Richard Feynman:

Electron waves are probability waves in the ocean of uncertainty.

Long Ago……

De Broglie Wavelength

346.626 10h x J s−= ⋅

hmv

λ =

112.4 10e x mλ −=

2 , 1, 2,3.....nL nn

λ = =Waves:

De Broglie: hp

λ =

Combine:

221

2 2pE mvm

→ = =

2 2

28nh nEmL

= Energy is Quantized!

Electron Waves leads to Quantum Theory

Assume electrons are waves. A wave function is defined that contains all the information about

the electron: position, momentum & energy. The wave density function is the square of the wave function.

The probability of finding an electron in a particular state is given by the square of the wave function.All we can know are probabilities.

Quantum Theory

2 2P ro b ab ility = ( , ) = (p o ss ib ility)x tΨ

Where do the Wave Functions come from???

2 2

22d ψ Uψ Eψ

m dx− + =

h

Solutions to the time-independent Schrödinger equation:

( )−=

h

2

2 2

2m U Ed ψ ψdx

OR

Double Slit is VERY IMPORTANT because it is evidence of waves. Only waves interfere like this.

Intensity

Superposition of Sinusoidal Waves• Assume two waves are traveling in the same direction,

with the same frequency, wavelength and amplitude• The waves differ in phase

• y1 = A sin (kx - ωt)• y2 = A sin (kx - ωt + φ)• y = y1+y2

= 2A cos (φ/2) sin (kx - ωt + φ/2)

Resultant Amplitude Depends on phase: Spatial Interference Term

1-D Wave Interferencey = y1+y2 = 2A cos (φ/2) sin (kx - ωt + φ/2)

Constructive Destructive Interference

0φ = φφ π=

( )

Resultant Amplitude: 2 cos2

Constructive Interference (Even ): 2 , 0,1, 2,3...Destructive Interference Odd : (2 1) , 0,1,2,3...

A

n nn n

φ

π φ ππ φ π

Δ⎛ ⎞⎜ ⎟⎝ ⎠

Δ = =

Δ = + =

Double Slit Interference

bright ( 0 1 2 ), ,λLy m md

= = ± ± K

dark1 ( 0 1 2 )2

, ,λLy m md⎛ ⎞= + = ± ±⎜ ⎟⎝ ⎠

K

Bright fringe: sin , 0,1, 2,3,...r d m mθ λΔ = = =

1Dark fringe : sin ( ) 0,1, 2,3,...2

r d m mθ λΔ = = + =

Constructive: r mλΔ =

1Destructive: ( )2

r m λΔ = +

2Phase Difference: rπφλ

Δ = Δ

small angle approx: sin tan /y Lθ θ≈ =

Light Waves: Intensity

E = Emax cos (kx – ωt) B = Bmax cos (kx – ωt)

=max

max

E cB

2 2max max max max

av 2 2 2I

o o o

E B E cBSμ μ c μ

= = = = I ∝ 2maxE

Intensity DistributionResultant Field

• The magnitude of the resultant electric field comes from the superposition principle– EP = E1+ E2 = Eo[sin ωt + sin (ωt + φ)]

• This can also be expressed as

– EP has the same frequency as the light at the slits– The amplitude at P is given by 2Eo cos (φ / 2)

• Intensity is proportional to the square of the amplitude:

2 cos sin2 2P oφ φE E ωt⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

I ∝ 2A 2max cos ( / 2)I I φ= Δ

Light Intensity

2 2max max

sin cos cosI I Iπd θ πd yλ λL

⎛ ⎞ ⎛ ⎞= ≈⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

2max cos ( / 2)I I φ= Δ

2Phase Difference: rπφλ

Δ = Δ

Bright fringe: sin , 0,1, 2,3,...r d m mθ λΔ = = =

Double Slit: Light

2I E∝

Wave Function : E AmplitudeIntensity: Amplitude squared

The intensity at a point on the screen is proportional to the square of the resultant electric field magnitude at that point.

Intensity is the REALITY for light waves.

What about for light particles?PROBABILITY ~ Intensity

Probability ConceptsProbability Concepts

The probability that a dart lands in A, B or C is 100%:

tot A B CP P P P= + +

Suppose you roll a die 30 times. What is the expected numbers of 1’s and 6’s?

A. 12B. 10C. 8D. 6E. 4

A. 12B. 10C. 8D. 6E. 4

Suppose you roll a die 30 times. What is the expected numbers of 1’s and 6’s?

Do Workbook 40.2 #1

Connecting the Wave and Photon ViewsConnecting the Wave and Photon ViewsThe intensity of the light wave is correlated with the probability of detecting photons. That is, photons are more likely to be detected at those points where the wave intensity is high and less likely to be detected at those points where the wave intensity is low. The probability of detecting a photon at a particular point is directly proportional to the square of the light-wave amplitude function at that point:

The figure shows the detection of photons in an optical experiment. Rank in order, from largest to smallest, the square of the amplitude function of the electromagnetic wave at positions A, B, C, and D. (The probability of detecting a photon !)

