Lecture 3: The Time Dependent Schrödinger Equation The ... - Cobalt

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 ,**)

setting up equation
Consider a particle of mass m that is moving in one
dimension. Let its position be given by x
V
Time Dependent Schrödinger Equation
O
V(X,t 1 ) V(X,t 2 )
X
Let the particle be
subject to the
potential V(x,t)
O
All properties of such a particle is in quantum mechanics
determined by the wavefunction Ψ(x,t) of the system

Time Dependent Schrödinger Equation
Vxt ( , ) setting up equation
X

Time Dependent Schrödinger Equation
setting up equation
A system that changes with time
is described by the time -
dependent Schrödinger equation
h δΨ( xt , )
− =
i δt
Hˆ Ψ( x, t)
Where Hˆ
is the Hamiltonian of
the system :
Hˆ h 2 δ
2 = − + Vxt ( , )
2m δx
2
for
1D - particle
according to postulate 6

Time Dependent Schrödinger Equation
setting up equation
2 2
− hδΨ( xt , ) = − h δ Ψ( xt , )
i δt
2m
2
+
δx
Vxt ( , ) Ψ( xt , )
The time dependent Schrödinger equation
The wavefunction Ψ(x,t) is also referred to as
The statefunction
Our state will in general change with time due to V(x,t).
Thus Ψ 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) = Ψ(x,t)
Ψ(x,t) dx
Probability density
Is the probability at time t to find the particle
between x and x + ∆x.
Ψ( x,t)Ψ * ( 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
Ψ(x,t) Ψ(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 Ψ( xt , ):
Let dn x denote the number of particle
which at the same time is found
between x and x + ∆x
Then :
dn x *
=Ψ( xt , ) Ψ ( xtdx , )
N

Time Dependent Schrödinger Equation
with time independent potential energy
The time - dependent Schroedinger equation :
2 2
− hδΨ( xt , ) =− h δ Ψ( xt , )
i δt
2m
2
δx
+
Can be simplified
in those cases where
the potential V only
depends on the
position : V(t,x) - >
V(x)
V
V(X)
O
Vxt ( , ) Ψ( xt , )

Time Dependent Schrödinger Equation
with time independent potential energy :
separation of time and space
We might try to find a solution of the form :
Ψ( xt , ) = ft ( ) ψ( x)
We have
δΨ(x,t)
δψ(
(x)f(t))
= = ψ(
x) δf(t)
δt δt δt
δ
and
2
2
Ψ(x,t) δψ( (x)f(t)) δψ(
x)
= = f(t)
2
2
2
δx δx δx
2

Simplyfied Time Dependent Schrödinger Equation
with time independent potential energy :
separation of time and space
hδΨ( xt , ) h δ Ψ( xt , )
− =− +
i δt
2m
2
δx
A substitution of
2 2
Ψ( xt , ) = ft ( ) ψ( x)
Vxt ( , ) Ψ( xt , )
into the Schrödinger equation thus affords :
2 2
− h
= − h
ψ
δ δψ(
( x) f(t) x)
f(t)
2
i δt 2m δx
+
Vx ( ) f(t) ψ(
x)

Simplyfied Time Dependent Schrödinger Equation
with time independent potential energy :
separation of time and space
2 2
h
h
− ψ
δ δψ(
( x) f(t) x)
= − f(t) +
2
i δt 2m δx
1
A multiplication from the left by affords :
f() t ψ( x)
2 2
h 1 δf(t)
h 1 δψ(
x)
− = − +
2
if() t δt 2mψ(
x) δx
Vx ( ) f(t) ψ(
x)
Vx ( )
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 :
h 1 δf(t)
− = E = constant
ift () δt
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.
h 2 1 δψ
2 ( x)
− + V( x) = E = constant
2
2m
ψ(
x) δx

We can now solve for f(t) :
h 1 δf(t)
− = E = constant
ift () δt
Or :
Simplyfied Time Dependent Schrödinger Equation
with time independent potential energy :
separation of time and space
δf(t)
i Et δ
f() t
=− h
Now integrating from time t=0 to t=to on both sides affords:
t
o
∫
o
δf(t)
o
i
=−∫ Et δ
f() t h
t
o

Simplyfied Time Dependent Schrödinger Equation
with time independent potential energy :
separation of time and space
t
o
∫
o
δf(t)
o
i
=−∫ Et δ
f() t h
t
o
i
ln[ ft ( o)] − ln[ fo ( )] = − Et [ o −0]
h
i
ln[ f( to)] =− Et o + ln[ f ( o )]
h
Cons
tan
t
ln[ f( t )]
o
i
=− Et o +
h
C

