FYS2010
Aktuell forskning i fysikk
F. Ould-Saada
Fysikk Institutt, Universitetet i Oslo
August 20061
1
This introduction is based on the books:
Invitation to Contemporary physics, Q.Ho-Kim, N. Kumar, C.S. Lam, World Scientific 2004.
Concepts of Modern Physics, Arthur Beiser, Mac Graw Hill International Edition 2003.
The New Quantum Universe, T. Hey, P. Walters, Cambridge University Press 2003.
...
• Goal:
– motivation and invitation into physics
• Arrangement:
– Introduction (Based on the appendices A and B of “Invitation to Contemporary Physics” - a book I previously based the lecture on - and extended
using material from several other books.)
– The New Physics for the twenty-first century is
Lively and accessible account of the hottest topics in physics
Written by leading international experts including Nobel prize winners
– Contents
Introduction Gordon Fraser;
Part I. Matter and the Universe:
1. Cosmology Wendy Freedman and Rocky Kolb;
2. Gravity Ronald Adler;
3. Astrophysics Arnon Dar;
4. Particles and the standard model Chris Quigg;
5. Superstrings Michael Green;
Part II. Quantum Matter:
6. Atoms and photons Claude Cohen-Tannoudji and Jean Dalibard;
7. The quantum world of ultra-cold atoms Christopher Foot and William
Phillips;
8. Superfluidity Henry Hall;
9. Quantum phase transitions Subir Sachdev;
Part III. Quanta in Action:
10. Quantum entanglement Anton Zeilinger;
11. Quanta, ciphers and computers Artur Ekert;
1
12. Small-scale structure and nanoscience Yoseph Imry;
Part IV. Calculation and Computation:
13. Nonlinearity Henry Abarbanel;
14. Complexity Antonio Politi;
15. Collaborative physics, E-science and the grid Tony Hey and Anne
Trefethen;
Part V. Science in Action:
16. Biophysics Cyrus Safinya;
17. Medical physics Nicolaj Pavel;
18. Physics and materials Robert Cahn;
19. Physics and society Ugo Amaldi.
– IMPORTNANT REMARKS:
∗ Goal is NOT to treat all subjects.
∗ You will be asked to read some chapters/sections that we discuss afterwards.
∗ There are interesting related web sites. You can use if you are interested
in some subject and in your project work.
∗ Possibility to work/read in small groups.
– Approach:
∗ top-bottom and not (traditional) bottom-top
• Exam: (“Vurderingsform”)
– 1 obligatory (project) work (ca. 30 % weight).
– 1 project (ca. 70 % weight)
∗ including a report (ca. 10-15 pages) (ca. 30 % weight)
2
∗ Semester ends with an oral presentation of the project (ca. 20 min.)
(ca. 40 % weight)
∗ Questions follow in presence of sensor.
3
Contents
0.1
0.2
0.3
0.4
0.5
0.6
0.7
General concepts in classical physics . . . . . . . . . . . . . . . . . . . . . .
0.1.1 The physical universe . . . . . . . . . . . . . . . . . . . . . .
0.1.2 Matter and motion . . . . . . . . . . . . . . . . . . . . . . . .
0.1.3 Waves and fields . . . . . . . . . . . . . . . . . . . . . . . . . .
General concepts in Quantum physics . . . . . . . . . . . . . . . . . . . . . .
0.2.1 Heisenberg’s uncertainty principle . . . . . . . . . . . . .
0.2.2 Wave function and probability: state of the system
0.2.3 Superposition of states and wave interference . . . .
0.2.4 Indistinguishability of identical particles . . . . . . . . .
0.2.5 Quantum mechanics and naive realism . . . . . . . . .
Special relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0.3.1 Popular effects of SR . . . . . . . . . . . . . . . . . . . . . . .
General relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Quantum gravity? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Superstring theory? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Thermal physics and statistical mechanics . . . . . . . . . . . . . . . . . . .
0.7.1 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . .
0.7.2 Statistical mechanics . . . . . . . . . . . . . . . . . . . . . . .
0.7.3 Dimensional analysis . . . . . . . . . . . . . . . . . . . . . . .
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1
1
6
10
18
18
20
34
36
39
40
41
46
54
54
55
55
57
68
Introduction
0.1
0.1.1
General concepts in classical physics
The physical universe
• Physics is the study of nature.
• Study of all forms of matter and energy the whole universe from smallest constituents of matter to largest bodies in universe.
– Observation of formation and evolution of all objects
∗ probe inner structure and interactions
– Search for basic rules:
∗ correlate seemingly disparate facts in a logical structure of thought,
∗ focus thinking along new directions,
∗ predict future observable events.
1
Size, mass and time are “fundamental”.
• International System of Units (SI):
Length : meters (m) ; Mass : kilograms (kg) ; Time : seconds (s)
also cgs: centimeter, gram, second
• Other example: Plancks’s units defined in terms of the speed of light (c), the
h
), and the Newtonian gravitational constant G:
quantum of action (h̄= 2π
Lp =
mp =
tp =
kilo(k) = 103
mega(M ) = 106
milli(m) = 10−3 micro(µ) = 10−6 ;
r
h̄G
3
rc
h̄c
rG
h̄G
c5
= 1.6 × 10−35m
= 2.18 × 10−8kg
= 5.4 × 10−44s
giga(G) = 109
tera(T ) = 1012 ;
nano(n) = 10−9 pico(p) = 10−12
peta(P ) = 1015
femto(f ) = 10−15
All other physical quantities can be derived as various combinations of powers of
the base units:
)
Pressure : pascal(P a = mkgs2 )
Force : newton(N = m kg
s2
2
Energy : joule(J = kgsm
electron volt (1 eV = 1.6 × 10−19J)
2 )
Temperature : kelvin(K ∝ Jk )
Speed of light in a vacuum
Gravitational constant
Planck constant
Boltzmann constant
Stefan-Boltzmann constant
Elementary charge
Electron mass
Proton mass
c
G
h
k
σ
e
me
mp
=
=
=
=
=
=
=
=
3.0 × 108 m/s
6.67 × 10−11 m3 /s2 kg
6.63 × 10−34 J s
1.38 × 10−23 J/K
5.67 × 10−8 J/sm2 K 4
1.60 × 10−19 C
9.11 × 10−31 kg
1.67 × 10−27 kg
2
Table A.1: Fundamental (or univer-
sal) physical constants (SI units).
Numerical values are results of various experiments. Unchanging in any
physical circumstances and constant
in time. Yet unexplainable by theories.
3
Figure A.1: Scales of length, mass and time of the physical universe.
• Specific phenomena explained in terms of facts.
• Facts generalised into rules that apply to larger classes of phenomena.
• Rules furher distilled into laws that cover observed and unobserved cases,
anywhere and anytime.
• Laws further organised into a coherent and logical structure, theory.
• In physics, 3 broad categories of theories:
– Mechanics: study of motion, plays a unique role central to all of physics
∗ classical mechanics,
∗ quantum mechanics and special relativity basis of all contemporary physics (modern technology and lifestyle).
– Theories of matter and interactions:
∗ Fundamental dynamical theories deal with forces (or interactions).
∗ Forces intimately related to matter.
·
All matter composed of a
few species of fundamental,
or elementary, particles.
