Transcript INTRODUCTION TO WAVE PACKETS

INTRODUCTION TO WAVE PACKETS
S.RAJAGOPALA REDDY
7/17/2015
1
1.Origin
2.Definition
3.Properties
4.Time evolution
Part 1
Origin of wave packets
Schrödinger Equation
In one dimension the Schrödinger equation is
Time dependent equation
By variable separable method this can be divided into two parts
On substitution of above equation in first, we will get
Time independent equation
Complete wave function
Applying Time independent Schrödinger equation for a Free particle
For free particle potential V(x) = 0
The time independent equation becomes
On solving above second order differential equation
The complete solution is
Importance of coefficients A & B
Put B=0
Applying momentum operator
on wave function
Put A=0
momentum
Energy of free particle
The above solution poses three severe problems
1. The probability densities corresponds to either solutions are
independent of x and t.
Heisenberg uncertainty principle give counter for this
Both momentum and energy precisely defined so we can't get
information of position and time.
2.The wave function is not square integrable
3.Speed of wave vs. particle
Speed of wave =
wavelength
Time period
Acc to Debrogle hypothesis
Time period is reciprocal of frequency
The particle travels twice as past as the wave which represents it
So, the plane waves are not solutions for Schrödinger equation
Part 2
Definition of wave packet
The remedy for all these problems come from Mathematical
expression called wave packet
Classical Mechanics – Particle - localized
Quantum Mechanics - Wave – not localized
A localized wave function is called wave packet
So, the function peaks at a certain value of x and disappear within a
small span .
Localized wave packet can be constructed by superposing waves of
slightly different wavelength but phases and amplitudes chosen
Constructively in desired region and destructively in other region
This can be done by Fourier Transformations
Where
Choosing t=0
Wave packet is given by
At x=0, exp(ikx)=1,so
As x
0, exp(ikx)
Peaks at x=0
1
In this case frequencies under go
constructive interference
Away from x=0 then destructive
interference takes place and
amplitude diminishes.
Examples
What we can get from wave packets
Part 3
How wave packet give answers for previous queries
1) Normalization
2) Since we are not precisely defining momentum and energy, we
will not loose information about position and time .
3) Wave velocity : To be explained in few minutes
Uncertainty relations
Gaussian wave packet
Other wave packets
Gaussian wave packet is called as minimum uncertainty
wave packet
Motion of wave packets
a) Non dispersive medium
Which implies
The wave packet travels right without any distortion
b) Dispersive medium
Angular frequency is function of k
Expanding angular frequency using Taylor series
Time evolution of wave packets
The motion of wave packet is
Group velocity is the velocity with which the wave
packet Propagates as a whole
Phase velocity is the velocity of individual waves
The angular frequency related to phase velocity by
For non dispersive medium
For a free particle
Group velocity nothing but classical velocity
Phase velocity has no physical significance
We can truncate the Taylor expansion of angular frequency
a) Linear approximation
This can be written as
travels right with phase velocity
is a curve which travels with group velocity and undistorted
In linear approximation the wave packet is undistorted and
undergoes a uniform translation
2) Quadratic approximation
Apply above formalism on Gaussian wave packet
Probability density
Width of wave packet
Width of wave packet in L.A
So, in Q.A wave packet broadens linearly with time
The Spreading of a Wave Packets
Probability density
Width of wave packet
vg
Δx0
References


“Quantum Mechanics – concepts and applications” by
Nouredine Zettili
“Molecular quantum mechanics” by Peter Atkins & Ronald
Friedman