Class 1. Quantum
mechanics: IA
Dr. Marc Madou Chancellor’s Professor
UC Irvine 2012
Schrödinger's Cat could not cope
with a lifetime of uncertainty.
In the IC and NEMS world, we are moving fast into the
realm of quantum mechanics. Moore’s law might remain
valid until about 2020 but by then the scale of
electronic components will be at the molecular/atomic
level, and hence can no longer be described by classical
mechanics. Quantum computing, nanotechnology,
including nanotubes, nanowires, biological nanostructures and quantum dots, all require some
grounding in quantum mechanics to be understood at
all. Quantum mechanics must now become a familiar
tool not only to physicists but also to materials
scientists, biologists and electrical, mechanical and
bioengineers.
Contents
 Classical Theory Starts
Faltering
 Quantum Mechanics to
the Rescue
 Schrödinger’s Equation
Classical Theory Starts
Faltering
 Electrical conductivity
 Heat capacity in
metals and insulators
 Temperature
dependence of
electrical conductivity
and heat capacity
 Hall effect
 Blackbody radiation
 Photoelectric effect
Paul Drude
(1863-1906).
Classical Theory Starts
Faltering
 Modern condensed
matter physics really
started with the discovery
of the electron by J.J.
Thompson in 1897.
 Drude describes electrons
in a metal as a free
electron gas (FEG) to
explain electrical
conductivity and Ohm’s
law with:   me
ne2
Classical Theory Starts
Faltering
 Using the lattice constant, a,
for the mean free path, and the
Maxwell-Boltzmann-Equation
at T=300K to calculate vth,
values for the resistivity of a
metal are obtained that are six
times too large.
 Temperature dependence is
wrong too. Expected was a √T
but what we get 
experimentally is:
  0  T
1
mevth
me 8kBT
 
 2
2

ne  ne a me
Classical Theory Starts
Faltering
 Since solids contain a number
of atoms and electrons with
similar density, why the large
conductivity differences?
 More intriguing yet, based on
the above why would carbon
allotropes come with such
highly varying electrical
conductivities?
Classical Theory Starts
 Based on the Drude assumption that
Faltering
electrons in a metal behave as a monoatomic gas of N classical particles, they  Atoms in a lattice usually have no
translational energy and as
should be able to take up a translational
temperature increases only vibrational
kinetic energy when the metal is heated
energy increases.
 According to the equipartition principle
 In the case of a monovalent metal, the
of energy in a gas there is an internal
lattice specific heat contribution, C
energy U =1/2kBT per degree of freedom
v,lat, equals 3R. The vibration of a
 So the electron contribution to the
classical 1-D harmonic oscillator is
U=kT –with 1/2 kT for kinetic and
specific heat capacity (expressed per
1/2kT for potential energy .
mole) is:
 From the latter we expect C v,lat to be
U 3N
3
3
constant at 3R or 25 J/mol K -the so
k BT = NAk BT = RT
called DuLong-Petit law.
n 2n
2
2

 U
3
Cv,el  ( )v  R
T n
2
Classical Theory Starts
Faltering
to be constant at 3R or 25
 We expect Cv,lat
J/mol K ( DuLong-Petit law). Experimentally,
Cv, the sum of vibrational and electronic
contributions (Cv= Cv,el+Cv,lat= 4.5 R), is
proportional to T3 at low T and approaches a
constant 3R, at high T.
 The total Cv for metals is found to be only
slightly higher than that for insulators, which
only feature lattice contributions. Where is the
electronic contribution? The absence of a
measurable contribution by electrons to the
CV was historically one of the major
objections to the classical free electron model.
Conduction electrons only contribute a small
part of the heat capacity of metals but they are
almost entirely responsible for the electrical
conductivity.
Classical Theory Starts
Faltering
 In 1853, long before Drude’s time, Gustav
Wiedermann and Rudolf Franz found the ratio
of thermal and electrical conductivities of all
metals has almost the same value at a given
temperature or one obtains Lorenz number L
as:
2
c v,elmv dx 3 k B 2