4
∗ Particles interact via 4 fundamental forces:
· gravitation that holds the planets and the stars together
· electromagnetic that controls much of the low-energy physics and
all of chemistry
· strong nuclear interactions that hold protons and neutrons together
in the nucleus
· weak nuclear interactions that are responsible for the radioactive
decay of unstable nuclei.
– Physics of large systems:
∗ thermodynamics,
∗ statistical mechanics.
5
Matter and motion
Space and time
0.1.2
• two basic physical concepts needed to describe observations or formulate theories,
• not defined but measured, which is absolutely essential to the development of
physics.
Space is
• three-dimensional (x, y, z)
• flat or Euclidean to a very good approximation: parallel lines never meet and
the sum of the interior angles of a planar triangle is 180◦, ... Euclidean geometry
breaks down in the immediate vicinity of very massive bodies - neutron star,
sun, ... - (general relativity, extra dimensions)).
Time
• flows in one direction only past-present-future. Time reversal never occurs in
the macroscopic world, although there are indications that this is not the case
in the microscopic world of particles.
• is absolute to a very high precision, according to classical physics. This is
not exact when rapid motions (special relativity) or gravitational effects
(general relativity) are involved.
Classical (or Newtonian) physics is not an exact theory and is based on postulates
that do not exactely hold in all situations:
• space is Euclidean, time is absolute,
• restriction to inertial reference frames.
Newtonian physics excellent approximation to real world.
6
The laws of motion
With space ~x, time t, define
• Velocity ~v =
d~x
dt
• Acceleration ~a =
d~v
dt
=
d2~x
dt2
• Force F~ defined by its effect, like stretching of spring.
Newton’s 3 laws of motion:
t
law: A body at rest (in uniform motion) remains at rest (in uniform motion) unless
acted by an unbalanced force.
d
law: Applied force F~ on a body of (inertial) mass m ⇒ Acceleration ~a
d
F~ = m ~a.
law: Action F~ ⇒ reaction −F~ .
Contrary to kinematics, dynamics involves mass in addition. Define:
• Momentum p~ = m ~v , such that (2nd law)
d~p
.
F~ =
dt
– p~ is constant in abscence of unbalanced force (1st law).
– p~tot is a constant of motion for an isolated 2-particle system (3rd law).
The law of conservation of momentum is a manifestation of the existence of
a space symmetry.
7
Angular momentum
• Angular momentum is a property characteristic of rotational motion.
• Just as force produces changes in (linear) momentum, torque (or “twist”)
~τ = ~r × F~
magnitude : τ = r · F · sin θ
produces changes in angular momentum:
~ = ~r × p~ ; magnitude :
Angular mom. : L
L=I ω
L = rp sin θ
moment of inertia × angular velocity
• Angular momentum is conserved if no torque exists.
The law of conservation of angular momentum arises from the rotational
symmetry of space, isotropy: the geometry of space is the same in all
directions.
– Kepler’s second law2 - the radial line segment from the sun to a planet
sweeps out equal areas in equal times - is an example.
• Circular (uniform) orbit (planet around the Sun) of period T :
Velocity : v = rω =
2πr
T
2
; Centripetal force : Fc = mac = m vr
2
Fc = 4πr2m × Const
Fc = G mr2M
G=
3
Const = Tr 2
4π 2 Const
−11 N m2
=
6.67
M
kg 2
−→ Newton’s (inverse) law of universal gravitation
m1 m2
F = G 2 .
r
Kepler’s first law: Each planet moves around the Sun in an elliptical orbit, with the Sun
at one focus. 2
3
Third law: TT12 = rr13 for any two planets.
2
2
8
Work and energy
Work :
W =F x
Kinetical energy :
Ek = 12 mv 2
mechanical energy
E = Ek + V
gravitational potential energy : V (z) = mgz
• Conservation of mechanical energy if dissipative forces, e.g., friction, are ignored.
• Some energy can dissipate in heat, such that E not conservated.
– However, loss only apparent: macroscopic point of view.
– Energy lost by body transmuted into disorderly E (“heat”) of the atoms
of body and surface.
• Other kinds of energy:
– Chemical energy: molecular or atomic.
– Nuclear energy: particles (protons and neutrons) in atomic nuclei.
– Electromagnetic energy: carried by all sorts of EM radiation, such as moonlight and TV signals.
– Mass equivalent energy: E = m c2 .
• The total amount of energy of any isolated system is conserved, although the
form of energy may change.
• This conservation law arises from a symmetry of space, the irrelevance of
the absolute measure of time.
9
0.1.3
Waves and fields
Late 19th century, the same laws of motion (Newton) explain
• motion of all bodies (on Earth and space)
• heat (kinetic theory of gases)
– But, what is the nature of light?
∗ Composed of microscopic particles (Newton)?
∗ A wave phenomenon (Huygens, Young, Fresnel)?
– How to incorporate the increasingly large number of electric and magnetic
phenomena (Oersted, Faraday, ...)?
– What to do with the action-at-a-distance through gravity and electric
force?
=⇒ Two new concepts: waves and fields.
Waves and wave propagation
• A Wave is a disturbance/pulse that propagates without transmission of matter.
– Energy and momentum are transmitted.
• Mechanical waves require a medium to propagate:
– Ripples spreading on water,
– Sound in air
• Electromagnetic waves propagate in vacuum.
10
• Example of a periodic wave, like vibrations of a string fixed at one end.
Figure 7.1(Hey): A sequence of “photographs” of a passing wave. The arrow points over the same crest,
showing how the wave moves to right.
– Wavelength, λ, is the distance of two adjacent peaks.
– Period, T , is the time needed for the peak to cover a wavelegth.
– frequency, ν, is the number of wavelengths sent out per second, given in
herz, 1Hz = 1cycle/s
ν=
1
= 2π ω
T
ω: angular velocity.
– Velocity, v, of the vave is determined by the medium
v=
λ
=λν
T
– Amplitude, A, is the maximum of displacement of the medium from its
equilibrium position.
11
• One distinguishes 2 kinds of traveling waves:
– Transverse wave: When a pulse travels along a string fixed at one end,
the vibration occurs in a direction perpendicular to the direction in which
the pulse moves.
– Longitudinal wave: Consider a row of particles connected by massless
springs.
∗ A disturbence (pull/push) producess vibrations - condensations and rarefactions of particles ∗ which oscillate in the same direction of the motion of the wave itself
∗ Many such identical strings forced to vibrate together by the same disturbance, leads to a wavefront.
• Other properties of waves are:
– Diffraction, refraction, ...
– Interference or principle of superposition: When two waves encounter
each other, they pass through each other unmodified
– → This is the reason why one can hear the telephone while listening to
radio, or distinguish the sounds of different instruments in a concert.
– → Two particles “would” collide.
∗ Where the two waves overlap, their amplitudes add up A = A1 + A2.
∗ Constructive interference: incident waves reinforce each other.
∗ Destructive interference: incident waves tend to cancel out.
– Waves of the same form and wavelength oscillating in phase (synchronously)
are said to be coherent (Fig. 7.2(Hey)). Incoherent waves have no definite
phase difference.
12
– Only coherent waves display the usual interference pattern.
Figure 7.2(Hey): Three pairs of waves with the same wavelength - top waves for reference and bottom waves
have different start times.
• Wave formula:
φ(x, t) = A cos 2π νt −
x
λ
= A cos (ωt − k · x) = A cos ω t −
x
v
is the wave number. In three dimensions, k · x → ~k · ~r;
where k = 2π
λ
normal to the wave front.
