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8/3/2019 Chem 373- Lecture 3: The Time Dependent Schrdinger Equation
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Lecture 3: The Time Dependent Schrdinger Equation
The material in this lecture is not covered in Atkins. It is required to understandpostulate 6 and11.5 The informtion of a wavefunction
Lecture on-lineThe Time Dependent Schrdinger Equation (PDF)The time Dependent Schroedinger Equation (HTML) The time dependent
Schrdinger 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 Schrdinger Equation
The Development of Classical MechanicsExperimental Background for Quantum mecahnicsEarly Development of Quantum mechanics
Audio-visuals on-linereview of the Schrdinger equation and the Born postulate (PDF)
review of the Schrdinger equation and the Born postulate (HTML)review of Schrdinger equation and Born postulate (PowerPoint **,
1MB)Slides from the text book (From the CD included in Atkins ,**)
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Consider a particle of mass m that is moving in onedimension. Let its position be given by x
O
X
Let the particle be
subject to thepotential V(x,t)
O
V
V(X,t1) V(X,t2)
All properties of such a particle is in quantum mechanicsdetermined by the wavefunction (x, t) of the system
Time Dependent Schrdinger Equationsetting up equation
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Time Dependent Schr dinger Equation
X
V x t( , ) setting up equation
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Time Dependent Schr dinger Equation
A system that changes with timeis described by the time -dependent Schr dinger equation
=h
ix tt
H x t
( , ) ( , )
according to postulate 6
Where H
H m x V x t
for
is the Hamiltonian of the system
1D - particle
:
( , )= +h 2 2
22
setting up equation
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= +h h
i
x t
t m
x t
x V x t x t
( , ) ( , )
( , ) ( , )
2 2
22The time dependent Schr dinger equation
The wavefunction is also referred to as(x,t)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 Schr dinger Equation
setting up equation
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The wavefunction does not have any physical interpretation.However :
P(x, t) = (x, t) (x, t) dx*
Probability density
ox
dx
( x, t)*( x, t) dx
Is the probability at time t to find the particlebetween x and x + x.
Time Dependent Schr dinger Equation
will change with time
Probability from wavefunction
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It is important to note that the particle is not distributedover a large region as a charge cloud
It is the probability patterns (wave function) usedto describe the electron motion that behaves likewaves and satisfies a wave equation
(x, t) (x, t) *
Time Dependent Schr dinger Equation
Probability from wavefunction
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Consider a large number N of identical boxes with identical
particles all described by thesame wavefunction ( , ) : x t
Then : dn N
x t x t dx x = ( , ) ( , )*
Let dn denote the number of particle
which at the same time is foundbetween x and x +
x
x
Time Dependent Schr dinger Equation
Probability from wavefunction
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= +h h
i x t t m
x t x
V x t x t
( , ) ( , ) ( , ) ( , )2 2 22
The time - dependent Schroedinger equation :
O
V
V(X)Can be simplifiedin those cases where
the potential V onlydepends on theposition : V(t, x) - >V(x)
Time Dependent Schr dinger Equation
with time independent potential energy
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We might try to find a solution of the form :
( , ) ( ) ( ) x t f t x= We have
(x, t) (x)f(t)) x) f(t) t t t
= =( (
2
2
2
2
2
2(x, t) (x)f(t))
f(t)x)
x x x= =( (
and
Time Dependent Schr dinger Equationwith time independent potential energy :separation of time and space
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= +
h h
i t m xV x
( ( ( ) (x) f(t) f(t) x) f(t) x)
2 2
22
A substitution of into the Schr dinger equation thus affords :
( , ) ( ) ( ) x t f t x=
Simplyfied Time Dependent Schr dinger Equation
= +h hi
x t t m
x t x
V x t x t
( , ) ( , ) ( , ) ( , )2 2 22
with time independent potential energy :separation of time and space
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Simplyfied Time Dependent Schr dinger Equation
A multiplication from the left by affords :1
f t x( ) ( )
= +h h
i f t t m xV x
1
2
12 2
2( ) (
(( )
f(t)
x)
x)
The R.H.S. does not depend on t if we now assume thatV is time independent. Thus, the L.H.S. must also be
independent of t
= +h h
i t m x V x
(
(( ) (x)
f(t)f(t)
x)f(t) x)
2 2
22
with time independent potential energy :separation of time and space
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= =h
i f t t E cons t1
( )tan
f(t)
Thus :
The L.H.S. does not depend on x so the R.H.S. must alsobe independent of x and equal to the same constant, E.
