Fundamentals of Physics by Eunil Won, Korea University 1

Eunil WonDepartment of Physics

Korea University

Ch 39 Photons and Matter Waves

Fundamentals of Physics by Eunil Won, Korea University 2

The Photon, the Quantum of LightIn 1905, Einstein proposed: electromagnetic radiation is quantized and exists in elementary amounts (quanta) called photons

h: Planck constant: h = 6.63 x 10-34 J s = 4.14 x 10-15 eV s

(energy of single photon)

The quantum of a light wave of frequency f has energy: E = hf f =

c

λ

ex) A lamp with 100 W power (wavelength=590 nm). How many photons are emitted per second?

# of photons per second = power / hf = power x c / h x wavelength

=(100 W )(590 × 10−9m)

(6.63 × 10−34J · s)(3.0 × 108m/s)

= 2.97 × 1020photons/s

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The Photoelectric Effect

First photoelectric experiment

1) incident light causes current

2) apply potential difference V : collector C is slightly negatively charged

3) At certain V, there will be no current V = Vstop (stopping potential)

Kmax : the kinetic energy of most energetic

electrons Kmax = eVstop

Kmax does not depend on the intensity of the

light source (inconsistent with wave nature)

If a beam of light is directed onto a clean metal surface, the light cause electrons to leave that surface

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The Photoelectric Effect

Photoelectric effect does not occur below a certain cutoff frequency f0

2nd Photoelectric Experiment: now we vary the frequency of the incident light and measure Vstop

λ0 =

c

f0

(cutoff wavelength)

To just escape from the target, e- must pick up a certain energy (properties of the target material: work function)

Φ

Einstein summed up the photoelectric experiments as:

hf = Kmax + Φ (photoelectric equation)

Vstop =Kmax

e=

h

cf −

Φ

eexplains the above plot

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Photons have MomentumIn 1916, Einstein extended his concept of light quanta: a quantum of light has linear momentum

(photon momentum)

Scattered x rays showed a shift in wavelength (Compton shift): a fraction of momentum is transfered

p =

hf

c=

h

λ

∆λ =h

mc(1 − cos φ)

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Light as a Probability WaveA fundamental mystery:

Light can be a wave in classical physicsIt is emitted and and absorbed as photons (in quantum physics)

Standard Version

: small photon detector tells us relative probability of single photon will be detected We take a concept of “probability wave”

Single-photon version

: A single-photon version of double-slit experiment (one photon at a time) -> Astonishingly interference fringes still build up,supporting the probability wave nature

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Electrons and Matter WavesMatter can behave wave?In 1924, Louis de Broglie suggested matter waves( A moving matter has wavelength)

λ =

h

p

ex) K=120 eV electron

p = mv, K =1

2mv2

= m

√2K

m=

√2mK

=√

2 × (9.11 × 10−31kg)(120eV )(1.6 × 10−19J/eV )

= 5.91 × 10−24kg · m/s

λ =h

p=

6.63 × 10−34J · s

5.91 × 10−24kg · m/s

= 1.12 × 10−10m = 112 pm

ex) Me running v=1m/s

λ =h

p=

6.63 × 10−34J · s

60kg × 1 m/s

10−35 m

X-ray and electron diffraction

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Wave and Particles

Schrodinger Equation : wave equation describes the matter wave

, i2 = -1

Ψ(x, y, z, t) = ψ(x, y, z)e−iωt

Matter wave: Ψ(x, y, z, t)

The probability (per unit time) of detecting a particle in a volume: |ψ|2

(space and time is separable in our case)

(one dimensional case)

For a free particle:

solution to this: ψ(x) = Aeikx + Be−ikx

Ψ(x, t) = Aei(kx−ωt) + Be−(ikx+ωt)

Choose B=0 to get a particle moving +x only

|ψ(x)|2 = constant

cannot predict the position of a free particle?

d2ψ

dx2+

8π2m

h2[1

2mv2]ψ = 0

d2ψ

dx2+

(2π

p

h

)2

ψ = 0

d2ψ

dx2+ k2ψ = 0

p

h=

1

λ,

2π

λ= k

d2ψ

dx2+

8π2m

h2[E − U0(x)]ψ = 0

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Heisenberg’s Uncertainty PrincipleThe position and the momentum of a particle cannot be measured simultaneously with unlimited precision

∆x · ∆px ≥ h̄

∆y · ∆py ≥ h̄

∆z · ∆pz ≥ h̄

h̄ =h

2π

Do not think that the particle really has a sharply defined position: I’m sure you are confused by now :-)

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Barrier Tunnelingelectron with energy E moving toward to a potential barrier (U0) when E<U0

classical physics: the electron is bounced off all the time

quantum physics: in some cases the electron penetrates the barrier

Transmission coefficient : the probability of tunneling of the electron(If T=0.020, 20 out of 1000 electrons will tunnel through)

T ≈ e−2kL

k =

√8π2m(U0 − E)

h2

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Crystalline quartz changes its dimension when an electric potential is applied (piezoelectricity): tip can be moved precisely

The Scanning Tunneling Microscope (STM)

Electrons from the sample can tunnel through to the tip : tunnel current can be measured and used as a microscope (STM)

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Summary

Light Quanta - PhotonsEnergy E = hf

Momentum p =

hf

c=

h

λ

Photoelectric Effect

Compton Shift

hf = Kmax + Φ

∆λ =h

mc(1 − cos φ)

Heisenberg’s Uncertainty Principle