Richard Feynman:

Electron waves are probability waves in the ocean of uncertainty.

Last Time We Solved some of the Problems with Classical Physics

•Discrete Spectra? Bohr Model but not complete. •Blackbody Radiation? Plank Quantized Energy of Atoms: •Photoelectric Effect? Einstein Quantized energy states of light photons: •Compton Effect? Light is a particle with momentum! •Stability of Atoms? deBroglie waves!

nE nhf=

E hf=

ehp

λ =

E hf hpc c λ

= = =

If photons can be particles, then why can’t electrons be waves?

ehp

λ =

Electrons are STANDING WAVES in atomic orbitals.

346.626 10h x J s−= ⋅

112.4 10e x mλ −=

1924: De Broglie Waves

Electron Diffraction

If electron were hard bullets, there would be no interference pattern.

In reality, electrons do show an interference pattern, like light waves.

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

Particle Wave Duality

Interference pattern builds one electron at a time. Electrons act like waves going through the slits but arrive at the detector like a particle.

Results of a double-slit-experiment performed by Dr. Tonomura showing the build-up of an interference pattern of single electrons. Numbers of electrons are 11 (a), 200 (b), 6000 (c), 40000 (d), 140000 (e).

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

Review on Waves: Superposition of Sinusoidal Waves

• Case 1: Identical, same direction, with phase difference (Interference)

• Case 2: Identical, opposite direction (standing waves)

• Case 3: Slightly different frequencies (Beats)

Superposition of Traveling 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 Interference y = 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

∆

∆ = =

∆ = + =

Standing Waves: Bound Systems Superposition of two identical waves moving in opposite directions.

1 2 sin ( - ) sin ( ) y A kx t y A kx tω ω= = +

(2 sin )cos y A kx tω=

•There is no kx – wt term, and therefore it is not a traveling wave! •Every element in the medium oscillates in simple harmonic motion with the same frequency, w: coswt •The amplitude of the simple harmonic motion depends on the location of the element within the medium: (2Asinkx)

Standing Waves on a String Harmonics

The possible frequency and energy states of an electron in an atomic orbit or of a wave on a string are quantized.

2vf nl

=

Strings & Atoms are Quantized

34

, n= 0,1,2,3,...

6.626 10nE nhf

h x Js−

=

=

2 2

W 4 m

Irπ

℘ =

The intensity of a wave, the power per unit area, is the rate at which energy is being transported by the wave through a unit area A perpendicular to the direction of travel of the wave and is proportional to the amplitude of the sound wave squared!

2max

1 ( )2

I v sρ ω=

2max

1 ( )2

E Av st

ρ ω∆℘= =

∆

Intensity of Sound Waves

Power Transmitted by Waves on Strings

Power Transmitted on a String is proportional to the square of the wave amplitude.

:

2 212

E A vt

µω∆℘= =

∆

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 c BSμ μ c μ

= = = = I ∝ 2maxE

Intensity is proportional to the square of the amplitude!

Double Slit Intensity Distribution Resultant Electric 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= ∆

Light Intensity for the Double Slit

2 2max max

sin cos cosI I Iπd θ πd yλ λL

= ≈

2max cos ( / 2)I I φ= ∆

2Phase Difference: rπφλ

∆ = ∆

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

Intensity is proportional to the square of the amplitude!

Intensity is proportional to the square of the amplitude! Probability is proportional to Intensity therefore: Probability is proportional to Amplitude Squared!!!

I ∝ 2A

In Sum…

Next…

Spontaneous Emission Transition Probabilities Fermi’s Golden Rule

Transition probabilities correspond to the intensity of light emission.

Double Slit: Light

2I E∝

Wave Function : E Amplitude Intensity: 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

Connecting the Wave and Photon Views The 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:

Probability Concepts

100tot A B CP P P P= + + =A tot AN N P=

Prob(AorB)Prob(AandB)

A B

A B

P PP P

= +=

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

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

Prob(AorB) A BP P= +

A tot AN N P=1 130( ) 106 6

N = + =

A. 12 B. 10 C. 8 D. 6 E. 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 Views The 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 > C D. 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 !)

Probability Density We 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 Electrons shows 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 function psi is “a wave-like function” that oscillates between positive and negative values that can be used to make probabilistic predictions 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?

2Prob(in x at x) (x) xδ δ= Ψ

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

2Prob(in x at x) (x) xδ δ= Ψ

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

A. x = 0 B. x = xA C. x = xB D. x = xC

2Prob(in x at x) (x) xδ δ= Ψ

A. x = 0 B. x = xA C. x = xB D. x = xC

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

Do Workbook 39.3

2Prob(in x at x) (x) xδ δ= Ψ

Normalization • 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 xR is 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 xR is:

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

Do Workbook 40.4

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 2Probability = ( , ) = (possibility)x tΨ

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 Principle

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

∆∆ =

x p h∆ ∆ >

Electron in a Box

~ ~ / /p p h h Lλ∆ =

~ /x p L h L h∆ ∆ ⋅ >

~ x L∆

If you squeeze the walls to decrease ∆x, you increase ∆p!

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

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

Electron in a Box

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!

Heisenberg Uncertainty Trying 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 the Uncertainty 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.

/ 4x p h π∆ ∆ >/ 4E t h π∆ ∆ >

The Heisenberg Uncertainty Principle

• 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 Principle

• 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 of an electron

Heisenberg Uncertainty Energy and Time

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

/ 4E t h π∆ ∆ >

Quantum Foam Virtual 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 Nothing Matter-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 Cosmology BIG 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!

In Sum…..

Where do the Wave Functions come from???

2 2

22d ψ Uψ Eψ

m dx− + =

Solutions to the time-independent Schrödinger equation:

( )−=

2

2 2

2m U Ed ψ ψdx

OR