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Lecture 3: The Time Dependent Schrödinger Equation
The material in this lecture is not covered in Atkins. It is required to understand
postulate 6 and
11.5 The informtion of a wavefunction
Lecture on-line
The Time Dependent Schrödinger Equation (PDF)
The time Dependent Schroedinger Equation (HTML) The time dependent
Schrödinger Equation (PowerPoint)
Tutorials on-line
The postulates of quantum mechanics (This is the writeup for Dry-lab-II
( This lecture coveres parts of postulate
6)
Time Dependent Schrödinger Equation
The Development of Classical Mechanics
Experimental Background for Quantum mecahnics
Early Development of Quantum mechanics
Audio-visuals 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 ,**)
Time Dependent Schrödinger Equation
setting up equation
Consider a particle of mass m that is moving in one
dimension. Let its position be given by x
X
O
V
V(X,t1)
O
V(X,t2)
Let the particle be
subject to the
potential V(x,t)
All properties of such a particle is in quantum mechanics
determined by the wavefunction (x,t) of the system
Time Dependent Schrödinger Equation
V(x, t)
setting up equation
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are needed to see this picture.
X
Time Dependent Schrödinger Equation
setting up equation
A system that changes with time
is described by the time dependent Schrödinger equation
(x,t) ˆ

 H(x,t)
i t
ˆ is the Hamiltonian of
Where H
the system :
2 2
ˆ  
H
 V(x, t)
2
2m x
for 1D - particle
according to postulate 6
Time Dependent Schrödinger Equation
setting up equation
(x,t)
 (x, t)


 V(x,t)(x,t)
2
i
t
2m x
2
2
The time dependent Schrödinger equation
The wavefunction (x, t) is also referred to as
The statefunction
Our state will in general change with time due to V(x,t).
Thus  is a function of time and space
Time Dependent Schrödinger Equation
Probability from wavefunction
The wavefunction does not have any physical interpretation.
However :
P(x, t) = (x, t)(x, t) dx
*
Probability density
Is the probability at time t to find the particle
between x and x + x.
( x, t)*( x, t)dx
will change with time
dx o
x
Time Dependent Schrödinger Equation
Probability from wavefunction
It is important to note that the particle is not distributed
over a large region as a charge cloud
(x,t)(x,t)
*
It is the probability patterns (wave function) used
to describe the electron motion that behaves like
waves and satisfies a wave equation
Time Dependent Schrödinger Equation
Probability from wavefunction
Consider a large number N of
identical boxes with identical
particles all described by the
same wavefunction (x,t) :
Let dnx denote the number of particle
which at the same time is found
between x and x + x
Then :
dnx
*
 (x, t) (x,t)dx
N
Time Dependent Schrödinger Equation
with time independent potential energy
The time - dependent Schroedinger equation :
(x, t)
 (x, t)


 V(x, t)(x, t)
2
i
t
2m x
2
2
V
Can be simplified
in those cases where
the potential V only
depends on the
position : V(t,x) - >
V(x)
V(X)
O
Time Dependent Schrödinger Equation
with time independent potential energy:
separation of time and space
We might try to find a solution of the form:
(x, t)  f (t) (x)
We have
(x,t) ((x)f(t))
f(t)

  (x)
t
t
t
and
 2 (x, t)  2 ((x)f(t))
 2  (x)

 f(t)
2
2
2
x
x
x
Simplyfied Time Dependent Schrödinger Equation
with time independent potential energy:
separation of time and space
(x, t)
 (x, t)


 V(x, t)(x, t)
2
i
t
2m x
2
2
A substitution of (x, t)  f (t) (x)
into the Schrödinger equation thus affords:
2
f(t)
 2 (x)
 (x)

f(t)
 V(x)f(t)(x)
2
i
t
2m
x
Simplyfied Time Dependent Schrödinger Equation
with time independent potential energy:
separation of time and space
2
f(t)
 2 (x)
 (x)

f(t)
 V(x)f(t)(x)
2
i
t
2m
x
1
A multiplication from the left by
affords:
f (t) (x)
1 f(t)
1  (x)


 V(x)
2
i f(t) t
2m (x) x
2
2
The R.H.S. does not depend on t if we now assume that
V is time independent. Thus, the L.H.S. must also be
independent of t
Simplyfied Time Dependent Schrödinger Equation
with time independent potential energy:
separation of time and space
Thus :
1 f(t)

 E  cons tant
i f (t) t
The L.H.S. does not depend on x so the R.H.S. must also
be independent of x and equal to the same constant, E.
1  2 (x)

 V(x)  E  cons tan t
2
2m (x) x
2
Simplyfied Time Dependent Schrödinger Equation
with time independent potential energy:
separation of time and space
We can now solve for f(t) :
1 f(t)