 ( ) =L
2
T
3ne
2 e
 This was obtained by:
3
1
3
cv,el  nkB and mv2dx  k BT
2
2
2
Lorenz number L in 10-8 Watt ohm/K2
Metal
273K
373K
Ag
2.31
2.37
Au
2.35
2.40
Cd
2.42
2.43
Cu
2.23
2.33
Ir
2.49
2.49
Mo
2.61
2.79
Pb
2.47
2.56
Pt
2.51
2.60
Sn
2.52
2.49
W
3.04
3.20
Zn
2.31
2.33
Classical Theory Starts
Faltering
 Drude using classical values for
the electron velocity vdx and
heat capacity Cv,el, somehow got
a number very close to the
experimental value. But how
lucky that Drude dude was: by a
tremendous coincidence, the
error in each term he made was
about two orders of
magnitude…. in the opposite
direction
Miscalculation
Classical Theory Starts
Faltering
 For most metals the Hall constant is negative. But for Be and
Zn, for example, RH is positive. Of course energy bands
were not heard of yet and the results of a positive RH were
baffling at the time: how can we have q > 0 (even for
metals!?).
Classical Theory Starts
 A blackbody of temperature T emits a
Faltering
continuous spectrum peaking at λmax. At
very short and very long wavelengths
there is little light intensity, with most
energy radiated in some middle range
frequencies. As the body gets hotter the
peak of the spectrum shifts towards
shorter wavelengths.
 Classical interpretation predicted
something altogether different; in the
classical Rayleigh-Jeans model, instead
of a peak in the blackbody radiation and
a falling away to zero at zero wavelength
the measurements should go off scale at
the short wavelength end -towards
“ultraviolet catastrophe”
 Max Planck took the revolutionary step 6.626 x 10-34 J.s
that led to quantum mechanics :E = h
Classical Theory Starts
Faltering
never appreciated how far
 Planck, interestingly,
removed from classical physics his work was. The
Planck constant h, was a bit like an ‘uninvited guest’
at a dinner table; no one was comfortable with this
new guest. Discontinuities in the nanoworld are
meted out in units based upon this constant. It is the
underlying reason for the perceived weirdness of the
nanoworld; the existence of a least thing that can
happen quantity-a quantum. The ubiquitous
occurrence of discontinuities in the nanoworld,
constantly upsets our common-sense understanding
of the apparent continuity of the macroscopic world.
Max Planck April 23, 1858 – October 4, 1947
Classical Theory Starts
 Photoelectric effect; no electrons are ejected, regardless of the
Faltering
intensity of the light, unless the frequency exceeded a certain




threshold characteristic of the bombarded metal (red light did
not cause the ejection of electrons, no matter what the
intensity).
The photoelectric phenomenon could not be understood
without the concept of a light particle, i.e., a quantum amount
of light energy.
Einstein's 1905 paper explaining the photoelectric effect was
one of the earliest applications of quantum theory and a major
step in its establishment.
The remarkable fact that the ejection energy was independent
of the total energy of illumination showed that the interaction
must be like that of a particle which gave all of its energy to
the electron!
Einstein reintroduced a modified form of the old corpuscular
theory of light, which had been supported by Newton but
which was long abandoned.
Classical Theory Starts
Faltering
 The electron charge was
determined by Robert Millikan
in 1909 and with that value
and the slope of the lines a
value for h of 6.626 x 10-34 J.s
can be calculated, identical to
the one derived from the
hydrogen atom spectrum and
blackbody radiation (see
above)
Classical Theory Starts
Faltering
 The young Einstein, in 1905, was the
first scientist to interpret Planck’s
work as more than a mathematical
trick and took the quantization of
light (E=h) for physical reality. He
gave the uninvited dinner guest -the
Planck constant h-a place at the
quantum mechanics dinner table.
What Einstein proposed here was
much more audacious than the
mathematical derivations by Planck
to explain the UV catastrophe away.
Classical Theory Starts
Faltering
 The three experiments that
made
the
quantum
revolution,
Black-body
radiation, the photo-electric
effect and the Compton
effect all indicate that light
consists of particles.
Arthur Harry Compton (1892-1962).
Quantum Mechanics to
the Rescue
 Einstein’s special
relativity allows one to
calculate the momentum,
p, of a photon starting
from:
E  (pc)  m0c
2
2

2 2
E 2  (pc)2  0 or E = pc


E h h
p = 

c
c 
 De Broglie, while studying for
his PhD in Paris in 1924,
postulated that this last
equation also applied to a
moving particle such as an
electron, in which case  is the
wavelength of the wave
associated with the moving
particle, i.e., a “matter wave.”
h

mv
Quantum Mechanics to
the
Rescue
 In his 1928 tests Thompson junior and
Reid observed interference patterns from
electrons reflecting from a thin
polycrystalline metal foil surface.
 Clinton Davisson and Lester Germer, at
Bell Laboratories, in 1927 found the
same experimental evidence: a beam of
electrons scattered from a single crystal
of nickel resulted in a diffraction pattern
fitting the Bragg diffraction law
 This established the wave character of
electrons, forming the basis of analytical
techniques for determining the structures
of molecules, solids and surfaces, such
as in LEED (low energy electron
diffraction) and SEM.
Davisson-Germer experiment (1927).
Quantum Mechanics to
the Rescue
 Louis-Victor, 7th duc de
Broglie (1892-1987)
discovered thus that the secret
of Planck’s and Einstein’s
quanta lay in a general law of
nature i.e., the dual character
of waves and particles.
 Einstein commented about this
fantastic insight: “de Broglie
has lifted the great veil.”
 Even buckyballs are wavy
Quantum Mechanics to
the Rescue
 Einstein gave, what we had come to
think of as a wave (light) a particle
character and de Broglie gave what
we thought of as a particle (electrons)
a wave character. Radiation has wave
character and particle character and
matter has particle and wave character
or at the nanoscale, nature presents
itself with a wave-particle duality.
h

