38. Photons and Matter Waves38. Photons and Matter Waves• Thermal Radiation and Black-Body Radiation

T

Thermal radiation : The radiation depends on the temperature and properties of objects

Color of a Tungsten filament as temperature increases– Black– Red– Yellow– White

Black-body Radiation

All the light is absorbed. But the radiation depends on the temperature of the inside wall.

KmT ⋅×= −2max 102898.0λ

Wien’s Displacement law

Classic Point of View

The thermal radiation was considered to be simply due to accelerated charged particles near the surface.⇒ Not right !

Wavelength

Inte

nsity Experimental

Classical theory

Ultraviolet catastrophe!!

38-2. Plank’s Theory, the Photon, the Quantum of Light • Plank --- Explain the black-body radiationwith two assumptions related to the oscillating charges.

1. The radiation energy is Quantized.

nhfEn = λ/cf =

2. The rasonators emit energy, the so-called photon.

hfE =

h = 6.63 ×10-34 J·s = 4.14 × 10-15 eV·s

photon energy

: Elementary quantity

Plank succeeded in reproducing the black-body radiation curve. But no body including Plank himself did not accept the quantum concept. -- Considered the assumptions unrealistic.

38-3. The Photoelectric Effect• photoelectric effect

Photoelectrons, Photoelectric current

• First Photoelectric ExperimentThe first discovery by Herz in 1887.

Vstop : Stopping potential (independent of the radiation intensity)

-Vstop

• Electrons having a kinetic energy K

stopmax eVK =

• Characteristics in the photoelectric effecti) Cutoff frequency, f0

⇒ No photoelectronsii) Kmax is independent of the light intensity.iii) iv) Photoelectric effect occurs instantaneously ( ~ 10-15 sec.)

0ff <

fK ∝max

00 f

c=λ

Cutoff wavelength

• Einstein (1905)

Extend the quantum concept of Plank’sEnergy of the electromagnetic waves⇒ PhotonsEach photon can give its energy to a single electron.

Φ−= hfKmax

Φ=0hf

Work function

Minimum energy bound in the metal (3 ~ 6 eV)

i) Cutoff frequencyii)iii) iv) The particle theory of light

0hf=ΦΦ−= hfKmax

fK ≤max Φ=

Φ==

hch

cfc

/00λ

Cutoff wave length

ef

ehV Φ

−⎟⎠⎞

⎜⎝⎛=stop

sV 101.4 15 ⋅×= −

eh

sJ 106.6)C10(1.6s)V 101.4( 34-1915 ⋅×=××⋅×= −−h

38-4. Photons have Momentum• Einstein

hfE = Photon Energy

λ/// hcEchfp === Photon Momentum

• H. Compton and P. Debye in 1923 carried an experiment to prove Einstein’s point-like particle concept. cEphfE == ,

The photoelectric effect (x-ray scattering): The total momentum of the photon-electron pair must be conserved.

λ = 71.1 pm

Doppler shift of scattered light varies with the scattered angle φ .

• Collision - Energy conservation Kfhhf +′=

)1( −+′

= γλλ

mchh

)1(2 −= γmcK )1(2 −+′= γmcfhhf

2)/(11

cv−=γ

- Momentum conservation

) ( sinsin0

) ( coscos

axisymvh

axisxmvhh

θγφλ

θγφλλ

−′

=

+′

=

shift)(Compton )cos1( φλλλ −=Δ≡−′mch

avelength)(Compton w nm 00243.0==cm

h

ecλ

38-5. Light as a Probability Wave• Light has a dual nature , Wave & Photon.

Low frequency : Long wavelength ⇒ More wave likeHigh frequency : Short wavelength ⇒ More particle like

Light can be a wave in classical physics but be photons in quantum mechanics.

• Young’s Double slit Experiment

: the evidence for the wave nature of light

but can be understood as a relative probability for a detection of a single photon.

38-6. Electrons and Matter WavesParticle also has a dual nature!!

In 1924, Louis Victor de Broglie postulated an electron also has a dual nature.Perhaps all forms of matter have wave as well as particle properties.

• Photon: hfE =phh

cEp =⇒== λ

λ

• Electron: mvp =

mvh

ph ==λ

hEf =

The wavelength of photon can be defined by the momentum.

frequency of matter

: de Broglie wavelegnthde Broglie wave

• The Double-Slit Experiment

2sin λθ =D Minimum

xph

=λxDp

hD 22

sin ==≈λθθ

The number of electrons detected at a certain spot is proportional to the intensity of two interfering matter waves.

How do we understand the wave-character of electrons?

Photon ⇒ EM Wave BErr

,

2EI ∝ ⇒ Interference effects

ψ : Wave function*2 ψψψ =∝I 21 ψψψ +=

φψψψψ

ψψψ

cos2 212

22

1

22

21

2

++=

+≠=I

• Which slit does the electron pass through? Slit 1 or Slit 2

• De Broglie (1923-4): All matters have a dual nature. Then an electron must exhibit diffraction and interference effects.

• Davisson-Germer Experiment (1927): Measure the wavelength of electrons.

Crystalized NiO target

Diffraction patterns due to electron beam.

Extended work on many single-crystalline targets

λhp =Conclude

• G. P. Thomson (1928)Electron diffraction pattern from electrons passing through a gold foil

• Helium atom, Hydrogen atom, Neutron also show the diffraction pattern.⇒ The matter wave is an Universal Nature

38-7. Schrödinger EquationΨ (x,y,z,t) : Wave function

tiezyxtzyx ωψ −=Ψ ),,(),,,( ω (=2πf): angular frequency of matter wave

|ψ|2 : probability density, and not ψ, has physical meaning.

• Schrödinger equation (1-dim):

[ ] 0)(82

2

2

2

=−+ ψπψ xUEh

mdxd U (x): Potential energy

( ) 02

88 2

2

2

2

22

21

2

2

2

2

=⎟⎟⎠

⎞⎜⎜⎝

⎛+=+ ψπψψπψ

mp

hm

dxdmv

hm

dxd

022

2

2

=⎟⎠⎞

⎜⎝⎛+ ψπψ

hp

dxd

022

2

=+ ψψ kdxd Schrödinger equation (free particle)

λπλ /2 ,/ == kph

General solution)()(),( and ,)( tkxitkxiikxikx BeAetxBeAex ωωψ +−−− +=Ψ+=

38-8. Heisenberg’s Uncertainty Principle1927 Werner Heisenberg• Heisenberg Uncertainty Principle

A measurement of position is made with precision Δx, anda measurement of momentum is made with precision Δpx.

h≥Δ⋅Δ xpx π2/h=h

It is fundamentally impossible to make simultaneous measurementsof a particle’s position and momentum with infinite accuracy.

Similarly h≥Δ⋅Δ tE ⇒ Life-time of a particle

λhp =

λhpx =Δ

λ=Δxhxpx =Δ⋅Δ⇒

Position of electron

h

h

≥Δ⋅Δ

≥Δ⋅Δ

z

y

pz

py

22

222

22

22

2

2

)()(

xx

xxx

xxxx

xxx

−=

+−=

+−=

−=Δ

222)( ppp −=Δ

A traveling electron (wave-pocket)

38-9. Barrier Tunneling

bLeT 2−≅ 1<<Tif

2

2 )(8h

EUmb b −=

π

1=

Transmittance

+ RT

• The Scanning Tunneling Microscope (STM)

where