Modern Physics (I)

Chap 3: The Quantum Theory of Light Blackbody radiation, photoelectric effect, Compton effect

Chap 4: The Particle Nature of Matter Rutherford’s model of the nucleus, the Bohr atom

Chap 5: Matter Waves de Broglie’s matter waves, Heisenberg uncertainty principle

Chap 6: Quantum Mechanics in One Dimension The Born interpretation, the Schrodinger equation, potential wells

Chap 7: Tunneling Phenomena (potential barriers)

Chap 8: Quantum Mechanics in Three Dimensions Hydrogen atoms, quantization of angular momentums

科學發展月刊 2005/11

Emission of electromagnetic radiation by solids – continuous spectra

(Cf. emission and absorption spectra of atoms – discrete spectra)

Wavelength of radiation near peak of emission spectrum determines color of object

500oC 700oC 1000oC 2500oC

Radiation at wavelength longer than optical

Increase fraction in optical wavelengths

Chapter 3: The Quantum Theory of Light

max constant5 B

hcT

k

The Problem to answer:

The problem is to predict the radiation intensity at a given wavelength emitted by a hot glowing “solid” at a specific temperature T (in thermal equilibrium)

To calculate the energy per unit volume per unit frequency of the radiation within the blackbody cavity

u = u (f, T )

1( ,T)d ( )u f f N f df

V

Energy density in frequency range from f to f+df

: average energy per mode

0

P d

/

/, ( )B

B

k Tk T

B

eP T A T e

k T

Boltzmann distribution Rayleigh-Jeans formula

/

0 0/

/

0 0

1

B

BB

nhf k T

nhf k T

nhf k T

P nhf Aehf

eP Ae

( = 1, 2, 3, ......)nhf n Planck’s blackbody radiation theory

V d)()d( N

Energy density in frequency range from f to f+df

B

2

/ k T3

1 8,T d ( ) d

c e 1hf

f hfu f f N f df f

V

/5

8 1,

1Bhc k T

hcu T d d

e

To fit the data by Coblentz (1916)

Planck obtained h = 6.5710-34 J · s

uT(

)(h = 6.626 10-34 J s)

h: a very small number which plays significant roles in microscopic worlds

Planck thought that his concept of energy quantization was merely a desperate calculational device and moreover a device that applied only in the case of blackbody radiation

Einstein elevated quantization to the level of a universal phenomenon by showing that light itself was quantized

Photoelectric effect

Radiant EM waves seems like many packages

(grains) of energy. Each has energy hf

Photoelectric Effect: Einstein’s quantization theory of light

Radiant energy is quantized and is localized in a small volume of space, and that it remains localized as it move away from source with a speed of c

Total radiant energy E = nhf

wave package

maxKE seV hf

When a light quantum hit an electron, it can either be absorbed

completely or no reaction

f0

cutoff VS

1/e

2/e

3/e

metals 1, 2, 31 < 2 < 3

Compton Effect (1922)

Between 1919–23, Compton showed that x-rays collide elastically with electrons, in the same way that two particles would elastically collide

What does this tell us ?

• Light “particles” (photons) carry momentum !

c

EP photon

E is the photon energy

c is the speed of light

• Earlier result that Ephoton= hf = hc/

h

Ph is Planck’s constant

is the wavelength of light

graphite (carbon)

Experimental details

A beam of x-ray of wavelength o is scattered through an angle by a metallic foil, the scattered radiation contains a well-defined wavelength which is longer than o

When 0, one more peak with > o appears. depends on

o=0.0709nm

oe

1 cosm c

h

0.00243 1 cos [nm]

Photon can scatter off matter

Collision of particles

after collision

x-ray

x-ray

e- e-

cP

E

o

o

o

o

hf

hf

0P

cmE

e

2ee

cP

E

hf

hf

' 2 4 2 2ee e

e

E m c P c

P 0

Conservation of momentum e

cos P cosc c

ohf hf esin P sin

c

hf

Conservation of energy 2 2 4 2 2o e e e

m c m c P chf hf

before collisionx

y

Summary:

Planck: energy quantization of oscillators in the walls of a perfect radiator

Einstein: extension of energy quantization to light in the photoelectric effect

Compton: further confirmation of the existence of the photon as a particle carrying momentum in x-ray scattering experiments

Rutherford’s model of the Nucleus

The Bohr Atom

Constituents of atoms (known before 1910)

There are electrons with measured charge and mass

There are positive charge to make the atom electrical neutral

The size of atom is known to be about 10-10 m in radius

Rutherford’s scattering experiment

Projectile: particle with charge +2eTarget: Au foil

KE 5 MeV !

