Wave Packet Etc

7/21/2019 Wave Packet Etc

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Wave packet etc



Plane Waves

• a plane wave is an ideal wave withsingle wavenumber k. The wave fronts(surfaces of constant phase) of a plane

wave are innite parallel planes ofconstant peak-to-peak amplitude normalto the phase velocit vector.

• !f a wave is of innite e"tent onl then itwill propagate as a plane wave. !n mostpractical cases waves are appro"imatelplane waves in a locali#ed region of space.



Plane Wave



Plane Wave



Phase $elocit of a PlaneWave



Plane Wave

Equation of a Plane% plane in &d-space has the e'uation

ax + by + cz = d,where at least one of the numbers a b c must be

non#ero.



Wave packet in bothposition and momentum space



$elocit of a Wave Packet

• !f the position of the wave packetchanges with time then the rate atwhich the ma"imum point moves is a

reasonabl good measure of thevelocit of the wave packet

∫ ∞

∞−

−= dk ek at x wt kxi )()(),(ψ



%ppro"imations•

!n order for the momentum of the particleto be reasonabl well dened

--- et us assume that the harmonic wavespresent in ψ ("t) have wave vectors lingwithin a small range centered around avalue *k+

k p =

)(

)()(

)()(

k k

k k k d

d k

k k dk

d

k k k

==

−

+=

−

+= =

ω ω

ω ω ω

ω

ω ω



%t the ,a"imum of Wave packet at ",%

0=

= MAX x xdx

d ψ

∫ ∞

∞−




%t the ,a"imum of Wave packet at ",%

∫ ∞

∞−


∫ =−0dk eik k a

t kxi MAX )(

))(( ω

0=

= MAX x xdx

d ψ



urther di/erentiating wrt to time

∫ =−0dk eik k a

t kxi MAX

)())((

ω

( )[ ]∫ =− −0dk e xk iik k a

t kxi

MAX

MAX )())((

ω

ω



0sing the %ppro"imation

( )[ ]∫ =− −0dk e xk iik k a

t kxi

MAX

MAX )())((

ω

ω

)()( k k k d

d k −

+= ω

ω ω

∫ ∫ =

−−

− −−

022

dk keik ak k d

d dk eik k a

k d

wd x

t kxit kxi

MAX

MAX MAX )()())(( ))((

ω ω ω

ω




∫ ∫ =

−−

− −− 022 dk keik ak

k d

d dk eik k a

k d

wd x t kxit kxi MAX

MAX MAX )()( ))(( ))(( ω ω ω

ω

∫ =−0dk eik k a

t kxi MAX )(

))(( ω

∫ =

− −

02

))(()(dk eik k a

k d

wd x

t kxi

MAX

MAX ω

0=

= MAX x xdx

d ψ




∫ =

− −0

2 ))((

)(dk eik k a

k d

wd x

t kxi

MAX

MAX ω

02

2

≠

= MAX x x

dx

d ψ

The term in the above integral ison!zero"



$elocit of the Peak

PacketWavetheof VelocityGroupcalledisG

G

MAX

MAX

MAX

V

k d

wd V

k d

wd

dt

dx

k d

wd

dt

dx

k d

wd x

=

=

=

−

=

−

0

0



Phsical 1ignicance of $2

• The $elocit of a free particle (bothnon-relativistic and relativistic) canbe written as

dp

dE v =



Phsical 1ignicance of $2

• The $elocit of a particle (both non-relativistic and relativistic) can bewritten as

dp

dE v particle =

k p & E == ω with

G particle V dk

d

dp

dE v ~

ω ==



The Particle Picture

G particle V dk

d

dp

dE v ~ ω ==

% small wave packet consisting of small band of3e 4roglie waves moves somewhat like a

classical free particle.



Wave-Particle 3ualit

G particle V dk

d

dp

dE v ~ ω ==

% small wave packet consisting of small band of3e 4roglie waves moves somewhat like a

classical free particle.

The appro"imation is valid onl to the e"tent that onecan ignore the fundamental limitations placed b thewave packet picture on the accurac with which boththe position and momentum of a particle can be

Wave Packet Etc

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