Lecture_09: Outline

Matter Waves

Wave packets

Phase and group velocities

Uncertainty relations

Problems

Wave packets

“Constructing” Particles From Waves:

• Particles are localized in space• Ideal wave is unlocalized in space.• It is possible to built “localized” entity

from waves– Such an entity must have a non-zero

amplitude only within a small region of space

• Such wave is called “wave packet”

Wave packets

Wave addition:

)cos(),( 111 txkAtx )cos(),( 222 txkAtx

),(),(),( 21 trtrtr

)22

cos()22

cos(2),( 2121 txkk

txk

Atx

21 kkk 21

Wave packets

Sum of two waves:

Wave packets

Sum of many waves:

• Multiple waves are superimposed so that one of its crests is at x = 0

• The result is that all the waves add constructively at x = 0

• There is destructive interference at every point except x = 0

• The small region of constructive interference is called a wave packet

Wave packets

Sum of many waves:

Wave packets

General Case:

0 0

)cos(),(),( dkdtkxkAtx

Amplitudes A(k, ω) determine how much each wave contributes to the packet and thus they determine the shape of the wave packet

Fixed moment of time t0:

0

00 )cos()(),( dktkxkAtx

Wave packets

1. A(k) is a very strong spike at a given k0, and zero everywhere else

• only one wave with k = k0 (λ = λ0) contributes; thus one knows momentum exactly, and the wavefunction is a traveling wave – particle is delocalized

2. A(k) is the same for all k • No distinctions for momentums, so

particle’s position is well defined - the wavefunction is a “spike”, representing a “very localized” particle

3. A(k) is shaped as a bell-curve• Gives a wave packet – “partially”

localized particle

Phase and group velocities

)22

cos()22

cos(2),( 2121 txkk

txk

Atx

dkkk 12

d 12

2,1kdk

2,1 d

)cos()22

cos(2),( 11 txktd

xdk

Atx

Phase and group velocities

Wave velocity:

EM wave:k

c

)cos()22

cos(2),( 11 txktd

xdk

Atx

Two velocities:

kvph

Phase

velocity dk

dvg

Group

velocity

)cos(),( max tkxEtxE

Wave packets

Sum of two waves:

Phase and group velocities

Group velocity:

dk

dvg

EhE 22

phpk 22 dp

dEvg

m

pE

2

2

Free particle: vm

pvg

pcE Photon: cvg

Uncertainty relations

•We want to know Coordinate and Momentum of a particle at time t = 0–If we know the forces acting upon the particle than, according to classical physics, we know everything about a particle at any moment in the future

–The answer in quantum mechanics is different and is presented by the Heisenberg’s Uncertainty Principle:

An experiment cannot simultaneously determine a component of the momentum of a particle (e.g., px) and the exact value of the corresponding coordinate, x.

The best one can do is2

))((

xpx

Uncertainty relations

1. The limitations imposed by the uncertainty principle have nothing to do with quality of the experimental equipment

2. The uncertainty principle does not say that one cannot determine the position or the momentum exactly– It refers to exact knowledge about both: e.g. if Δx = 0,

then Δ px is infinity, and vice versa

3. The uncertainty principle is confirmed by experiment, and is a direct consequence of the de Broglie’s hypothesis

Problems

Velocity of a 100 g bullet is measured to be 1000 m/s with an accuracy of 0.01%. What is the uncertainty of its position?

Problems

Kinetic energy of an electron is measured to be 4.9 eV with an accuracy of 0.01%. What is the uncertainty of its position?

Uncertainty relations

Heisenberg-Bohr thought experiment:

• It shows that a measurement itself introduces the uncertainty

• When we “look” at an object we see it through the photons that are detected by the microscope– These are the photons that are

scattered in the angle < 2θ

– Momentum of electron is changed

Uncertainty relations

Heisenberg-Bohr thought experiment:sin2max

phph pp

sin2 phphelectron ppp

Trying to determine the electron position, we introduce the uncertainty of the momentum

sin2

sin2h

pp

hp

phelectron

ph

Uncertainty relations

Heisenberg-Bohr thought experiment:

Image is not the point, but the diffraction pattern

The uncertainty of the position is approximately the width of the central maximum

sinx

hh

xp 2sin

sin2

Uncertainty relations

Uncertainty energy-time:

2))((

tE

m

pE

2

2

Free particle: pvpm

pp

dp

dEE tvx

2))(())(

1())((

2

tEtvE

vxp

Energy conservation law can be violated but for the short time interval!

Problems

How long is the time interval during which you disappear but nobody notices?

Problems

What is the linewidth for the emission from the electron level with 5 ns lifetime?