interpretation of the wavefunction
► In the wave theory of light, the square of the amplitude of an electromagnetic
wave in a particular region is interpreted as its intensity in that region.
► In quantum mechanics, the square of the wavefunction at some point is
proportional to the probability of finding a particle at that point.

2
or  *
Interpretation



2
2
2
or  *
probability density
or  *
no negative or complex probability densities
or  *
written

2
Both large positive and large negative regions of a wavefunction correspond
to a high probability of finding a particle in those regions.
Normalization constant
► A normalization constant is found which will insure that the probability of
finding a particle within all allowed space is 100%.

N 
2
2
dx  1
1
N


 

2

dx 

1
2
Born interpretation of the wavefunction
► There are several restrictions on the acceptability of wavefunctions.
► The wavefunction must not be infinite anywhere.
It must be finite
everywhere.
► The wavefunction must be single-valued.
It can have only one value at
each point in space.
► The 2nd derivative of the wavefunction must be well-defined
everywhere if it is to be useful in the Schrodinger wave equation.
Born interpretation of the wavefunction
► The 2nd derivative of a function can be taken only if it is continuous (no
sharp steps) and if its 1st derivative is continuous.
► Wavefunctions must be continuous and have continuous 1st derivatives.
► Because of these restrictions, acceptable solutions to the Schrodinger
wave equation do not result in arbitrary values for the energy of a system.
► The energy of a particle is quantized. It can have only certain energies.
Expectation value
ˆ   
ˆ  d

*
► Let’s find the average score on a quiz. There were 5 problems on the quiz,
worth 20pts each (no partial credit). The scores for 10 students are given
below.
80, 80
60, 60, 60, 60
40, 40, 40
20
2
(80)  16
On this particular quiz, 1/5 of the students
4 (60)  24 received a score of 80. On future similar
10
quizzes, we could say the probability of
3 (40)  12 getting a score of 80 is 1/5.
10
10
1 (20)  2
10
What is the average score?
54
Expectation value
2
80, 80
10
(80)  16
(60)  24
10
3 (40)  12
10
1 (20)  2
10
4
60, 60, 60, 60
40, 40, 40
20
► We calculated the average by multiplying each score by the
probability of receiving that score and then found the sum of all
those products.
ˆ   
ˆ  d

*
Heisenberg’s uncertainty principle
► It is impossible to specify simultaneously, with arbitrary precision, both the
momentum and the position of a particle.
► If the momentum of a particle is specified precisely, it is impossible to
predict the location of the particle.
► If the position of a particle is specified exactly, then we can say nothing about
its momentum.
p x 

2