Matter-Waves and Electron Diffraction

“The most incomprehensible thing about the world is that it is comprehensible.”

– Albert Einstein

Day 18, 3/31: Questions? Electron DiffractionWave packets and uncertainty

Next Week:Schrodinger equation

The Wave FunctionExam 2 (Thursday)

2

Recently: 1. Single-photon experiments2. Complementarity and wave-particle duality3. Matter-Wave Interference

Today: 1. Electron diffraction and matter waves2. Wave packets and uncertainty

Exam II is next Thursday 3/31, in class.HW09 will be due Wednesday (3/30) by 5pm in my mailbox.HW09 solutions posted at 5pm on Wednesday

Double-Slit Experiment

• Pass a beam of electrons through a double-slit apparatus.

• Individual electrons are detected as points on the screen.

• Over time, a fringe pattern of dark and light bands appears.

Double-Slit Experiment with Single Electrons (1989)

A. Tonomura, J. Endo, T. Matsuda, T. Kawasaki and H. Ezawa, "Demonstration of Single-Electron Buildup of an Interference Pattern,“Amer. J. Phys. 57, 117 (1989).

http://www.hitachi.com/rd/research/em/doubleslit.html

cos( )E A kx t

Double-Slit Experiment with Light

Recall solution for Electric field (E)from the wave equation:

cos( )E A kx

exp( ) cos( ) sin( )ikx kx i kx

cos( ) Re[ cos( ) sin( )] Re[ exp( )]E A kx A kx iA kx A ikx

exp( )E A ikx

Let’s forget about the time-dependence for now:

Euler’s Formula says:

For convenience, write:

exp( )A ikx

1 2Total

1 1 1exp( )A ikx 2 2 2exp( )A ikx

1 1 2 2exp( ) exp( )A ikx A ikx

Double-Slit Experiment with Electrons

Before Slits

After Slits

&

1 1 2 2exp( ) exp( )A ikx A ikx

Double-Slit Experiment

2 *[ ] | ( ) ( ) ( )P x x x x

1 1 2 2 1 1 2 2exp( ) exp( ) exp( ) exp( )A ikx A ikx A ikx A ikx

1 1 1 1 2 2 2 2exp( ) exp( ) exp( ) exp( )A ikx A ikx A ikx A ikx

1 1 2 2 2 2 1 1exp( ) exp( ) exp( ) exp( )A ikx A ikx A ikx A ikx

2 21 2 1 2 2 1 2 1exp( ( )) exp( ( ))A A A A ik x x ik x x

2 21 2 1 2 2 12 cos ( )A A A A k x x

Double-Slit Experiment2 *[ ] | ( ) ( ) ( )P x x x x

2 2 21 2 1 2 2 1( ) 2 cos ( )x A A A A k x x

2 2 21 2 1 2 2 1

2( ) 2 cos ( )x A A A A x x

2 21 1 1 1 1 1( ) exp( ) exp( )x A ikx A ikx A

2 22 2 2 2 2 2( ) exp( ) exp( )x A ikx A ikx A

How do we connect this with what we observe?

Double-Slit Experiment2 2 2

1 2 1 2 2 12[ ] ( ) 2 cos ( )P x x A A A A x x

21 2

2 INTERFERENCE!

• In quantum mechanics, we add the individual amplitudes andsquare the sum to get the total probability

• In classical physics, we added the individual probabilities to getthe total probability.

2 2 212 1 2 1 2 1 2[ ] [ ] [ ]P x P x P x

21

22

21 2

2 21 2

Double-Slit Experiment

1

2D

sinx D D m

1 2

sin

L

H

D

1x2x

2 1x x x m 0,1,2,...m

Determining the spacing between the slits:

sinH L L

mD

m LHD

Spacing between bright fringes

L

H

D

1x2x

m LHD

Distance to first maximum for visible light:

Let , ,1m 500 nm 7

4

5 10 0.001 5 10

radD

45 10 mD

If , then 3 mL 3 mmH L

L

H

D

1x2x

Double-Slit Experiment2 *[ ] ( ) ( ) ( )P x x x x

2 2 21 2 1 2 2 1

2( ) 2 cos ( )x A A A A x x

For a specific λ, describes where we findthe interference maxima and minimaof the fringe pattern.

2( )x

What is the wavelength of an electron?

As a doctoral student (1923), Louis de Broglie proposedthat electrons might also behave like waves.

• Light, often thought of as a wave, had been shown to have particle-like properties.