• Wave equation in one space dimension
∂2φ
∂x2
In 3 dimensions,
∂2φ
∂x2
2
+ ∂∂yφ2 +
∂2φ
∂z 2
=
=
~k is
1 ∂2φ
v 2 ∂t2
1 ∂2φ
v 2 ∂t2
• Solutions of the wave equation maybe a single traveling pulse, a train of waves
of constant A and λ, a train of superposed waves of the same (or different) A
and λ, a standing wave (string fixed at both ends), ...
13
– Solutions must be of the form:
φ = F t±
x
v
,
which represent waves travelling in the +x (minus sign) and −x (plus sign)
directions.
– A more general wave formula involves complex quantities:
x
φ(x, t) = Ae−iω(t− v ) = Ae−i(ωt−k·x) = A cos ω t − xv − iA sin ω t − xv
Nature of light
• Is light particle-like or wave-like?
→ more later
Figure A.2: (a) Refraction of light pulse going from
air to water; wavefronts are right angles to the light
ray and delineate successive maxima. In (a) it is
assumed that the speed of light is smaller in water
than in air, and so wavefronts travel closer together
in water; in (b) light is supposed to move faster in
water.
Electric and magnetic fields
• Coulomb, 1785, electric force between two stationary charges q and q 0 separated by a distance r
q q0
F =K 2
r
K '9×
14
Nm
109 2
C
2
K is a numerical constant. F repulsive (attractive) for like sign (opposite sign)
charges.
• Oersted, 1820, moving charges induce an extra force, the magnetic force.
– Electric current through wire caused a magnetic compass needle to rotate
by 90◦.
I[A]
Ampere’s law: magnetic field at a distance r from the wire, B[T ] = k r[m]
,
where k = 2 × 10−7 TAm .
• Faraday, , moving a bar magnet across a wire induces an electric current in the
wire (electromagnetic induction).
• Faraday, concept of field - a quantity that can be defined at each point in space
- as an intuitive way of representing the electric interaction between charges
(Fig. A.3(a)). Idem for magnetic field (Fig.A.3(b)).
– Scalar fields have a magnitude at every point, like average temperature.
– Vector fields have both magnitude and direction.
~ and magnetic, B,
~ fields.
• Charges and magnets create electric, E,
~ and B
~
• Electromagnetic force on charge q moving with velocity v in E
~ + q ~v × B
~
F = qE
• Maxwell’s Equations: equations of motion of EM
– summarise all previous work of Coulomb, Oersted, Ampere, Faraday, ...
– extended relationships and symmetries between electricity and magnetism
15
Figure A.3: (a) Representative electric field lines
generated by two charges equal in magnitude and
opposite in sign. (b) Representative magneic
field lines around a straight wire carrying an
electric current. The current emerges perpendicular to the plane of the page.
The electric/magnetic field is tangential to the
line. The density of the lines is proportional to
the field strength.
Gauss0 law for electricity :
Gauss0 law for magnetism :
Faraday0s law of induction :
Ampere0 s law :
~ ·E
~
∇
~ ·B
~
∇
~ ×E
~
∇
~ ×B
~
∇
=
ρ
=
0
~
= − ∂∂tB
~
= ~j + ∂∂tE
• Maxwell’s theory predicts that
~ and B
~ fields at right angles and perpen– EM waves consist of changing E
dicular to the direction of propagation, (Fig. 2.1(Beiser)),
– EM waves propagate with the same speed of light, c
c =
√1
o µo
= 2.998 × 108 ms
o and µo are the electric permitivity and magnetic permeability of free
space.
• Unification of not only electricity and magnetism but also of light and optics
in one single theory,
• Herz, 1888: generation of EM waves in laboratory.
=⇒ The picture of particles interacting at a distance gives place to a picture
of interaction by contact through a field, such that energy and momentum are
16
Figure 2.1(Beiser): Electro~
magnetic waves consist of E
~ fields that vary toand B
gether at right angles and
perpendicular to the direction of propagation of the
wave.
overall conserved.
17
0.2
General concepts in Quantum physics
Quantum mechanics
• describes all physical phenomena at the microscopic level,
• generally disagrees with classical mechanics
– numerical disagreement pronounced in the domain of very small
– deviations practically negligeable at the macroscopic level.
• led to a large number of pratical applications:
– laser, electron microscope, silicon chip, superconductivity, ...
– new applications are still being found!
– ... We are just at the beginning ...
0.2.1
Heisenberg’s uncertainty principle
• Classical mechanics:
– State of particle at any instant t specified by momentum p~ = m~v and
position ~x, simulnateously
– + Knowledge of acting forces F~ =⇒ trajectory completely determined
through Newton’s laws of motion.
∗ In general, accuracy of prediction limited by accuracy of initial data →
physics of chaotic systems
∗ In principle, nothing prevents to predict trajectory with absolute precision
– Particle assumed to have well defined trajectory (unobserved objective reality)
18
• Quantum mechanics
– p~ = m~v and ~x cannot be determined simulnateously with arbitrary high
degrees of precision, even in principle.
– Heisenberg’s uncertainty principle
h̄
h̄ ' 10−34J ṡ
2
const. h = 2πh̄ was introduced by Planck to explain black-body radiation.
∆x · ∆p ≥
∗ Measurement involved in determining any of p or x perturbs the other
∗ Particle of mass m in a box (∆x):
h̄
· ∆p ∼ ∆x
· Associated “zero-point” kinetic energy:
h̄ 2 1
∆x
2m
· For ∆x ∼ 1Å: ∆EK ∼ 3 eV for electron, corresponding to pressure
on the walls several million times Patm!
· the lighter the particle, the greater its “0-point” energy
· this energy prevents inert gases of light atoms from solidiying even
at 0◦K (Helium remains liquid down to lowest T known)
· this energy prevents the e− from collapsing onto the proton in H
– Other pairs of dynamical variables:
∆E · ∆t ≥
h̄
2
∆θ · ∆L ≥
19
h̄
2
0.2.2
Wave function and probability: state of the system
• State of particle given by a quantum mechanical wave function ψ(x, t)
– ψ is a complex function of position and time:
∗ ψ = A + iB =| ψ | eiθ
– ψ has a statistical interpretation:
∗ | ψ(x, t) |2 is probability (density) of finding particle at x and t
– ψ to be determined by solving a differential equation - the Schrödinger
equation - replacing Newton’s equations of motion
∗ For the simplest case of freely moving particle,
i
· wave function is a plane wave: ψ = Ae h̄ p~·~r
· with well defined momentum p~ and completely uncertain position
(| ψ(x, t) |2 the same for all points)
• Wave-particle duality
2
p
– Matter wave, or de Broglie wave, carries energy 2m
and has a wavelength
h
λ= p
∗ For a fee electron of energy 1 eV , λ ∼ 12 Å λlight.
=⇒ higher spatial resolution → electron microscope
∗ high energy accelerator resolve distances down to 10−20m!
– In water waves, the quantity that varies periodically is the height of the
water surface.
In sound waves, it is pressure.
In light waves, electric and magnetic fields vary.
For matter waves, the quantity that varies is the wave function, which is
related to the likelihood of finding the particle at position x and time t.
20
• Wave equation for matter-waves?