 E  cons tant
i f (t) t
Or :
f(t)
f (t)
i
  Et
Now integrating from time t=0 to t=to on both sides affords:
to
f(t)
o
f (t)

to

o
i
Et
Simplyfied Time Dependent Schrödinger Equation
with time independent potential energy:
separation of time and space
to
f(t)
o
f (t)

to

i
Et
o
i
ln[f(t o )]  ln[f(o)]   E[t o  0]
i
ln[ f (to )]   Eto  ln[ f (o)]
i
ln[ f (to )]   Eto  C
Cons tant
Simplified Time Dependent Schrödinger Equation
with time independent potential energy:
separation of time and space
i
Or:
ln[ f (to )]   Eto  C
 i

 i 
f (t)  Exp  Et  C  ExpCExp  Et 




E 
E 


f (t)  ExpC (cos t  i sin t )
 
 
Simplified Time Dependent Schrödinger Equation
with time independent potential energy:
separation of time and space
E 
E 


f (t)  ExpC (cos t  i sin t )
 
 
+
t 0
-i
i
-


t ( / E) t  ( / E)
2
i
+
t 3 ( /E) t  2( / E)
2
Change of
sign of f(t)
with time
Simplified Time Dependent Schrödinger Equation
with time independent potential energy:
separation of time and space
Time independent Schrödinger equation
The equation for (x) is given by
1  (x)

 V(x)  E
2
2m (x) x
2
2
Or :
 (x)

 (x)V(x)  E(x)
2
2m x
2
2
Simplified Time Dependent Schrödinger Equation
with time independent potential energy:
separation of time and space
Time independent Schrödinger equation
 (x)

 (x)V(x)  E(x)
2
2m x
2
2
This is the time-independent Schroedinger Equation
for a particle moving in the time independent potential V(x)
It is a postulate of Quantum Mechanics that E is
the total energy of the system
Part of QM postulate 6
Simplified Time Dependent Schrödinger Equation
with time independent potential energy:
separation of time and space
Time independent Schrödinger equation
The total wavefunction for a one-dimentional particle in
a potential V(x) is given by
(x, t)  f (t) (x)
 Exp[C]Exp[i
 AExp[i
E
E
t] (x)
t] (x)
Simplified Time Dependent Schrödinger Equation
with time independent potential energy:
separation of time and space
Time independent Schrödinger equation
If  (x) is a solution to
 (x)

 (x)V(x)  E(x)
2
2m x
2
2
So is A (x)
Lecture 2
 (A(x))


A(x)V(x)

AE(x)
2m
x 2
2
2
Simplified Time Dependent Schrödinger Equation
with time independent potential energy:
separation of time and space
time independent probability function
 2 (A(x))

 A(x)V(x)  AE(x)
2
2m
x
2
or :
 '(x)

 '(x)V(x)  E'(x)
2
2m x
2
2
with  (x) = A (x)
'
Simplified Time Dependent Schrödinger Equation
with time independent potential energy:
separation of time and space
time independent probability function
Thus we can write without loss of generality for a
particle in a time-independent potential
(x, t)  Exp[i
E
t] (x)
This wavefunction is time dependent and complex.
Let us now look at the corresponding probability density
(x, t) (x, t)
*
Simplified Time Dependent Schrödinger Equation
with time independent potential energy:
separation of time and space
time independent probability function
We have :
(x, t) (x, t)  Exp[i
*
Exp[i
E
E
t](x)
t] (x)   (x)  (x)
*
*
Thus , states describing systems with a time-independent
potential V(x) have a time-independent (stationary)
probability density.
Simplified Time Dependent Schrödinger Equation
stationary states
(x, t) (x, t)  Exp[i
*
Exp[i
E
E
t](x)
t] (x)   (x)  (x)
*
*
This does not imply that the particle is stationary.
However, it means that the probability of finding
a particle in the interval x+ -1/2x to x + 1/2x is
constant.
Simplified Time Dependent Schrödinger Equation
stationary states
 (x) (x)dx
*
Independent of time
We say that systems that can be described by
wave functions of the type
(x, t)  Exp[i
Represent
Stationary
states
E
t] (x)
Simplified Time Dependent Schrödinger Equation
Postulate 6
The time development of the
state of an undisturbed system
is given by the time - dependent
Schrödinger equation
(x,t) ˆ

 H(x,t)
i t
ˆ is the Hamiltonian
where H
(i.e. energy) operator
for the quantum mechanical system
What you should know from this lecture
1. You should know postulate 6 and the form of the
time dependent Schrödinger equation
(x,t) ˆ

 H(x,t)
i t
2. You should know that the wavefunction for
systems where the potential energy is independent of
time [V(x,t)  V(x)] is given by
E
(x, t)  Exp[i t](x)
Where (x) is a solution to the time - independent
Schrödinger equation : H(x)  E(x),
and E is the energy of the system.
What you should know from this lecture
3. Systems with a time independent potential
energy [V(x, t)  V(x) ] have a time - independent
probability density :
E
E
*
(x, t) (x,t)  Exp[i t](x)Exp[i t] * (x)
= (x) * (x).
They are called stationary states