• For photons, we know how to relate momentum to wavelength:

Matter Waves

hcE hf

E p c

hp

From the photoelectric effect:

From special relativity:

Combined:

de Broglie wavelength:

hp

Confirmed experimentally byDavisson and Germer (1924).

Matter Waves

de Broglie wavelength: hp

In order to observe electron interference, it would be best to perform a double-slit experiment with:

A) Lower energy electron beam.B) Higher energy electron beam.C) It doesn’t make any difference.

Matter Waves

m LHD

de Broglie wavelength: hp

In order to observe electron interference, it would be best to perform a double-slit experiment with:

A) Lower energy electron beam.B) Higher energy electron beam.C) It doesn’t make any difference.

Matter Waves

m LHD

Lowering the energy will increase the wavelength of the electron.

Can typically make electron beams with energies from 25 – 1000 eV.

de Broglie wavelength: hp

For an electron beam of 25 eV, we expect Θ (the angle betweenthe center and the first maximum) to be:

A) Θ << 1B) Θ < 1C) Θ > 1D) Θ >> 1

45 10 mD

D

Use

and remember:

&

Matter Waves

2

2pEm

m LHD

346.6 10 J sh

de Broglie wavelength: hp

For an electron beam of 25 eV:

Matter Waves

34

131 19 2

(6.6 10 J s)

[2 (9.1 10 kg) (25 1.6 10 / )]

x

x eV x J eV

2

22pE p mEm

m LHD

7 6(3 )(4.9 10 ) 1.5 10 1.5H L m m m

de Broglie wavelength: hp

346.6 10 J sh For an electron beam of 25 eV:

for 45 10 mD

3410

24

6.6 10 2.4 10 2.7 10

h mp

74.9 10D

Too small to easily see!

Matter Waves

m LHD

de Broglie wavelength: hp

For an electron beam of 25 eV, how can we make the diffraction pattern more visible?

A) Make D much smaller.B) Decrease energy of electron beam.C) Make D much bigger.D) A & BE) B & C

Matter Waves

m LHD

de Broglie wavelength: hp 346.6 10 J sh

For an electron beam of 25 eV, how can we make the diffraction pattern more visible?

A) Make D much smaller.B) Decrease energy of electron beam.C) Make D much bigger.D) A & BE) B & C

25 eV is a lower bound on the energy of a decent electron beam.

Decreasing the distance between the slits will increase Θ.

Matter Waves

Slits are 83 nm wide andspaced 420 nm apart

S. Frabboni, G. C. Gazzadi, and G. Pozzi, Nanofabrication and the realization of Feynman’s two-slit experiment, Applied Physics Letters 93, 073108 (2008).

LHD

If we were to use protons instead of electrons,but traveling at the same speed, what happens to the distance to the first

maximum, H?A) IncreasesB) Stays the sameC) Decreases

L

H

D

1x2x

2

2pEm

hp

Which slit did this electron go through?

A) LeftB) RightC) BothD) NeitherE) Either left or right, we just can’t know which one.

Each electron passes through both slits, interferes with itself, then becomes localized when detected.

• For a low-intensity beam, can have as little as one electron in the apparatus at a time.• Each electron must interfere with itself• Each electron is “aware” of both paths.• Asking which slit the electron went through disrupts the fringe pattern.

Screen

Describe photon’s EM wave spread out in space.

High Amplitude

Low Amplitude

Light Waves

2probability of photon detection (amplitude of EM wave)

Light Waves

x

E-field x-section at screen

Photons

ρ(x) =probability density

2probability of photon detection (amplitude of EM wave)

32

Electron double slit experiment. Display=Magnitude of wave function

Large Magnitude (|Ψ|)= probability of detecting electron here is high

Small Magnitude (|Ψ|)= probability of detecting electron here is low

Follow same prescription for matter waves:

• Describe massive particles with wave functions:

• Interpret (wave amplitude)2 as a probability density:

( )x

2 *( ) ( ) ( ) ( )x x x x

What are these waves?EM Waves (light/photons)

• Amplitude = electric field• tells you the probability

of detecting a photon.• Maxwell’s Equations:

• Solutions are sine/cosine waves:

2EE

2

2

22

2 1tE

cxE

( , ) sin( )E x t A kx t

( , ) cos( )E x t A kx t

Matter Waves (electrons/etc)

• Amplitude = matter field• tells you the probability

of detecting a particle.• Schrödinger Equation:

• Solutions are complex sine/cosine waves:

2

ti

xm

2

22

2

( , ) expx t A i kx t cos( ) sin( )A kx t i kx t

Matter Waves

xL-L

• Describe a particle with a wave function.

