Chem 373- Lecture 3: The Time Dependent Schrödinger Equation

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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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


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.


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) *




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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




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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)


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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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



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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)




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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




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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




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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



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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)


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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+ =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





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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)


Lecture 2





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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)


time independent probability function




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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






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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.





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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


=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

Documents

Chem 373- Lecture 3: The Time Dependent Schrödinger Equation