A. D > C > B > A B. A > B > C > D C. A > B = D > C D. C > B = D > A

A. D > C > B > A B. A > B > C > D C. A > B = D > CD. C > B = D > A

The figure shows the detection of photons in an optical experiment. Rank in order, from largest to smallest, the square of the amplitude function of the electromagnetic wave at positions A, B, C, and D. (The probability of detecting a photon !)

Spontaneous EmissionTransition ProbabilitiesFermi’s Golden Rule

Transition probabilities correspond to the intensity of light emission.

Probability DensityProbability DensityWe can define the probability density P(x) such that

In one dimension, probability density has SI units of m–1. Thus the probability density multiplied by a length yields a dimensionless probability.

NOTE: P(x) itself is not a probability. You must multiply the probability density by a length to find an actual probability.

the photon probability density is directly proportional to the square of the light-wave amplitude:

Do Workbook 40.2 #2

The probability of detecting a photon within a narrow region of width δx at position x is directly proportional to the square of the light wave amplitude function at that point.

In Sum: Light

2Prob(in x at x) A(x) xδ δ∝

Wave model: Interference pattern is in terms of wave intensity

Photon model: Interference in terms of probability

Probability Density Function: 2( ) A(x)P x ∝

The probability density function is independent of the width, δx , and depends only on x. SI units are m-1.

Double Slit for Electronsshows Wave Interference

Interference pattern builds one electron at a time.

Electrons act like waves going through the slits but arrive at the detector like a particle.

There is no electron wave so we assume an analogy to the electric wave and call it the wave function, psi, :

The intensity at a point on the screen is proportional to the square of the wave function at that point.

( )xΨ

The Probability Density Function is the “Reality”!

A light analogy…..

2( ) ( )P x x= Ψ

Double Slit: Electrons

WARNING!

There is no ‘thing’ waving! The wave funtion psi is “a wave-like function” that oscillates between positive and negative values that can be used to make probalisiticpredictions about atomic particles.

The probability of detecting an electron within a narrow region of width δx at position x is directly proportional to the square of the wave function at that point:

Probability: Electrons

2Prob(in x at x) (x) xδ δ= Ψ

Probability Density Function:

2( ) (x)P x = ΨThe probability density function is independent of the width, δx , and depends only on x. SI units are m-1. Note: The above is an equality, not a proportionality as with photons. This is because we are defining psi this way. Also note, P(x) is unique but psi in not since –psi is also a solution.

Where is the electron most likely to be found? Least likely?

At what value of x is the electron probability density a maximum?

This is the wave function of a neutron. At what value of x is the neutron most likely to be found?

A. x = 0B. x = xAC. x = xBD. x = xC

A. x = 0B. x = xAC. x = xBD. x = xC

This is the wave function of a neutron. At what value of x is the neutron most likely to be found?

In Sum…..

Where do the Wave Functions come from???

2 2

22d ψ Uψ Eψ

m dx− + =

h

Solutions to the time-independent Schrödinger equation:

( )−=

h

2

2 2

2m U Ed ψ ψdx

OR

NormalizationNormalization• A photon or electron has to land somewhere on the

detector after passing through an experimental apparatus. • Consequently, the probability that it will be detected at

some position is 100%.• The statement that the photon or electron has to land

somewhere on the x-axis is expressed mathematically as

• Any wave function must satisfy this normalization condition.

The Probability that an electron lands somewhere between xL and xRis the sum of all the probabilities that an electron lands in a narrow strip i at position xi:

2

1 1( ) = ( )

N N

i ii i

P x x x xδ ψ δ= =∑ ∑

2Prob( ) ( ) ( )R R

L L

x x

L Rx x

x x x P x dx x dxψ≤ ≤ = =∫ ∫

2 ( ) ( ) 1P x dx x dxψ∞ ∞

−∞ −∞

= =∫ ∫

Normalization

The Probability that an electron lands somewhere between xL and xRis:

Total Area must equal 1.

The value of the constant a is

A. a = 0.5 mm–1/2.B. a = 1.0 mm–1/2.C. a = 2.0 mm–1/2.D. a = 1.0 mm–1.E. a = 2.0 mm–1.

A. a = 0.5 mm–1/2.B. a = 1.0 mm–1/2.C. a = 2.0 mm–1/2.D. a = 1.0 mm–1.E. a = 2.0 mm–1.

The value of the constant a is

De Broglie Wavelength

346.626 10h x J s−= ⋅

hmv

λ =

112.4 10e x mλ −=

Wave Packet: Making Particles

out of Waves

Superposition of waves to make a defined wave packet. The more waves used of different frequencies, the more localized.