Simplified Time Dependent Schrödinger Equation
with time independent potential energy :
separation of time and space
Or:
ln[ f ( t i
o)]
=− Et o + C
h
i i
f ()= t Exp
⎡ − Et + C
⎤ Exp C Exp Et
⎣⎢ ⎦⎥ = [ ] ⎡ ⎣⎢
− ⎤
h
h ⎦⎥
E E
f ( t ) = Exp[ C] ⎡ ⎤ (cos t isin t )
⎣⎢ ⎦⎥ − ⎡ ⎤
h ⎣⎢ h ⎦⎥

Simplified Time Dependent Schrödinger Equation
with time independent potential energy :
separation of time and space
E E
f ( t ) = Exp[ C] ⎡ ⎤ (cos t isin t )
⎣⎢ ⎦⎥ − ⎡ ⎤
h ⎣⎢ h ⎦⎥
+
-
i
i
- i +
t = 0 t =π (h/ E) t = π(h/ E) t
2 = 3π
2 (h /E) t= 2π(h/ E)
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 ψ(x) is given by
h 2 1 δψ
2 ( x)
− + Vx ( ) =
2m
ψ(
x)
2
δx
E
Or :
h 2 2 2
δψ(
x)
− + ψ( x) Vx ( ) = Eψ(
x)
2m δx

Simplified Time Dependent Schrödinger Equation
with time independent potential energy :
separation of time and space
Time independent Schrödinger equation
h 2 2 2
δψ(
x)
− + ψ( x) Vx ( ) = Eψ(
x)
2m δx
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
Ψ( xt , ) = ft ( ) ψ( x)
= Exp[ C] Exp[ −i E t] ψ( x)
AExp[ i E h
= − t] ψ( x)
h

Simplified Time Dependent Schrödinger Equation
with time independent potential energy :
separation of time and space
Time independent Schrödinger equation
If ψ(x) is a solution to
h 2 2 2
δψ(
x)
− + ψ( x) Vx ( ) = Eψ(
x)
2m δx
So is A ψ(x)
Lecture 2
h 2 δ
2 ( Aψ(
x))
− + Aψ( x) V( x) = AEψ(
x)
2
2m δx

Simplified Time Dependent Schrödinger Equation
with time independent potential energy :
separation of time and space
time independent probability function
h 2 δ
2 ( Aψ(
x))
− + Aψ( x) V( x) = AEψ(
x)
2
2m δx
or :
h 2 2 2
δψ'(
x)
− + ψ'( x) Vx ( ) = Eψ'(
x)
2m δx
with ψ
' (x) = A ψ(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
Ψ( x, t) = Exp[ −i E t] ψ( x)
h
This wavefunction is time dependent and complex.
Let us now look at the corresponding probability density
Ψ( xt , ) Ψ
*
( xt , )

Simplified Time Dependent Schrödinger Equation
with time independent potential energy :
separation of time and space
time independent probability function
We have :
*
Ψ( xt , ) Ψ ( xt , ) = Exp[ −i E t] ψ( x)
h
* *
× Exp[ i E t] ψ ( x) = ψ( x) ψ ( x)
h
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
*
Ψ( xt , ) Ψ ( xt , ) = Exp[ −i E t] ψ( x)
h
* *
× Exp[ i E t] ψ ( x) = ψ( x) ψ ( x)
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 is
constant.

Simplified Time Dependent Schrödinger Equation
stationary states
*
ψ( x) ψ ( x)
dx Independent of time
We say that systems that can be described by
wave functions of the type
Ψ( x, t) = Exp[ −i E t] ψ( x)
h
Represent
Stationary
states

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
h δΨ( xt , )
− =
i δt
Hˆ Ψ( x, t)
where H ˆ is the Hamiltonian
(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
h δΨ( xt , )
− =
i δt
Hˆ Ψ( x, 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
Ψ( x, t) = Exp[ −i E t] ( x)
h
ψ
Where ψ( x)
is a solution to the time - independent
Schrödinger equation : Hψ( x) = Eψ(
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 :
* *
Ψ( xt , ) Ψ ( xt , ) = Exp[ −i E t] ψ( x) Exp[ i E t] ψ ( x)
h
h
= ψ( x) ψ
*
( x).
They are called stationary states