– Wave formula (ω = 2πν and v = λν):
x
x
ψ(x, t) = Ae−iω(t− v ) = Ae−2πi(νt− λ )
– E = hν = 2πh̄ν and λ =
h
p
=
2πh̄
p
i
ψ(x, t) = Ae− h̄ (Et−px)
– Wave equation:(equivalent of
∂2φ
∂x2
∂2ψ
p2
=
−
ψ
2
∂x
h̄2
∂ψ
iE
∂t = − h̄ ψ
– Total energy: E =
p2
2m
=
1 ∂2φ
)
v 2 ∂t2
2
=⇒ p2ψ = −h̄2 ∂∂xψ2
=⇒ Eψ = − h̄i ∂ψ
∂t
+ V (x, t)
p2 ψ
+Vψ
Eψ =
2m
– Substitution of E and p2 above
=⇒ Schrödinger equation in one spatial dimension
h̄2 d2ψ
∂ψ
−
+
V
ψ
=
ih̄
2m dx2
∂t
21
Quantum probability amplitude for a particle in a box
Schrödinger equation for particle of mass m and energy E, confined to a 1-dim box
by potential V :
h̄2 d2ψ(x)
+ V (x)ψ(x) = Eψ(x)
−
2m dx2
Assume that outside box V infinitely high so that probability of finding particle in
box (0 ≤ x ≤ L) is 1.
Particle is free inside, so that V (x) = 0 for 0 < x < L:
Figure 4.8(Hey): Energy levels of quantum particle in a box labelled by the quantum number n, wave patterns
and corresponding probabilities (square of wave amplitudes).
h̄2 d2ψ(x)
= Eψ(x).
−
2m dx2
22
Or, with the new variable k
2m E
d2ψ(x)
2
.
k =
=⇒ −k ψ(x) =
dx2
h̄2
Equation for simple harmonic motion, well known in classical physics.
Most general form of the wave function:
2
ψ(x) = A sin kx + B cos kx.
A, B: arbitrary constants.
Boundary conditions, that quantum amplitude vanishes at the two ends of the box:
ψ(0) = B = 0 ,
ψ(L) = A sin kL = 0
=⇒ k =
nπ
L
, n = 1, 2, 3, ...
n=1,2,3,... Condition for k means that energy is quantised with a magnitude determined by the main quantum number n:
π 2h̄2
.
E = n
2mL2
2
Determine constant A.
Probability of finding the partice in the box is 100%
Z
L
0
ψ(x)2 dx = 1.
2
Insert ψ(x) = A sin nπ
L x, use relation sin X =
A2 L
2
= 1 =⇒ A =
1−cos 2X
2
r
and integrate
2
L.
Quantum wave functions:
ψn(x) =
v
u
u
u
t
2
nπ
sin x
L
L
23
n = 1, 2, 3, ...
Wavefunctions together with a scale indicating the corresponding particle energy.
Two important comments:
• The lowest energy of the particle is not 0.
– This is in agreement with Heisenberg’s uncertainty principle,
h
,
L
requiring that, even at the ground state, the particle is jiggling around with
a so-called “zero-point energy”.
∆x ≤ L =⇒ ∆p ≥
• The probability amplitude for the electron in the box vanishes not only at the
ends (n = 1), but also at other places for the higher energy states (n > 1).
24
Hydrogen atom
• Electron bound to proton by Coulomb attraction
2
e
– electric potential energy V (r) = − 4π
0r
– Schrödinger equation in cartesian coordinates:


h̄2  ∂ 2ψ ∂ 2ψ ∂ 2ψ 

−
+
+ 2  + V ψ = Eψ
2m ∂x2 ∂y 2
∂z
√
In spherical polar coordinates: r = x2 + y 2 + z 2 , θ = cos−1 zr ,
φ = tan−1 xy
"
h̄2 1 ∂
− 2m
r 2 ∂r
r2 ∂ψ
∂r
+
1
∂
r 2 sin θ ∂θ
sin θ ∂ψ
∂θ
+
∂2ψ
1
2
2
r sin θ ∂φ2
#
+ V ψ = Eψ
– Set of stationary states is discrete to only allowed values of energy, angular
momentum and its component along a chosen direction
ψ(r, θ, φ) = Rnl (r)Θlm(θ)Φm (φ) = ψnlm
∗ n = 1, 2, 3, .. is the principal quantum number leading to quantisation
of energy
1
E1
E1 = 1 Rydberg = 13.6 eV
n2
∗ l = 0(s), 1(p), 2(d), 3(f ), ..., n − 1 is the orbital quantum number giving
the angular momentum in units of h̄
En = −
L2
h̄2
El =
=−
l(l + 1)
2mr
2mr
r
L = l(l + 1)h̄ is the electron angular momentum
Atomic electron states:
25
n = 1, l = 0(1s);
n = 2, l = 0(2s), l = 1(2p);
n = 3, l = 0(3s), l = 1(3p), l = 2(3d), etc...
√
∗ An electron in l = 2 has L = 6h̄ = 2.6 × 10−34J · s.
By contrast, the orbital angular momentum of the Earth is 2.7×1040J ·s.
The angular momentum is so large that the separation into discrete
angular momentum states cannot be experimentally observed.
∗ −l ≤ m ≤ l = 0, 1, 2, ..., n − 1 is the magnetic quantum number
· gives the component of angular momentum along any arbitrary chosen axis.
· leads to the quantisation of space: Lz = mh̄.
(Below) Idea of an atom,
with electrons looping
around the nucleus, before the advent of quantum mechanics. (Right)
Some plots of where it
is most likely to find an
electron in a hydrogen
atom (the nucleus is at
the center of each plot).
– Stern-Gerlach, 1922:
26
∗
∗
∗
∗
Atomic particles possess an intrinsic angular momentum, or spin,
completely quantum mechanical property, no classical equivalent
Spin is quantized
Magnetic resonance imaging (MRI):
· Under certain conditions the spin of hydrogen nuclei can be ’flipped’
from one state to another.
· By measuring flip location, a picture can be formed of where H atoms
(mainly as a part of water) are in a body.
· Since tumors tend to have a different water concentration from the
surrounding tissue, they would stand out in such a picture.
• Electromagnetic waves, classically described by Maxwell’s equations, must also
be quantised.
Electron
transition from 1st ES of
hydrogen (n = 2) to GS
(n = 1) with emission of
a photon.
Dashed line is ionisation energy, where an
electron has enough energy to overcome the
attraction of the proton
and move away from the
atom, leaving a positively charged hydrogen
ion or proton.
Figure
4.14(Hey):
– The resulting quantum is a “packet” of energy - a photon
27
∗ Photon carries energy and momentum
hc
h
= hν ;
p=
λ
λ
∗ Interaction of quantised radiation with matter involves
· absorption of a quantum accompanied by an electronic transition
from a lower energy state 1 to a higher energy state 2
E=
E2 − E1 = hν
· emission of a quantum accompanied by an electronic transition from
a higher energy state 2 to a lower energy state 1
E1 − E2 = hν
28
Figure 4.15(Hey):
Hydrogen energy level diagram: various series of spectral lines.
Electron jumps (arrowed
lines)
lead to photon emissions Eγ = hν.
Balmer series λ1 = R 212 − n12 , R = 1.097 × 107m−1 , lead to visible photons.
Lyman series (transtions to n = 1) lead to HE ultraviolet photons.
Paschen and Brackett series lead to LE infrared photons.
