TodayToday

• Subatomic physics

• Photons, quanta & quantum physics

� Photoelectric effect

� Photon momentum

� Probability waves

� Matter waves

� Uncertainty Principle

� Model of atom

Photons & QuantaPhotons & Quanta

• Definition of quantized

� Something quantized comes in discrete

amounts, and larger quantities of that

something arise when N of the its quanta are

added together

• Example

� Physical U.S. currency is quantized in

pennies

Photons and QuantaPhotons and Quanta

• Einstein proposed that light is quantized & exists in discrete amounts called photons

� This is explicitly contrary to the wave-like point of view!

• Light of frequency “f” is made up of photons of energy

� E = hf

• h = Plancks’ constant = 6.63 x 10-34 J⋅⋅⋅⋅s� This is a VIC—very important constant

• The minimum amount of energy a light wave of frequency f can have is hf

� In general, Etot = Nγhf

� We can never have Etot equal to, say, 3.8hf

Photoelectric EffectPhotoelectric Effect

• If you shine light of short enough wavelength on a surface of, say, metal, electrons pop off� recall: shorter wavelength

means higher frequency means greater energy

• The experiment is shown in the figure

• Photoelectric effect� What is the dependence of

Kmax = eVstop on incident light intensity?

Photoelectric EffectPhotoelectric Effect

• Below a certain cutoff frequency, fo, no electrons are ejected at all

• This cannot be explained by “classical” (non-quantum) physics:� classically:

• no matter how low the f, with a bright enough light source we should be able to provide enough energy to electrons to eject them

• the existence of a cutoff frequency contradicts this

� quantum-mechanically:• electrons are bound in the target

material by electric forces (otherwise they would always be dripping off)

• electrons therefore need some minimum energy before they will get ejected at all (i.e., ejected with zero kinetic energy)

Photoelectric EffectPhotoelectric Effect

• To just escape from the target, electrons need a certain minimum energy

� call this energy the “work function”

� abbreviation: Φ

• The work function is a property of the target material

• If hf>Φ, electron is ejected; otherwise it is not.

Photoelectric EffectPhotoelectric Effect

• Einstein summarized this as follows:� hf = Kmax + Φ

� in words, the energy of the incoming photon goes into freeing the electron and then whatever energy remains goes into giving the electron some kinetic energy

• this kinetic energy is maximal when the electron is near the surface of the material. Otherwise some energy is also expended beyond that needed to overcome Φ, and the electron has K<Kmax when it is ejected

� Basically, this is a statement of conservation of energy

• Amusingly, this is what Einstein got a Nobel prize for (not relativity!)

8

Aside: Energy UnitsAside: Energy Units

• For the tiny energies we will deal with now, we use the “electron volt” as the unit of energy

� abbreviated “eV”

� corresponds to the K.E. of an electron after it has been accelerated from rest through a potential difference of 1.0 Volts

� 1eV = 1.6x10-19 Joules

• use to convert to MKS

� We also may use KeV, MeV, GeV,…

Photoelectric EffectPhotoelectric Effect

What is the work function

Φ of sodium given this

plot? How much KE does

an electron have at a

frequency of 8 x 10^14

Hz?

)(3.25.563.6 1434eVhfo =⋅==Φ −

Photon MomentumPhoton Momentum

• We have already shown that radiation can exert pressure, so it should come as no surprise that photons carry linear momentum

• Back in 1916, when Einstein suggested it, it was a more radical concept

� a photon with energy hf has momentum p = hf/c = h/λλλλ

• This means that when a photon interacts with matter, momentum (and energy) are transferred as though the interaction was a collision in the classical sense (which it wasn’t)

Photon Momentum: Experimental ProofPhoton Momentum: Experimental Proof

• Send a beam of x-ray

photons of wavelength λλλλat a carbon target

• See what wavelength

photons were scattered at

various angles

Arthur ComptonArthur Compton

(1892(1892--1962)1962)

Nobel PrizeNobel Prize

(1927)(1927)

Photon Momentum: Experimental ProofPhoton Momentum: Experimental Proof

• Quantum mechanics to the rescue� We can interpret the

scattering of x-rays from carbon in terms of momentum (and energy) transfers between x-rays and carbon atoms’ electrons

� In the process, the x-ray loses energy and is re-emitted at a lower wavelength, and the electron goes off w/some K.E.

