Download Quantum mechanics

PHYS541000
Quantum Mechanics(I)(II)
Chung-Yu Mou
Unique subjects in
-- quantum mechanics and
physics
cosmology
Cosmology：
closest subject to God、can
understand how God creates the w
Quantum Mechanics：
Describing the world by using the w
that electrons can pass two slits
at the same time
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http://www.phys.nthu.edu.tw/~mou/teach.htm
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http://www.phys.nthu.edu.tw/~mou/teach/16Fall_QuantumMech.ht
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s
s
Quantum mechanics –
The new way that was developed at the
beginning of the 20th century to interpret
& predict behaviors of microscopic objects
such as atoms, electrons, ..
Purpose of this course
Learn to calculate and think in
the quantum mechanical way
Conceptual problem will not be fully
discussed in this course.
In fact, as Richard Feynman remarked:
s
I think that it is fair to say that no one
understands the quantum theory…
Two tracks of discoveries
h
Spectroscopy
(discrete)
Photoelectric
Effect (Einstein)
Theory of H
Atom (Bohr)
Matter wave
(De Broglie)
Matrix Mechanics
(Heisenberg)
Wave Mechanics
(Schrodinger)
Equivalent
(shown by Schrodinger)
(based on the general formalism of Dirac)
Third Formulation
Richard Feynman
Path Integral
Equivalent to the above two formulations
but may be more general
Track 2: particle-wave
controversy
Huygens: light is wave
Newton: light is particle
Hook: wave
Maxwell: wave
Young: wave
Planck, Einstein, and
Compton: particle
..
The Maxwell’s theory of
light
E
Light intensity (I)
~ |E|2
B
Wave-like properties of
light
Hygen’s principle
Interference
Diffractio
n
Particle-like properties
of light
Pierre Gassendi and Newton
Reflection:
elastic collision of
particles
Refraction：
1
If q1 > q 2
 1y
u y = u x cot q
u 1y < u 2y \u 1 < u 2
2
 y2
However, using n1 sinq1 = n2 sinq 2
n1 < n2 (u = c / n)
Radiation and black-body radiation:
all objects radiate
electromagnetic waves
Thermodynamics of
Radiation
El =
Energy of emitted EM wave with l
per unit area ×sec
Al = energy absorption rate for wavelength l , depending on properties of the object.
Il =
El = Al I l
Energy of incident wave with l
= energy flux for wavelength l
per unit area ´ sec
El
=I l is independent of properties of the object
Al
Black Body
Radiation
El
is universal, independent of the composition
Al
and geometry of the object.
If Aλ=1, Eλ= f(λ,T) is universal
=> black body radiation
Ideal black body ＝ cavity
Universe » cavity: cosmic microwave background
T = 2.715K
Max Planck
For a given n , there are about n 2 waves with different directions
Classically, the averaged energy for each wave is kBT
Expt & Planck
Planck’s consideration is based on statistical
mechanics not dynamics
In an equilibrium system
The probability of finding a subsystem
with the energy E is e
(e = 2.71828..)
-
E
k BT
Plank: energy of light is not continuous
basic unit is hn ,
average of energy = hn e
Þ e=
hn
e
hn /k BT
-1
- hn /k BT
+ 2hn e
-2hn /k BT
+ 3hn e
-3hn /k BT
+ ×××
h = Planck constant
=6.626×10-34 joule-sec
Solid State Version : C versus T
3
3
U =N  k BT (kinetic energy)  N  k BT (potential energy)=3Nk BT
2
2
dU
C 
 3Nk B
dT
Quantum
region
Photoelectric effect:
metal
Wave-like interpretation of radiatio
electron
λ
x
Electric field experienced by electrons
Expectation from wave
properties
electron

E
x
Electric field experienced by electrons
• The larger E is, the higher the intensity is. It is easier to
shake off electrons with larger current. This is
independent of wavelength/frequency.
•Any frequency of light can yield
photoelectrons.
•Need sufficient time to accumulate energy to
Photoelectric
effect experiment
Einstein: E=nhν is dynamical!
Light is composed of light quanta, the energy of each light quantum is hn
h
W

W = work function
1
m 2  h - 
2
1
2
eV0  h - W  mmax
 min   f 0  W / h
2
Compton effect:

for waves: l =l ¢



Compton：experimental results are
consequences of particle’s collision
Light quanta collides with
electron elastically:
After collision, the
change of momentum for
light quanta changes its
E = hn
E = p 2 c 2 + (mc 2 )2 = pc
hn h
\p=
=
c l
Viewed as collision of particles
h
h
=
cosq + pe cosf
li l f
h
sinq = pe sinf
lf
hni =hn f + Ee ® l i > l f
Ee =
p 2 c 2 + (me c 2 )2
Is light particle or wave?
1924 de Broglie joined the
debate
Is the fundamental
particle –electron wave or
particle?
Ph.D. thesis of de Broglie：
electrons are also wave-like
h
l=
p
What is particle? Wave?
particle：
(1) One grain by one
grain, discontinous
(2) trajectory
Wave：
(1)interference
(2)diffraction
Both satisfy
momentum
and energy
conservation
Need more clear and direct
Expectation for electrons being particles:
expectation for partic
electro
n
source
32
Expectation for electrons being waves:

33
Electronic version of Young’s experiment
electr
on
source
Interference of waves
  1   2
Destructive
Constructiveinterfer
interfr
1
/2
2
Quantitative analysis
L
d
θ
dsinθ
(m+1/2)λ
dsinθ==mλ
m constructive
destructive
interferenceinteference
y
sin     tan  
L
L
y  
d
y
d