However, the more frequencies used, the less the momentum is known.

/p hf c=hpλ

= c fλ=

You make a wave packet by wave superposition and interference.The more waves you use, the more defined your packet and the more defined the position of the particle. However, the more waves you use of different frequencies (energy or momentum) to specify the position, the less you specify the momentum!

Heisenberg Uncertainty Principlex p hΔ Δ >

Heisenberg Microscope

Small wavelength (gamma) of light must be used to find the electron because it is too small. But small wavelength means high energy. That energy is transferred to the electron in an unpredictable way and the motion (momentum) becomes uncertain. If you use long wavelength light (infrared), the motion is not as disturbed but the position is uncertain because the wavelength is too long to see the electron. This results in the Uncertainty Principle.

~ /

xp h

λλ

ΔΔ =

x p hΔ Δ >

A wave packet in a square well (an electron in a box) changing with time.

Electron in a Box

Electron in a Box

~ ~ / /p p h h LλΔ =

~ /x p L h L hΔ Δ ⋅ >

~ x LΔ

If you squeeze the walls to decrease Δx, you increase Δp!

The possible wavelengths for an electron in a box of length L.

Improved technology will not save us from Quantum Uncertainty!Quantum Uncertainty comes from the particle-wave nature of matter

and the mathematics (wave functions)

used to describe them!

EXAMPLE 40.4 Creating radioEXAMPLE 40.4 Creating radio--frequency pulsesfrequency pulses

EXAMPLE 40.4 Creating radioEXAMPLE 40.4 Creating radio--frequency pulsesfrequency pulses

Wave PacketsWave PacketsSuppose a single nonrepeating wave packet of duration Δtis created by the superposition of many waves that span a range of frequencies Δf. Fourier analysis shows that for any wave packet

We have not given a precise definition of Δt and Δf for a general wave packet.The quantity Δt is “about how long the wave packet lasts,”while Δf is “about the range of frequencies needing to be superimposed to produce this wave packet.”

/ 4x p h πΔ Δ >/ 4E t h πΔ Δ >

The Heisenberg Uncertainty The Heisenberg Uncertainty PrinciplePrinciple

• The quantity Δx is the length or spatial extent of a wave packet.

•Δpx is a small range of momenta corresponding to the small range of frequencies within the wave packet.

• Any matter wave must obey the condition

This statement about the relationship between the position and momentum of a particle was proposed by Heisenberg in 1926. Physicists often just call it the uncertainty principle.

The Heisenberg Uncertainty The Heisenberg Uncertainty PrinciplePrinciple

• If we want to know where a particle is located, we measure its position x with uncertainty Δx.

• If we want to know how fast the particle is going, we need to measure its velocity vx or, equivalently, its momentum px. This measurement also has some uncertainty Δpx.

• You cannot measure both x and px simultaneously with arbitrarily good precision.

• Any measurements you make are limited by the condition that ΔxΔpx ≥ h/2.

• Our knowledge about a particle is inherently uncertain.

EXAMPLE 40.6 The uncertainty EXAMPLE 40.6 The uncertainty of an electronof an electron

Heisenberg UncertaintyTrying to see what slit an electron goes

through destroys the interference pattern.

Electrons act like waves going through the slits but arrive at the detector like a particle.

Which Hole Did the Electron Go Through?

Conclusions:• Trying to detect the electron, destroys the interference pattern.• The electron and apparatus are in a quantum superposition of states.• There is no objective reality.

If you make a very dim beam of electrons you can essentially send one electron at a time. If you try to set up a way to detect which hole it goes through you destroy the wave interference pattern.

Feynman’s version of theUncertainty Principle

It is impossible to design an apparatus to determine which hole the electron passes through, that will not at the same time disturb the electrons enough to destroy the interference pattern.

Heisenberg UncertaintyEnergy and Time

Particles can be created with energy ΔE can that live for a time Δt. They are called virtual particles but they can become real.

/4E t h πΔ Δ >

Quantum FoamVirtual Particles are constantly popping in and out of the

quantum vacuum, making a Quantum Foam. In QED, virtual particles are responsible for communicating forces.

Making Matter out of NothingMatter-Antimatter Pair Production

Antimatter: Same mass, different charge.

When matter and antimatter collide they annihilate into pure energy: light.

Quantum Fluctuations

Casimir Effect: The Zero Point Energy of the Quantum Vacuum

Quantum CosmologyBIG BANG!

The Universe Tunneled in from Nothing

In Sum: The Quantum Vacuum

The Universe came from nothing and is being accelerated by nothing!

Due to Quantum Uncertainty, the nothingness of empty space is actually full of energy!

Virtual Particles pop in and out of the Quantum Vacuum in matter-antimatter pairs. If they have enough energy they can become real!