� See figure

Photon Momentum: Experimental ProofPhoton Momentum: Experimental Proof

• Doing the (somewhat complicated) algebra we get� ∆λλλλ = (h/mc)(1 – cosφφφφ)

• This agrees exactly with Compton’s plots

• h/mc is called the Compton wavelength� note that m is a particle mass, and that therefore a particle

(e.g., an electron) appears to have a wavelength associated with its motion…

• Note: the peak at the incident wavelength is due to Compton scatters of x-rays off an effective mass equal to the entire carbon atom� the resulting shift is tiny, so the resulting peak is extremely

close to the incident wavelength

Photon Momentum: Experimental ProofPhoton Momentum: Experimental Proof

∆l = (h/mec)(1 – cosφφφφ)

∆l = (h/matomc)(1 – cosφφφφ)

Photon Momentum: Sample ProblemPhoton Momentum: Sample Problem

• X-rays of wavelength 22pm are scattered with a

Compton shift of 2.2pm. What percentage of

the initial x-ray photon energy is transferred to

an electron in such a scattering?

%1.921.222

21.2

'

''=

+=

∆+

∆=

−=

−=

λλ

λ

λ

λλ

hf

hfhffrac

Light as a Probability WaveLight as a Probability Wave

• Back to the two-slit experiment

� What if the incident light is so weak that only one photon at a time heads towards the slits?

• We STILL see an interference pattern!

• Does each photon pass through both slits and then interfere with itself???

Light as a Probability WaveLight as a Probability Wave

• How do we explain this?� Treat light as a probability wave, represented by a wave

function, Ψ• Ψ behaves like any other wave behaves

• probability of observing the light goes as Ψ2

� The photon is produced as a particle, then travels as a wave (of probability) Ψ through the slits, undergoing interference, diffraction, etc.

� It is then absorbed as a photon on the screen• absorption is more probable to occur at bright fringes than at dark

ones

• Put another way:

� We cannot specify with certainty at which point a particular

photon will be absorbed

� But we can specify the probability of it being absorbed at a

particular point

18

Einstein vs BohrEinstein vs Bohr

God doesn’t God doesn’t

play dice!play dice!

Einstein, stop telling Einstein, stop telling

God what to do.God what to do.

Matter WavesMatter Waves

• What happens if we

send electrons through

a double slit

apparatus?

� a) initially, the pattern

looks random

� b) start to see

interference

� c&d) characteristic

interference pattern

pas.

roch

este

r.ed

u

Light Is Not AloneLight Is Not Alone

Assign wave vector (wavelength) to each particle

� k=2ππππp/h or λλλλ = h/p

� Other particles also behave like light, even particles

we never before thought might be waves at all, such

as electrons

Louis de Louis de BroglieBroglie

(1892(1892--1987)1987)

Nobel PrizeNobel Prize

(1929)(1929)

Heisenberg’s Uncertainty Principle (recall diffraction)Heisenberg’s Uncertainty Principle (recall diffraction)

• It is impossible to measure simultaneously the momentum and position of a particle with unlimited precision

� ∆x·∆px ≥ h/2π (similarly for y & z)

• Meaning:

� If you measure px extremely precisely, such that ∆px is very small, then ∆xwill have to be large enough to satisfy Heisenberg’s inequality. And vice-versa.

• ∆t⋅⋅⋅⋅∆E ≥ h is another form of the Heisenberg Uncertainty principle. Implies that we can violate conservation of energy, but only for a very short time. This actually happens!

Werner HeisenbergWerner Heisenberg

(1901(1901--1976)1976)

Nobel PrizeNobel Prize

(1932)(1932)

22

Why do atoms not collapse?Why do atoms not collapse?

Earnest RutherfordEarnest Rutherford

(1885(1885--1962)1962)

Nobel PrizeNobel Prize

(1908)(1908)

According to Rutherford, electrons orbit the nucleus.

Classical Electrodynamics tell us that accelerating

electrons must radiate EM-waves, i.e. loose energy.

This would lead to the collapse of the Hydrogen Atom in 10-13 s

Why are we still alive?Why are we still alive?

The uncertainty principle saves the world!The uncertainty principle saves the world!(the closer electron gets to nucleus, the more uncertain its mom(the closer electron gets to nucleus, the more uncertain its momentum)entum)

23

Bohr’s Model of AtomBohr’s Model of Atom

only integral numbers of wavelengths allowed

deBroglie's wavelength of a particle: λ = h/p

DeBroglie'sDeBroglie's Vision of Bohr's AtomVision of Bohr's Atom

electron wave packet

standing wave vibrating in "orbital"

around a nucleus (4 wavelengths in picture)

NielsNiels BohrBohr

(1885(1885--1962)1962)

Nobel PrizeNobel Prize

(1922)(1922)

RecapRecap

• Subatomic physics

• Photons, quanta & quantum physics

� Photoelectric effect

� Photon momentum

� Uncertainty Principle

� 1D well

hf = Kmax + Φ

p = h/λλλλ

∆x·∆px ≥ h/2π

∆t⋅⋅⋅⋅∆E ≥ h

Next LectureNext Lecture

Black Holes!