• Wave does not describe the path of the particle.

• Wave function contains information about the probability to find a particle at x, y, z & t.

( , )x t

( , 0)x t ( , , , )x y z t( , , , )r t

In general:

Simplified:

&( , )x t ( )x2 *( ) ( ) ( ) ( )x x x x

Matter Waves

xL-L

L-L x

( , 0)x t

2( , 0) ( , 0)x t x t

Probability of finding particlein the interval dx is ( )x dx

Wavefunction

Probability Density

2[ ] ( ) ( ) 1P x x dx x dx

Normalized wave functions

36

Below is a wave function for a neutron. At what value of xis the neutron most likely to be found?

A) XA B) XB C) XC D) There is no one most likely

place

Matter Waves

L

a b c ddx

x

( )x ( ) from to xx x L x LL

0 elsewhere

How do the probabilities of finding the electron near (within dx) of a,b,c, and d compare?

A) d > c > b > aB) a = b = c = dC) d > b > a > cD) a > d > b > c

An electron is described by the following wave function:

An electron is described by the following wave function:

L

a bdx

x

( ) from to xx x L x LL

0 elsewhere

( )x

How do the probabilities of finding the electron near (within dx) of a,b,c, and d compare?

A) d > c > b > aB) a = b = c = dC) d > b > a > cD) a > d > b > c

c d

( )x

2 22

2( ) ( ) xx xL

Plane WavesPlane waves (sines, cosines, complex exponentials)

extend forever in space:

1 1 1( , ) expx t i k x t

4 4 4( , ) expx t i k x t

3 3 3( , ) expx t i k x t

2 2 2( , ) expx t i k x t

Different k’s correspond to different energies, since2 2 2 2

22

12 2 2 2

p h kE mvm m m

If and are solutions to a wave equation,

then so is1( , )x t 2 ( , )x t

1 2( , ) ( , ) ( , )x t x t x t

Superposition (linear combination) of two waves

We can construct a “wave packet” by combining many

plane waves of different energies (different k’s).

Superposition

fourier_en.jnlp

41

Superposition ( , ) expn n n

n

x t A i k x t

Plane Waves vs. Wave Packets ( , ) expx t A i kx t

( , ) expn n nn

x t A i k x t

Which one looks more like a particle?

Plane Waves vs. Wave Packets ( , ) expx t A i kx t

( , ) expn n nn

x t A i k x t

For which type of wave are the position (x) and momentum (p) most well-defined?

A) x most well-defined for plane wave, p most well-defined for wave packet.B) p most well-defined for plane wave, x most well-defined for wave packet.C) p most well-defined for plane wave, x equally well-defined for both.D) x most well-defined for wave packet, p equally well-defined for both.E) p and x are equally well-defined for both.

Uncertainty Principle

Δx

Δxsmall Δp – only one wavelength

Δxmedium Δp – wave packet made of several waves

large Δp – wave packet made of lots of waves

Uncertainty Principle

In math:

In words:The position and momentum of a particlecannot both be determined with complete precision. The more precisely one is determined, the less precisely the other is determined.

What do (uncertainty in position) and(uncertainty in momentum) mean?

2x p

xp

• Photons are scattered by localized particles.

• Due to the lens’ resolving power:

• Due to the size of the lens:

Uncertainty Principle

2xhx p

x

2xhp

A RealistInterpretation:

•Measurements are performed on an ensemble of similarly prepared systems.• Distributions of position and momentum values are obtained.• Uncertainties in position and momentum are defined in terms of the standard deviation.

Uncertainty Principle

A StatisticalInterpretation:

2

1

1 N

x ii

xN

2x p

• Wave packets are constructed from a series of plane waves.• The more spatially localized the wave packet, the less uncertainty

in position.• With less uncertainty in position comes a greater uncertainty in momentum.

Uncertainty Principle

A WaveInterpretation:

Matter Waves (Summary)• Electrons and other particles have wave properties

(interference)• When not being observed, electrons are spread out in space

(delocalized waves)• When being observed, electrons are found in one place

(localized particles)• Particles are described by wave functions:

(probabilistic, not deterministic)• Physically, what we measure is

(probability density for finding a particle in a particular place at a particular time)

• Simultaneous measurements of x & p are constrained by the

Uncertainty Principle:

( , )x t

2( , ) ( , )x t x t

2x p