L
L 1m
l 700nm
l 0.17nm
d
101 mm  104 m
L
y  
d
 7mm
y  1.7  m
37
The Feynman Lectures on Physics (III) p. 1-4~1-5
…This experiment has never been done
in just this way. The trouble is that the
apparatus would have to be made on an
impossible small scale … We are doing
a “thought experiment”…
Reference:
Davisson and Germer: diffraction of electrons
wavelength: 0.165nm(1.65 Å, 50eV)
38
Tonomura et al.
American Journal of Physics
57, 117(1989)
λ = 0.054Å (50kV), Va = 10V
a = 0.5μm, b = 5mm
40
41
Matter waves and mechanical properties
h = Planck constant
(6.626×10-34 joule-sec)
DeBroglie:
λ=h/p
Einstein:
E=hν=p2/2m

p
3
h
= k BT Þ lth =
2m 2
3mk BT
2
free electron: lth (300K ) = 6.2nm
atom: l (300K ) £ 0.2nm
Bulk size
L
l
L
Nano size
L
Neutron
Reviews of Modern Physics 60, 1067 (1988)
Helium atoms
Physics Review Letters 66, 2689 (1991)
Macromolecules
C60
http://www.quantum.univie.ac.at/research/c60/index.html
Nature 409, 304(2001)
Nature 401,680 (1999)
(1) Diffration grating is SiNx grating (period 100 nm) with
width 0.1 m.
(2) C60 is thermal ionized by a laser. The ions are then accelerated
and directed towards a conversion electrode. The ejected electrons
are subsequently counted by a Channeltron electron multiplier.
48
Other atoms:
Na, Phys. Rev. 66, 2693 (1991)
Bio-molecules
3D structure of tetraphenylporphyrin C44H30N4(TPP)
3D structure of the fluorofullerene C60F48
L Hackermuller et al. Phys. Rev. Lett. 91 090408, (2003)
Interference of 106 Na atoms
Science 275, 637 (1997)
Conclusions
(i) Number of counting ~ probability
of finding the particle
(ii) When there are many exclusive choices, 1,2,3,..
P (total probability) ≠ P1+P2+P3+ …
Particle-like behavior: P = P1+P2+P3+ …
53
How do we describe the results ?
Experience from electromagentism:
(i) Light intensity (I) ~ |E|2
(ii) For multiple choices:
I (total intensity) ~ |E1+E2|2 ≠ |E1|2+|E2|2
A way out to escape the classical sum rule
of probability P (total probability) = P1+P2
Max Born’s probability interpretation
Ψ(x,y,z,t) =
wave function of matter
wave (complex)
E(x,y,z,t) = Electric
field
(i) Occurrence of events (particle’s appearance)
= Probability density ~ |Ψ |2
(ii) For many alternative routes (choices), each
alternative route is represented by Ψi
total probability ~ |Ψ1+ Ψ2+ …. |2
(principle of superposition)
Difference between classical and quantum:
|Ψ1+ Ψ2|2 – (|Ψ1|2+ |Ψ2|2)= Ψ1Ψ2 *+ Ψ1*Ψ2 =
interference term
 1  eikr
1
r1
 2  eikr
2
r2
|  1   2 | |1  e
2
ik ( r1  r2 ) 2
 2  2 cos k (r1  r2 )
k(r1 - r2 ) =
2p
l
(r1 - r2 ) = 2p
path difference
l
|
Particle-wave duality
• When the material is detected,
it is a whole particle that is being detected.
It is particle-like.
• The matter wave commands the particle where
to go and arrive in different positions with
the associated probability.
pilot wave
New concepts
classical: |Ψ1|2+ |Ψ2|2
quantum: |Ψ1+ Ψ2|2
electron
source
The electron does not pass 1 or 2
1
“It can pass 1 and 2 at one time.”
2
Ref: The ghost in atom
Which-way experiment
Electron source
Nature 395, 33(1998)
PRL70,2359(1993)
Once which-way is known...
Particle-like behav
Electron
source
Which-way experiment
198Hg+
polarized
Phys. Rev. Lett. 70,2359(1993)
Ground state: 6s2S1/2, Excited state: 6p2P1/2
degenerate: mJ
Photons: σ & π polarized
π : ΔmJ=0 two atoms are in the same state
σ: |Δ mJ|=1 two atoms are not in the same state
π polarized
σ polarized
Five quantum effects (to be covered)
•
•
•
•
•
Interference
Quantization
Tunneling effect
Spin
Entanglement
Entanglement/糾纏
(Quantum telepathy)
Entangled photon-pair experiment
photon 1
photon 2
?
focal plane
焦平面上Detector之距離
Dopfer, B., 1998
Zeilinger, Rev. Mod. Phys. S288, (1999)
Einstein: Spooky action at a distance!
Spooky action:
“ghost-like behavior”
Due to the nature of probabi
the propagation of informati
does not exceed speed of lig
67
s
I think that it is fair to say that no one
understands the quantum theory…
Description of plane waves
x
λ
n = 1/ T c = ln
x
t
T
t
sin(2  2 )  sin(kx  t )

T
ei ( kx t )  cos(kx  t )  i sin(kx  t )
phase
Sinusoidal wave in higher
dimension: being directional
sin(kx - w t) Þ sin(k × r - w t)
k
y
(x, y)


l y sinq = l
l x cosq = l
k sinq =
l
k = kx2 + k y2 + kz2
x
2p
k=
2p
= ky
ly
2p
k cosq =
= kx
lx
phase:kx x + k y y = k × r
= 2p
x
lx
+ 2p
y
ly
Counting number of waves
c    k
z
Number of possible
directions for a given ν

x
» 4pn µ n
2
y
2