+ = =
h 2 2
221
m xV x E cons t
((
( ) tanx)
x)
Simplyfied Time Dependent Schr dinger Equation
with time independent potential energy :separation of time and space
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= =h
i f t t E cons t1
( )tan
f(t)
We can now solve for f(t) :
Or :
f(t)
f t
i E t
( )
= h
Now integrating from time t=0 to t=to on both sides affords:
o
t
o
t o o
f ti E t =
f(t)( ) h
Simplyfied Time Dependent Schr dinger Equation
with time independent potential energy :separation of time and space
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o
t
o
t o o
f ti E t =
f(t)( ) h
ln[ ( )] ln[ ( )] [ ]f t f oi
E to o = h 0
ln[ ( )] ln[ ( )] f t
i Et f o
o o= +
h Cons ttan
ln[ ( )] f ti Et C o o= +h
Simplyfied Time Dependent Schr dinger Equation
with time independent potential energy :separation of time and space
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ln[ ( )] f t
i Et C
o o= +
hOr:
f t Exp
i Et C Exp C Exp
i Et( ) = + = [ ]
h h
f t Exp C E
t iE
t( ) (cos sin )= [ ] h h
Simplified Time Dependent Schr dinger Equation
with time independent potential energy :separation of time and space
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with time independent potential energy :separation of time and space
Simplified Time Dependent Schr dinger Equation
Change of sign of f(t)with time
t =2(h / E )
+
t = (h / E ) t =32 ( h / E ) t = (h / E )
-
i- i +
t =0 2
i
f t Exp C
E t i
E t( ) (cos sin )= [ ]
h h
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+ =
h 2 2
22
1
m xV x E
(
(( )
x)
x)
The equation for is given by ( x)
Simplified Time Dependent Schr dinger Equation
Time independent Schr dinger equation
with time independent potential energy :
separation of time and space
+ =
h 2 2
22m xV x E
( ( ( ) (x) x) x)
Or :
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Simplified Time Dependent Schr dinger Equation
Time independent Schr dinger equation
with time independent potential energy :
separation of time and space
+ =h 2 2
22m xV x E ( ( ( ) (x) x) x)
This is the time-independent Schroedinger Equationfor a particle moving in the time independent potential V(x)
It is a postulate of Quantum Mechanics that E isthe total energy of the system
Part of QM postulate 6
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The total wavefunction for a one-dimentional particle ina potential V(x) is given by
( , ) ( ) ( )
[ ] [ ] ( )
[ ] ( )
x t f t x
Exp C Exp iE
t x
AExp iE
t x
=
= =
h
h
Simplified Time Dependent Schr dinger Equation
Time independent Schr dinger equation
with time independent potential energy :separation of time and space
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If is a solution to (x)
+ =
h 2 2
22m xV x E
( ( ( ) (x) x) x)
So is A (x)
+ =
h 2 2
22mA
xA V x AE
( ( ( ( ) (x)) x) x)
Simplified Time Dependent Schr dinger Equation
Lecture 2
Time independent Schr dinger equation
with time independent potential energy :
separation of time and space
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or :
+ =
h 2 2
22m x V x E
' (
' ( ( ) ' (
x)
x) x)
with ' (x) = A (x)
+ =
h 2 2
22mA
xA V x AE
( ( ( ( ) (x)) x) x)
Simplified Time Dependent Schr dinger Equation
time independent probability function
with time independent potential energy :
separation of time and space
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Thus we can write without loss of generality for aparticle in a time-independent potential
( , ) [ ] ( ) x t Exp iE
t x= h This wavefunction is time dependent and complex.
Let us now look at the corresponding probability density
( , ) ( , )*
x t x t
Simplified Time Dependent Schr dinger Equation
time independent probability function
with time independent potential energy :
separation of time and space
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( , ) ( , ) [ ] ( )[ ] ( ) ( ) ( )
*
* *
x t x t Exp iE
t x
Exp iE
t x x x=
=h
h
We have :
Thus , states describing systems with a time-independentpotential V(x) have a time-independent (stationary)probability density.
Simplified Time Dependent Schr dinger Equation
time independent probability function
with time independent potential energy :separation of time and space
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( , ) ( , ) [ ] ( )
[ ] ( ) ( ) ( )
*
* *
x t x t Exp iE
t x
Exp iE
t x x x
= =
h
h
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 isconstant.
Simplified Time Dependent Schr dinger Equationstationary states
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( ) ( )* x x dx Independent of timeWe say that systems that can be described bywave functions of the type
( , ) [ ] ( ) x t Exp i E t x= h
RepresentStationarystates
Simplified Time Dependent Schr dinger Equationstationary states
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Postulate 6The time development of thestate of an undisturbed systemis given by the time - dependentSchr dinger equation
Simplified Time Dependent Schr dinger Equation
=h
ix tt
H x t
( , ) ( , )
where H is the Hamiltonian(i.e. energy) operatorfor the quantum mechanical system
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What you should know from this lecture
=h
i
x t
tH x t
( , ) ( , )
1. should know postulate 6 and the form of thetime dependent Schr dinger equation
You
2.
( , ) [ ] ( )
( ) ( ) ( ),
should know that the wavefunction for
systems where the potential energy is independent of time [V(x, t) V(x)] is given by
Where is a solution to the time - independentSchr dinger equation : HE is the energy of the system.
You
x t Exp iE
t x
x x E xand
=
=
h
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What you should know from this lecture
3.
( , ) ( , ) [ ] ( ) [ ] ( )
( ) ( ).
* *
*
Systems with a time independent potentialenergy [V(x, t) V(x) ] have a time - independentprobability density :
=are called stationary states
= x t x t Exp i E t x Exp i E t x
x xThey
h h