Transitions corresponding to capture of a free electron at the ionisation energy
(dashed lines).
29
∗ This is the origin of energy spectrum characterictic of atoms, molecules,
...
· line spectra extensively used in laboratories to fingerprint atoms
· Astronomers determine the elemental composition of distant stars
from spectrum of light received.
– Similarly, the sound wave-like oscillatory motion of atoms in a solid also
must be quantised. The resulting quantum is the phonon
30
31
Quantum tunneling
One of processes forbidden classically but allowed quantum mechanically
• Particle with E < U of a potential barrier
– reflected back classically, cannot escape over the barrier
– can tunnel through it due to waviness of the quantum wave function
– de Broglie waves (ψI+) corresponding to the particle are partly reflected
(ψI− ) and partly transmitted (ψIII+)
– particle has a finite chance of penetrating the barrier.
Figure 5.9(Beiser)
• The transmission probability for a particle to pass through the barrier is proportional to e−L :
√
2m(U −E)
T = e−2k2L ; k2 =
L: barrier thickeness
h̄
32
– Tunneling allows electrons to jump across thin insulating layers separating
metallic or superconducting electrodes
→ Scanning Tunneling Microscope, ...
→ Nuclear α-decay, fission, fusion reactions in stars
∗ An α particle of energy a few M eV escapes a radioactive nucleus whose
potential well is ∼ 25 M eV high.
∗ T ∼ 10−38, small but possible!
Potential of 2 protons is a combination of a repulsive electric potential
and a mainly attractive nuclear potential. To get closer together - fusion, they
must traverse a classically forbidden region (Coulomb barrier) by quantum
tunneling.
Figure 8.4:
33
0.2.3
Superposition of states and wave interference
Characteristic feature of quantum mechanical waves
• If ψ1 and ψ2 are two possible states, solutions of Schrödinger equation
– A linear combination (superposition) is also a solution
ψ = a 1 ψ1 + a 2 ψ2
where a1 and a2 are constants
– ψ1 and ψ2 - like other waves - obey principle of superposition
∗ Wave functions add
– Probability densities (assume a1 = a2 = 1):
P1 =| ψ1 |2= ψ1∗ψ1
P2 =| ψ2 |2= ψ2∗ψ2
∗
∗
ψ1} 6= P1 + P2
ψ
+
ψ
P = P1 + P2 + ψ
2
2
1
|
{z
in general 6=0
∗ Probabilities do not add!
34
• Probability waves interfere and diffract as any other waves (mechanical, light)
→ phenomenon of electron and neutron diffraction
∗ Of great practical use in studying crystal structures
→ Young’s double slit experiment
Double slot experiment
for electrons: Distribution of electrons when
both slots 1 and 2 are
open. The interference
pattern disappears when
only slot1 or 2 is open.
Double slot experiment
for light, first carried out
by Young in 1801. The
interference pattern was
interpreted as evidence
for the wave nature of
light.
∗ the electron can propagate through the 2 slits as alternatives
∗ the interference is between these 2 alternatives.
35
0.2.4
Indistinguishability of identical particles
• Classically,
– identical particles remain distinguishable
∗ spatially separated
∗ trajectories can be continuously followed, in principle
• Quantum mechanically,
– identical particles are indistinguishable
∗ Wave function of the pair: ψ(r1, r2)
∗ | ψ(r1, r2) |2 is probability density of finding one at r1, the other at r2
∗ No specification which one!
– under interchange of 1 and 2, wave function
∗ either remains the same - symmetrical WF
∗ or changes sign - antisymmetrical WF
• Fermions with odd half-integer spins ( 21 , 32 ,
5
2
...): e, p, n, 3He
– only one fermion can exist in a particular quantum state of the system
– Do obey Pauli’s exclusion principle.
– Antisymmetric wave function, changes sign upon the exchange of any pair
(1,2) of them
ψF =
√1
2
[ψa(1)ψb (2) − ψa(2)ψb (1)]
(a) and (b) denote quantum states. ψF = 0 for a = b!
36
• Bosons with 0 or integral spins (1, 2, ...): 4He, γ
– Do not obey Pauli’s exclusion principle
– Any number of bosons can exist in a particular quantum state of the system.
– Symmetric wave function, not affected by the exchange of any pair (1,2) of
them
ψB =
√1
2
[ψa (1)ψb(2) + ψa(2)ψb (1)]
37
All particles have a property called spin, having to do with
what the particle looks like from different directions.
The higher the spin, the smaller the fraction of a complete revolution necessary to have the particle look the same.
Spin- 21 particles only look the same if one turns them through 2 complete revolutions.
Figure 2.12 (Hawking)
38
0.2.5
Quantum mechanics and naive realism
• Quantum Mechanics
– Forbids the simultaneous measurement of position x and momentum p
=⇒ no trajectories
– Denies their objective existence, independent of measurement
∗ e− in a room in a state of potentiality
∗ described by probabilistic wave function
∗ state “collapses” to a certain point when detected
• Philosofically disturbing
– Attempts to introduce “hidden” variables, such that
∗ observed particles behave apparently probabilistically
∗ while entire system particle+hidden variables is deterministic in spirit
of classical statistical mechanics.
– ruled out by crucial experiments
∗ Bell’s inequalities and Aspen’s experiment
∗ Entanglement and Quantum teleportation.
• There are still
– paradoxes,
– philosofical uneasiness
• BUT: Great agreement of QM predictions with experiments
39
0.3
Special relativity
• Einstein, 1905
• Postulates of Special Relativity (SR)
1. Principle of relativity: The laws of physics are the same in all inertial
frames of reference.
2. The speed of light in free space has the same value in all inertial frames of
reference: c = 2.998 × 108 ms .
• Maxwell’s theory consistent with SR
• Newtonian mechanics
– agrees with SR for relative speeds v c
– fails at higher speeds v ∼ c
=⇒ must be replaced by SR
• SR replaces 3-dim Euclidean space and time (x, y, z; t) by 4-dim Minkowski
space-time (x, y, z, t)
– t is treated as another coordinate to label the events.
– absolute space invariant r12 and time interval t12 between 2 events 1 and 2
replaced by single absolute interval s12



2
r12
2
2
= (x1 − x2) + (y1 − y2 ) + (z1 − z2

 t
12 = t1 − t2
40


)2 


2
−→ s212 = r12
− c2t212
• Galilean transformation replaced by Lorentz transformation:
0
x =
rx−vt
2
1− v2
c
= γ (x − vt)
;
0
t =
t− vx2
r
c
2
1− v2
c
=γ t−
vx
c2
Frame S 0 moves in the +x direction with speed v relative to S.
γ=
relativistic factor :
– Ultra-relativistic case: v → c =⇒ β =
1
= √1
.
r
2
1− v2
c
v
c
→ 1 =⇒ γ → ∞
1−β 2
– non-relativistic case: v → 0 =⇒ γ → 1, x0 → x − vt, t0 → t
– Velocity addition:
Vx =
Vx0 +v
V 0v
1+ x2
c
– If light is emitted in moving frame S 0 , Vx0 = c, then Vx = c also from frame
S.
v c =⇒ Vx = Vx0 + v
0.3.1
Popular effects of SR
• Time dilatation
– A moving clock ticks more slowly than a clock at rest.