Rutherford’s -particle Scattering Experiment (1911)

To probe the distribution of the positive charge with a suitable projectile

How is the mass of the positive charge distributed within the atom?

Experimental results: (Geiger and Marsden)99% of deflected particles have deflection angle 3o

However, there are 0.01% of particles have larger angle > 90o

Rutherford’s model of the structure of the atom to explain the observed large angle scattering

A single encounter of particle with a massive charge confined to a volume much smaller than size of the atom

Nucleus

All positive charges and essentially all its mass are assumed to be concentrated in the small region

-14 4~ 10 m 10r R

(m,v)

b

Ze+ Trajectory of particle (r, )

(r, )

Deflection due to Coulomb interaction:

22

2 2

2ˆ ˆ

4 o

e Ze d r dr m r r

r dt dt

Rutherford’s scattering model

b: impact parameter

fast, massive particles

( )

( )

r r t

t

( )P d

d

4

1( ) ( )

sin / 2DN d NP b db

# of particles detected by detector at scattering angle

# of particles detected by detector at scattering angle

4

1( )

sin / 2DN d

ND

2/sin 4

2 (2 )( )

2 4 o

mv e Ze

D

In the case when the KE of the particle is so high that the equation begins to fail, this distance of the closest approach is approximately equal to the nuclear radius

155 10 mD

(Rutherford assumed that particles do not penetrate the nucleus)

Rutherford Scattering:

Rutherford’s calculations and procedures laid the foundation for many of today’s atomic and nuclear scattering experiments

By means of scattering experiments similar in concepts to those of Rutherford, scientists have elucidated (1) the electron structure of the atom, (2) the internal structure of the nucleus, and even (3) the internal structure of the nuclear constituents, protons and neutrons

Einstein:

Splitting the atom by bombardment is like shooting at birds in the dark in a region where there are few birds

Schematics of energy levels and radiated spectrum of H atom

1890

Rydberg & Ritz formula

2 2

1 1 1

nm

Rn m

7 11.0968 10 mR

n, m integers with n < m

Bohr’s quantum model of the Atom (1913)

Four postulates：

1. An electron in an atom moves in a circular orbit about the

nucleus under the influence of the Coulomb attraction

between the electron and the nucleus

2. The allowed orbit is a stationary orbit with a constant energy E

3. Electron radiates only when it makes a transition from one stationary

state to another with frequency i fE Ef

h

4. The allowed orbit for the electron: . The quantum

number n labels and characterizes each atomic state2

hL n n

n = 1, 2, 3, …… (“quantum number”)

Bohr atom Consider an atom consists of nucleus with +Ze protons and a single electron –e at radius r

Ze

e-

r 2 2

2

1

4 o

Ze mv

r r

Coulomb attraction Centripetal force

L n r m v

L nv

mr mr

n = 1, 2, 3, …Orbital angular momentum

2 2 2

2

4 oo

n nr a

Z e m Z

Radius of allowed orbit:

ao Bohr radius = 0.529 Å

For n = 1 and Z = 1,

r = ao = 0.529 10-10 m Correct prediction for atomic size !!

Quantized orbits !!

Ze

e-Total Energy of the electron

2 21

2 4 o

mv zeE KE U

r

21

8 o

Ze

r

2 2 2

2 208n o

o

Z e ZE E

n a n

13.6 eVoE

2

o

nr a

Z

Good to describe the observed spectra of any Hydrogen-like atom

with nucleus charge +Ze and a single orbital e-

H, He+, Li2+, …

(3)2

2 2

1 1i f o

f i

E E Z Ef

h h n n

22 2

1 1 1

f i

Z Rn n

c

7 11.097 10 moER

hc

Rydberg constantAllowed transition

The Bohr atom—

“Bohr’s original quantum theory of spectra was one of the most revolutionary, I suppose, that was ever given to science, and I do not know of any theory that has been more successful …… I consider the work of Bohr one of the greatest triumphs of the human mind.” (Lord Rutherford)

“This is the highest form of musicality in the sphere of though.” (Einstein)

「他不但具有關於細節的全部知識，而且還始終堅定的注視著基本原理。」 (Einstein)

Franck-Hertz Experiment (1914)To observe current I to collector as a function of accelerating voltage Va

