Understanding Wavepackets and Heisenberg Uncertainty Principle: A Review, Slides of Microwave Engineering and Acoustics

Download Understanding Wavepackets and Heisenberg Uncertainty Principle: A Review and more Slides Microwave Engineering and Acoustics in PDF only on Docsity!

Wavepacket 1

FREE PARTICLE GAUSSIAN

WAVEPACKET

Wavepacket 2

• Time dependent Schrödinger equation
• Energy eigenvalue equation (time independent SE)
• Eigenstates
• Time dependence
• (Connection to separation of variables)
• Mathematical representations of the above

GAUSSIAN WAVE PACKET - REVIEW

Wavepacket 4 We have already discussed the principle of superposition & the time evolution of that superposition in the context of the discrete quantum mechanical states of the infinite potential energy well. We now discuss how build a wave packet from harmonic waveforms (with a continuous frequency distribution). We use the case of superposition of quantum mechanical states of the free particle, which are no longer discrete, and we choose to weight different frequency components more heavily than others.

Wavepacket 5

E

k

2

k

2

2 m

" k

( x ) = e

ikx

-0. 0

1 ! -2 (^) x 0 2

Free particle eigenstates

! k

( x , t ) = e

ikx

e

" i Ek ! t

• Oscillating function
• Definite momentum p = hk/2π
• Subscript k on ω reminds us that ω depends on k
• What are E and ω in terms of given quantities?

k " E k ! = ! k 2 2 m " k ( x ) # $ k

( x ,0)

Wavepacket 7

c

n

" A

k

" A ( k )

-0. 0

1 ! -2 (^) x 0 2 Superposition of eigenstates (Fourier integral)

-0. 0

1 -2 0 2

-0. 0

1 -2 0 2

-0. 0

1 "( x ) = dk A ( k ) e -2 0 2 ikx #$ $ %

-0. 0

1 ! -2 (^) x 0 2 Localized particle Indefinite momentum Definite momentum Extended position

n

" # $ dk

n " k

**_+

+_**

Wavepacket 8

-0. 0

1 ! -2 (^) x 0 2 Superposition of eigenstates How does it develop in time?

-0. 0

1 -2 0 2

-0. 0

1 -2 0 2

-0. 0

1 -2 0 2 "( x , 0 ) = dk A ( k ) e ikx #$ $ %

-0. 0

1 ! -2 (^) x 0 2 Definite momentum Extended position "( x , t ) = dk A ( k ) e ikx

e

iEkt /!

#$ $ %

E

k

2

k

2

2 m

**_+

+_**

Localized particle Indefinite momentum

Wavepacket 10 Gaussian wave packet Localized particle

A ( k ) =!

k

t = 0

= dx!

k

( x ) !( x , 0 )

"#

!( x , 0 ) =

2

1 4

e

• " 2 x 2 x → ! 2 2 "

A ( k ) =!

k

t = 0

" dx e

ikx

e

$

2 x 2 #% %

dy e vy e " uy 2 "#

$ =^

% u e v 2 4 u We did this integral (by hand). using:

Wavepacket 11 Gaussian wave packet

!( x , 0 ) =^ Localized particle

2

1 4

e

• " 2 x 2 x → A ( k ) =! k ! t = 0 " dx e

_ikx

e_

$

2 x 2 #% % & A ( k )! e " k 2 4 # 2

Wavepacket 13 A ( k ) is the projection of ϕ( x ) on the momentum eigenstates e ikx , and thus represents the amplitude of each momentum eigenstate in the superposition. We need the contribution of a wide spread of momentum states to localize a particle. If we have the contribution of just a few, the location of the particle is uncertain

"( x ) # e $ % 2 x 2 A ( k )! e " k 2 4 # 2

Wavepacket 14 ! "( x ) 2

e

$ 2 % 2 x 2 x → ! 2 " x

A ( k )

2

" e

2 k

2 4 $ 2 ! 2 " k k → To define "uncertainty" in position or momentum, we must consider probability, not wave function.

Wavepacket 16 HEISENBERG UNCERTAINTY PRINCIPLE

" p " x = !" k " x = !#

" p " x =

! "( x ) 2

e

$ 2 % 2 x 2 x → ! 2 " x

A ( k )

2

! e

" 2 k 2 4 # 2 ! 2 " k k →

Wavepacket 17 Next, we ll ask how a general wave packet propagates, And deal with the particular example of the Gaussian wavepacket. In short, we simply attach the exp(-iE(k)t/hbar) factor to each eigenstate and let time run. Difference to non-dispersive equation: not all waves propagate with same velocity. Packet does not stay intact! Need to invoke group velocity to follow the progress of the bump.

Wavepacket 19 Superposition of eigenstates How does it develop in time?

-0. 0

1 ! -2 (^) x 0 2 "( x , t ) = dk A ( k ) e Localized particle ikx

e

i $ kt

#% % &

k

! k

2 m

The quantity dω/d k may (does!) vary depending on the k value at which you choose to evaluate it. So it must be evaluated at a particular value k 0 that represents the center of the packet. The next few pages spend time deriving the basic result. The derivation is not so important. The result is important:

v

group

d!

dk

k 0

! k

0

m

Wavepacket 20 Particular example of the Gaussian wavepacket.