∗ Measurements of time intervals affected by relative motion
∗ Time dilatation:
t = r t0 v2 = γt0
1− 2
c
· t0 = proper time (clock at rest); t = relative time (clock in motion);
v = speed of relative motion.
• Length contraction
– Faster means shorter.
41
∗ In the direction of motion: length L of an object in motion w.r.t an
observer appears to the observer to be shorter than the proper length
L0 (at rest w.r.t observer):
r
2
∗ Lorentz length contraction: L = L0 1 − vc2 = Lγ0
• Variation of mass with energy
– Redefinition of mass:
m(v) =
r
m = proper or rest mass.
– Redefinition of momentum:
p~ =
m
= γm
2
1− v2
c
v
r m~
2
1− v2
c
= γm~v
Comparison
between the classical and relativistic expressions for the
momentum p of an object
moving at the velovity v relative to an observer. m is the
mass at rest.
Figure 1.14(Beiser):
– Relativistic “second law”
F~ =
d~
p
dt
d
dt
=
F =
∗ If constant force, a =
F
m
1−
(γm~v )
m
3
3 = γ ma
3
v2 2
c2
42
2
1− v2
2
c
decreases when v increase
∗ v → c =⇒ a → 0
∗ Massive objects can never reach speed of light!
• Equivalence of mass and energy
– Kinetic Energy:
R mv
R
R
=
EK = 0s F · ds = 0s d(γmv)
0 vd (γmv) , ...
dt
EK = (γ − 1) mc2
∗ At low speeds:
v c =⇒ EK ∼ 12 mv 2
Comparison between the classical and relativistic expressions for the ratio between the kinetic energy
KE of a moving body and its
rest energy mc2.
At low speeds the 2 formulas
give the same result, but they
diverge at speeds approaching
that of light.
According to relativistic mechanics, a body would need ∞
kinetic energy to travel with
v = c, whereas in classical mechanics
it would need KE =
mc2
2 .
Figure 1.16(Beiser):
– Total Energy:
– Rest Energy:
E = γmc2 =
E0 = mc2
rmc
2
2
1− v2
43
c
E 2 = m 2 c4 + p 2 c2
– Energy and Momentum:
E 2 − p2c2 is invariant.
– Massless particles:
E = p · c
If E [eV ], p eVc and m eV
.
c2
• Doppler shift
– The speed of light emitted from a moving source is always c,
– wheras energy of individual photons depends on the velovity of the source.
∗ Consider a source of photons E = p · c in rest frame S of source.
∗ In a frame S 0 moving towards the photon source with a speed v = βc,
the observed energy E 0 is
0
E = γ (E + βp c) = γE (1 + β) = E
E=
hc
λ
; E0 =
hc
λ0
=⇒
λ0
λ
=
s
s
1+β
1−β
1−β
.
1+β
=⇒ blue shift: Photon wavelength seen from S 0 is shorter.
E 0 > E =⇒ ν 0 > ν =⇒ λ0 < λ.
∗ If photons are measured in a frame moving away from the photon source
with v = βc,
0
s
1−β
1+β
=
s
E = γ (E − βp c) = γE (1 − β) = E
λ0
λ
=
s
1+β
1−β
ν
ν0
=⇒
E0
E
=
1+β
1−β .
=⇒ red shift: Photon wavelength seen from S 0 is longer.
E 0 < E =⇒ ν 0 < ν =⇒ λ0 > λ.
44
s
1−β
1+β
The Doppler shift.
The photons from a source
at rest have a wavelength λ.
When there is relative motion
of the source towards (away
from) the observer with speed
v = βc, the wavelength of the
photons is observed to be blueshifted (red-shifted)
to λ0 =
!
s
s
1−β
1+β
0
1+β λ = 1−β .
Figure:
• Doppler also
– well-known acoustic analogue
– radar used to check for overspeeding
– red-shift of light from distant galaxies lead to Expansion of Universe:
Hubble, 1929.
– tunability of light-atom interaction allows adaptive and selective Laser
cooling of atoms down to µK
45
0.4
General relativity
• In special relativity, observers’ perceptions of time and space modified by factors
depending on their relative velocities.
=⇒ During acceleration(changing velocity) an observer’s scales of time and space
must become distorted.
• Principle of equivalence:
– An acceleration is identical to the effects of a gravitational field.
– Gravitational field must give rise to a distortion of space-time.
• General relativity suggests that bodies, instead of moving under the influence of a gravitational force, they are rather moving freely through a
warped, or curved, space-time. The force of gravity is reduced to the curved
geometry of space-time.
– Einstein’s general relativity
explains warping of space-time
around a mass-source.
– Effect of gravity on trajectory
of a passing particle is analogous to rolling a marble across
the curved rubber sheet.
46
• Main features of General Relativity, Einstein 1917:
– Space and space-time are
∗ not rigid arenas in which events take place
∗ have form and structure which are influenced by the matter and energy
content of the universe.
– Matter and energy tell space-time how to curve.
– Space-time tells matter how to move.
∗ Small objects travel along the straightest possible lines in curved spacetime
∗ In curved space the rules of Euclidean geometry are changed.
Parallel lines can meet, and the sum of the angles in a triangle can be
more, or less than 180 degrees, depending on how space is curved.
• Einstein’s interpretation of gravity as geometry
– Field equations of general relativity:
Gµν
= 8πG ×
Tµν
(geometry of space − time) = 8πG × (mass and energy) .
– Metrics: dr 2
2M G
2
2
2
2
2
2
2
ds = −c 1 − c2r dt + 1− 2M G + r dθ + sin θdφ .
c2 r
Remember:
2
2
2
2
2
2
Spherical coordinates: dl = dr + r dθ + sin θdφ .
Special relativity: ds2 = dr2 − c2dt2.
47
• Successes of Einstein’s theory
– Deflection of light in gravitational field (photons massless but have energy)
∗ Confirmed in 1919 solar eclipse
∗ Gravitational lensing.
Source (galaxy, quasar, ...) emits
light (or radio waves).
Lens (dense concentration of
mass/energy) deflects light.
Observer sees two or more identical apparent images.
First observation: 1979.
– Explains why objects fall independent of their mass:
∗ they all follow the same straightest possible line in curved space-time.
48
– Correct prediction for the perihelion shift of Mercury.
∗ perihelion precesses ∼ 1.6◦ per century.
∗ ∼ 4000 not due to attractions of other
planets.
∗ discrepancy thought to be undiscovered
planet (Vulcan)
∗ Einstein’s prediction: 4300 !
– instantaneous gravitational force replaced by curvature of spacetime.
∗ Theory predicts existence of gravitational waves - ripples - propagating
through space.
∗ Moving a mass (or some changes in a mass-source, such as collapse of a
star or black hole) causes ripples to form in the curvature.
∗ Ripples travel with speed as light
∗ A distant mass would not feel any instantaneous change in the gravitational force (SR not violated)
∗ considerable experinental effort devoted to attempts to detect such waves
∗ no success sofar, probably disturbances are too small.
49
• Photons and gravity:
– Other way to approach gravitational behaviour of light.
∗
∗
∗
∗
Photon massless but interacts as though it has inertial mass: m =
√
Mass m from height H: 21 mv 2 = mgH =⇒ v = 2gH
Photon that “falls” from H cannot get faster (v = c)!