Acceleratingvoltage (0–40 V)

Retarding voltage (1.5 V)

(6 V)

When the tube is filled with low pressure of mercury vapor, there are collisions between some electrons and Hg atoms Peaks in current I with a period of 4.9 V

4.9 Vdrop oV n V

Low-energy electrons ( a few tens of volt)

Incoming electron

nuclear

Orbital e-

Scattered electron

Inelastic collision, 4.9 eV of KE of incident electron raises Hg electron from the ground state to the first excited state

Inelastic collision leaves electron with less than Vs, so the electron cannot contribute to current

Ground state

2nd excited state

4.9 eV

6.7 eV

10.4 eV E = 0

Energy levels of outer electron of Hg atom

1st excited state

1240 eV nm253 nm

4.9 eV

Confirmed by emission of single photons !!

Significance of the Franck-Hertz Experiment

The Franck-Hertz provided a simpler and more direct experimental proof of the existence of discrete energy levels in atoms

The experiment confirmed the universality of energy quantization in atoms, because the quite different physical processes of photon emission (optical line spectra) and electron bombardment yielded the same energy levels

Summary:

Rutherford’s scattering of particles from gold atoms

Bohr’s model provides the explanation of the motion of electrons within the atom and of the rich and elaborate series of spectral lines emitted by the atom

D

de Broglie’s intriguing idea of “matter wave” (1924)

Extend notation of “wave-particle duality” from light to matter

For photons,

E hf h

Pc c

Suggests for matter,

h

P de Broglie wavelength

E

fh

de Broglie frequency

P: relativistic momentum

E: total relativistic energy

The wavelength is detectable only for microscopic objects

Chapter 5: Matter Waves

The Davisson-Germer Experiment (1927)a clear-cut proof of wave nature of electrons

21

2 2e

e e

h hm v eV

m v eVm

sind q

50o

a constructive peak

in excellent agreement with the de Broglie formula !!

Applying Ni atoms as a reflecting diffraction grating

Kept detector at a fixed angle and varied the accelerating voltage V

Experiment of Davisson and Germer confirmed that low-energy electrons with mass (v << c) do have wave-like properties

1sin qd q

1

const.2

q

hV q q

me

Constructive peaks occur at wavelengths: q = 1 /q

(q = 1, 2, 3, ……)

1

2q

q

h

q meV

Particle

Large probability to be found in a small region of space at a specific time t

Wave representation

“Wave group” or summed collection of waves with different wavelengths: amplitudes and relative phases chosen to produce constructive interference in small region

the group velocity of the matter wave = the velocity of the particle

Wave groups and Dispersion ( 波群與色散 )

Toward a Wave description of Matter

m v

vg = v

Δx

“wave packet”

波數值越密集，波包在空間的

週期性越大

( ) ikxa k f x e dx

( ) ikxf x a k e dk

Matter waves are represented by wavefunctions: (x,y,z,t ) (a solution for the Schrodinger equation)

Matter waves is not measurable; they require no medium for propagation

(x,y,z,t ) is a complex number and is used to calculate the probability of finding the particle at a given time in a small volume of space

The statistical view (Max Born): the probability of finding a particle is directly proportional to ||2 =

The Heisenberg Uncertainty Principle (1927)It is impossible to determine simultaneously with unlimited precision the position and momentum of a particle

If a measurement of position x is made with an uncertainty x and a simultaneous measurement of momentum Px is made within an uncertainty Px, then the precision of measurement is inherently limited by

Px x /2 (momentum-position uncertainty)

Similarly,

E t /2 (energy-time uncertainty)

Double-slit electron diffraction experiment

While the electrons are detected as particles at a localized spot at some instant of time, the probability of arrival at that spot is determined by finding the intensity of two interfering matter waves

wave properties

particle properties

first minimum:

minsin / 2D

Accumulated results with each slit closed half the time

The experimental result contradicts this sum of probability !!

Once one measures unambiguously which slit the electron passes through, the act of measurement disturbs the electron’s path enough to destroy the interference pattern

A thought experiment: Measuring through which slit the electron passes

py

Summary

The existence of matter waves (de Broglie)Davisson-Germer experiment (electron diffraction from Ni crystal)

Constructing “wave packets” by superposition of matter waves with different frequencies, amplitudes, and phases

Uncertainty principles

Wave-particle duality; double-slit electron diffraction experiment

Need a new mechanics that incorporates both wave and particle natures of subatomic objects