Increase of mgH in its energy by increase in frequency ν → ν 0
· ∆ν extremely small in laboratory
experiment
gH
· hν 0 = hν + mgH = hν + hν
c2
0
· Photon energy after falling height H: hν = hν 1 +
gH
c2
hν
.
c2
∗ Increase in energy of falling photon first observed in 1960 (Pound, Rebka).
· H = 22.5 m, ν = 7.3 × 1014Hz =⇒ ∆ν = 1.8 Hz
50
• Gravitational red shift:
– Frequency of photon moving towards earth increases
=⇒ Frequency of photon moving away from it should decrease!
∗ Potential energy of mass m on star’s
m
surface: U = − GM
.
R
∗ For a photon of “mass”
hν
:
c2
hν
U = − GM
.
2
c R
GM hν
c2 R
GM
c2 R
· Total energy: E = hν −
= hν 1 −
· Far from strong gravitational field (photon arriving on earth where
weak field negligeable): E = hν 0
0
GM
0
· hν = hν 1 − c2R =⇒ νν = 1 − GM
c2 R
= GM
∗ Gravitational red shift: ∆ν
ν
c2 R
· Photon leaving Star has lower frequency when at earth.
· Visible region of spectrum shifted towards red end
· Different from Doppler red shift in spectra of distant galaxies (due to
apparent recession from earth)
negligeable.
∗ For most stars M
R
∗ Observed for white dwarfs (limit of measurement)
A white dwarf is an old very small star with collapsed electron structures.
Typically of size of earth with mass of sun.
51
• Black holes (BH): → chapter 8
– Star so dense that
GM
c2 R
≥1
∗ No photon can ever leave the star (this would require more energy than
initial hν).
∗ Star does not radiate, is invisible
∗ Black hole in space.
– Correct criterion:
GM
c2 R
≥
1
2
– Schwarzschild radius: Rs =
∗
∗
∗
∗
2GM
c2
Body is a BH if all its mass inside sphere with radius Rs.
Boundary of black hole: event horizon.
Escape speed from a BH is v = c at Rs
Star with m = m, RS = 3 km
– (Invisible) BH detection
∗ BH member of double-star system
· gravitational pull on the other star
· attraction of matter from the other star followed by
compression and heating =⇒ X-ray emission
· Cygnus X-1: m = 8 m, R ∼ 10 km
∗ Very heavy stars end as BH
∗ Lighter stars → white dwarfs or neutron stars → BH, once they attracted enough dust and gas.
∗ BH in core of galaxies
· motion of nearby bodies + amount and type of emitted radiation
52
· Stars close to galactic center observed to move so rapidly that only
gravitational pull of an immense mass could keep them in their orbits
instead of flying off.
· m ∼ 109m , copious radiation
∗ Quasars (quast-stellar radio sources)
· powerful sources of radio waves
· believed that BH (m ∼ 108m) at the heart of every quasar
∗ “Mini BH” in high energy collisions vs extra space dimensions
53
0.5
Quantum gravity?
• Einstein’s GR still a classical theory;
– does not account for gravity in quantum-mechanical regime.
• A successful quantum theory of gravity not yet been formulated
– reconciliation of general relativity with quantum mechanics is one of the
major outstanding problems in theoretical physics.
• First few steps towards such a theory (Analogy with Quantum ElectroDynamics, QED).
– Gravitational field consists of microscopic quanta called gravitons
∗ Gravitons must be massless (to accommodate the infinite range of gravity)
∗ of spin 2 (for consistency with general relativity).
– Gravitational force between two masses can be described as an exchange of
gravitons between them.
– Problems arise:
∗ unlike QED, certain graviton sub-processes diverge (always seem to occur with an intinite probability)
∗ Quantum Gravity is not renormalisable.
0.6
Superstring theory?
54
0.7
0.7.1
Thermal physics and statistical mechanics
Thermodynamics
Heat is a form of energy and flows from hot spot to cold.
• Heat energy in a macroscopic body carried
– by random motion of its microscopic constituents (molecules, atoms,
elementary particles,...)
– through collisions
• There is sharing of energy until thermal equilibrium is reached.
• Absolute temperature measured on the Kelvin scale (K),
K = C + 273.15◦
– Water feezes at 0◦C or 273.15◦K.
– Room temperature is 26.85◦C or 300◦K.
– At 0◦K all random motions stop and no more heat energy is left in body.
• A system tends to have less random motion at low temperature, T ,
– but the ordered nature of it is not completely determined by T .
∗ At 0◦C, ice (orderly crystal structure) and water (less order) can coexist,
although at same T .
• Orderliness (or lack of it) of constituents of matter is determined by another
thermodynamic quantity, entropy, S.
55
– At constant T = 0◦C,
∗ to freeze water completely,
∗ must extract its latent heat ∆Q (freezer)
∗ System more orderly with less Entropy S.
– Definition of entropy:
∆Q = T S
• Free energy, F replaces mechanical energy in abscence of heat, U
F = U − TS
– F minimal for thermodynamic stability.
• A substance can have different phases
– H2 O has 3 phases:
∗ Ice: T < 0◦C; Water: 0 < T < 100◦C; Steam: T > 100◦C.
∗ Phase transitions occur at transition T = 0◦C and 100◦C.
• The lower the temperature, the more orderly the phase
• Phase at low T has less Symmetry than that at high T
• Other important phase transitions:
– Change from normal metal to Superconductor at low T
– As the universe cooled down from the Big Bang towards current 2.7◦K,
various phases took place =⇒ Cosmo − Astro − Particlephysics
– Unification scales in Particle Physics
– Higgs mechanism of Spontaneous Symmetry Breaking, responsible for
generating masses of elementary particles in the Standard Model
56
0.7.2
Statistical mechanics
At thermal equilibrium, constituents carry same amount of energy on average.
• Statistical mechanics
– studies how overall behaviour of a system is related to properties of its
constituents.
– is not concerned about the motions and interactions of individual particles.
– determines the energy distribution, the most probable way a certain energy
is distributed among the constituents.
• Number of particles with energy :
n() = g() f ()
– g(): statistical weight, number of states of energy ,
– f (): distribution function, probability of occupancy of each state of energy
.
Consider 3 types of (identical) particles:
• Distinguishable particles sufficiently far from each other (molecules in a gas)
– The total number of particles in the macroscopic system is small compared
to the number of quantum states available.
– Wave functions nearly do not overlap,
→ unlikely that 2 particles occupy the same quantum state
57
– obey the Maxwell-Boltzmann distribution function (Fig. C.1)
dN
d3 p
= fM B () = C e− kT
which gives the number of particles per unit momentum volume ddN
3 p with an
energy at a given temperature T . C is a normalisation factor determined
by the total number of particles in the system.
MaxwellBoltzmann distribution
The average energy per
point particle is 23 kT .
Figure
C.1:
– Average energy per degre of freedom for a non-relativistic particle is
E =
1
2k
T
k = 1.38 × 10−23J/K is Bolzmann constant, T absolute temperature.
∗ At room temperature T = 300◦K, E = 2.07 × 10−21J = 0.013 eV .
∗ Each degree of temperature carries E = 86 µeV .
· Point particle, 3 d.o.f, E = 23 kT
· Rigid body, 6 d.o.f, E = 3kT
58
• Undistinguishable particles with overlapping wave functions are of 2 groups:
– Fermions with odd half-integer spins ( 12 , 23 ,
∗
∗
∗
∗
5
2
...): e, p, n
Do obey Pauli exclusion principle
only one fermion can exist in a particular quantum state of the system.
Antisymmetric wave function, changes sign upon permutation.
obey the Fermi-Dirac distribution function
dN
d3 p
= fF D () =
1
(−F )
e kT
+1
F is the Fermi energy ≡ energy of the last occupied state:
Fermi-Dirac
distribution with Fermi
energy F .
Figure C.2(a):
F ≡ kTF =
h̄2
2m
!
2
3π 2 N 3
V
At T = 0,
fF D ( = F ) =
1
2
fF D ( < F ) = 1
fF D ( > F ) = 0
N
V
is the density of free particles of mass m.
∗ At T = 0,
59
· all energy states up to F are occupied and none above (solid line of
Fig. C.2(a))
∗ When T > 0◦K,
· distribution smears out (Fig. C.2(a)), as long as T T0,
T0: degeneracy temperature, below which quantum quantum effects
become important:
T0 = T
2
N 3
N0
= 2π
2
N 3 h̄2
V
mk
∼
TF
1.31
∗ As T T0, the distribution approches the Boltzmann distribution
(Fig. C.1).
22
−3
◦
Proton: mH ∼ 2 × 10−24g, N
V = 10 cm , T0 ∼ 1 K
Photon: mγ = 0, T0 → ∞ and photons are always degenerate.
idem for e− in metals: me ∼ 10−27g, N
= 1022−23 cm−3, T0 ∼ 16 − 20 ×
V
103K.
– Bosons with 0 or integral spins (1, 2, ...): 4He, γ
∗ Do not obey Pauli exclusion principle
∗ Any number of bosons can exist in a particular quantum state of the
system.
∗ Symmetric wave function, not affected by permutation.
∗ obey the Bose-Einstein distribution function
dN
d3 p
= fBE () =
1
eα e kT −1
· At T = 0, particles occupy the lowest possible quantum state (vertical
bar of Fig. C.2(b))
· As T rises, more and more particles are pumped to higher energy
states (dashed line of Fig. C.2(b))
60
Bose-Eisnstein
distribution, where the
heavy vertical line shows
Bose condensation.
Figure C.2(b):
· For T < Tc, condensation temperature (Tc ∼ 0.527 T0), enough
particles remain in the lower energy state. Tc ∼ 2.19 − 3.1 K for
helium.
· QM phenomenon of Bose-Einstein condensation
Notice that there is No BE condensation of photons (Eγ 6= 0!), so
they must disappear absorbed by container wall.
· Superfluidity and Superconductivity (e− e− ≡ boson)
61
Figure 9.5(Beiser): A comparison of the three statistical distribution functions.
Applies to systems of
Category of particles
Properties of particles
Examples
Properties
Maxwell-Boltzmann
identical, distinguishable
particles
Bose-Einstein
Fermi-Dirac
identical, indistinguishable identical, indistinguishable
particles that do not
particles that obey
obey exclusion principle
exclusion principle
Classical
Bosons
Fermions
any spin, particles far
Spin 0,1,2, ...;
spin 21 , 32 , 52 , ...;
enough apart so WFs
WFs symmetric to
WFs antisymmetric to
do not overlap
interchange of particles
interchange of particles
molecules of a gas
Photons in a cavity;
free electrons in a metal;
phonons i a solid; liquid
electrons in a white dwarf
He at low temperature
no limit to number of
no limit to number of
never more than 1 particle
particles per state
particles per state; more
per state; fewer particles per
particles per state than
state than fM B at LEs;
fM B at LEs; approaches
approaches fM B at HEs
fM B at HEs
LE: low energies; HE: high energies; WF: wave function
62
=⇒ Bose-Einstein distribution for photons known as Planck’s distribution,
• Blackbody Radiation:
– Blackbody at T > 0◦K emits EM radiation with continuous spectral distribution(intensity versus wavelength or frequency).
63
– Thermal radiation called “blackbody radiation”
– Mathematical form of the spectral energy density: Max Planck’s black-body
radiation law → Energy per unit volume per unit frequency/wavelength:
ρ(ν) =
8πh ν 3
c3 hν
e kT −1
ρ(λ) =
8πhc
λ5
1
hc
e λkT
−1
∗ h is Planck’s constant (1900) which connects energy and frequency:
E = hν = h λc ,
· first appearance of quantum theory in physics,
· Seeds for quantum mechanics.
∗ c is speed of light,
∗ k is Boltzmann’s constant: converts T units → E through E ∼ k T
· (particle ←→ statistics).
• Distribution fixed for given T =⇒ its measure determines the temperature of
a (even unreachable) body.
– Temperature in a steel furnace
– Temperature of a distant star
– Temperature of the Microwave Background Radiation
64
65
• Two special aspects of Planck’s distribution
– Wien’s law.
∗ λmax at the peak of Fig. C.3 is inversely ∝ T :
Wiens 0s law : λmax T =
hc
4.965 k
hc
kT λmax
= 4.965
= 0.2898 cm · K
∗ At room temperature T = 300 K, λmax ∼ 10 µm λvis ∼ 0.4−0.7µm
∗ Tsun ∼ 6000 K, λmax ∼ 0.5 µm and we can see all colors of rainbow at
sunrise and sunset!
Figure C.3: Planck distribution
dρ
dλ = f (λ) at a given temper-
ature T .
The dotted line joins the
peaks at different temperatures (Wien’s law).
66
– Stefan-Bolzmann’s law:
∗ Total energy density ρ of the radiation in a box is given by the integral
of the energy density over all frequencies:
ρ =
R∞
0
ρ(ν)dν =
8π 5 k 4 4
T
15c3 h3
= a T4
a = 7.56 × 10−16J · m−3K −4 = 4.73 × 10−3eV · cm−3K −4 .
∗ The energy ρ radiated by an object per second per unit area is also ∝
T4
Stef − Boltz law : f = eσT 4
Stefan 0s const :
σ = ac4 = 5.67 × 10−8W · m−2K −4,
emissivity 0 < e < 1 depends on nature of radiating surface, 0 for a
perfect reflector and 1 for a blackbody.
∗ Other form - Number of photons per unit volume:
nγ = b T 3, b = 20.4cm−3K −3.
– Wien’s law tells what color the radiation has;
– Seffen-Botzmann’s law tells how bright the radiation is.
– Hot object are so bright
67
0.7.3
Dimensional analysis
Every quantity in physics carries a dimension made up of 3 fundamental units:
cm, g, s or m, kg, s, ...
• Dimensional analysis:
– Whatever quantity we consider must have the right units.
– If we also know on physical grounds what the quantity depends on, this
yields valuable information.
• Deduce that photon number density nγ ∝ T 3:
nγ
1
cm3
,c
−3 −3
3
= f (T K = erg(k = 1) =
3
2
g cm
s2
– Only combination leads to 1/cm : c h̄ T
3
nγ = b0 h̄T3c3
cm
s
"
, h̄ erg · s =
cm −3
(erg
s
b0 : dimensionless
– Argument valid for relativistic particles (m 68
kT
).
c2
· s)
2
g cms
−3
erg
3
#
=⇒ Fundamental research vs Applied research?
=⇒ Simple vs Complex?
=⇒ Microcosmos vs Macroscosmos?
=⇒ Theory vs Experiment?
=⇒ Science vs Science fiction?
=⇒ Allowed vs Forbidden?
69