mv
Quantum Mechanics to
the Rescue
 The wave-particle duality
introduced in the previous
section forced physicists to
reconsider their description of
the position and momentum of
very small particles and is at the
core of the Heisenberg’s
uncertainty principle (HUP).
 In the nanoworld, the
Heisenberg principle states that
there are physical parameters in
quantum physics whose values
cannot be known accurately
simultaneously.


Quantum Mechanics to
the Rescue
h
px x 

2
h
Et 

2
Quantum Mechanics to
the Rescue
 The uncertainty about the energy of a particle depends on the time
interval t that the system remains in a given energy state. Importantly
this also means that conservation of energy can be violated if the time is
short enough.
 From the uncertainty principles it is possible that empty space locally
does not have zero energy but may actually have sufficient E for a very
short time t to create particles and their antiparticles. This can be
demonstrated through the Casimir effect.
 This effect is also responsible for “lifetime broadening” of spectral lines.
Short-lived excited states (small t) possess large uncertainty in the
energy of the state (large E). As a consequence, shorter laser pulses
(e.g., femto and attosecond lasers) have broader energy (therefore
wavelength) band widths.
Quantum Mechanics to
the Rescue
 Based on the idea that the 'vacuum' of
space is actually a seething foam of
quantum fluctuations of different
frequencies, Casimir proposed that if
two electrically conducting, but
uncharged parallel plates were
mounted a small distance apart in a
vacuum, they would tend to be drawn
together. An important point is that the
plates carry no electrical charge so that
any interaction between the plates
must come from some other source.
 Using MEMS devices the Casimir
force has been measured
Quantum Mechanics to
the Rescue
 The existence of a Zero-Point
–Energy: vibrational energy
cannot be zero even at T=0K
is also a consequence of the
Heisenberg principle. If the
vibration would cease at
T=0K, then the position and
momentum would both be 0,
violating the HUP.
Schrödinger’s Equation
 Schrödinger, after attending a seminar
on Einstein’s and de Broglie’s ideas
that wavelike entities can behave like
particles and vice versa, thought that
there must be a wave equation,
(x,t), to describe particles.
 Schrödinger’s picture of the atom has
the electron standing waves vibrating
in their orbitals much like the
vibrations on a string - but in three
dimensions instead of one. In this
figure
a
two
dimensional
representation of Schrödinger waves,
like vibrations on a drum skin, is
shown.
Schrödinger’s Equation
 A concept that plays an important role in both classical and quantum
theory is that of the Hamiltonian of a system.
 H=En(Total Energy ) = KE (Kinetic Energy-depends on v) +PE
(Potential Energy V-depends on position)
 Example 1: free moving object: V=0, knowing x0 and p, we can
predict xt at t in other words we can predict the trajectory at all
2
p
times later. H = E =  V(x) and since V(x)= 0, E = E k
2m
v = dx/dt =
dp
d2x
F =
 ma = m 2
dt 
dt
2Ek
m
2E k
2p2
p
xt  x0 
t=
t
=
t = vt
2
m
2m
m
Schrödinger’s Equation
 Example 2: harmonic oscillators,d 2 x
k


x
Hooke’s law (F=-kx)
dt 2
m
 As the amplitude (A) can take
any value, this means that the
energy (E) can also take any
value – i.e., energy is continuous.
Any energy value is allowed by
simply changing the force
constant k.
p 2 kx 2
mv 2 kx 2
E=

or also E=


2m
2
2
2
2


 k 
 k 
k
cos 
t 
k  Asin 
t 
 mA
m
m
m



 

 
=
 
2m
2
2
Schrödinger’s Equation
 In the late 18th century the mathematician Pierre Simon de Laplace (17491827) encapsulated classical determinism as follows: “…if at one time we
knew the positions and motion of all the particles in the Universe, then we
could calculate their behavior at any other time, in the past or the future.” In
classical physics, particles and trajectories are real entities and it is assumed
that the universe exists independently from the observer, that it is predictable
and that for every effect there is a cause so experiments are reproducible.
 Heisenberg’s uncertainty principle destroyed all this. In quantum physics the
measured and unmeasured particle are described differently. The measured
particle has definite attributes such as position and momentum, but the
unmeasured particle does not have one but all possible attribute values, as Nick
Herbert in his book Quantum Reality writes …somewhat like a broken TV that
displays all its channels at the same time.
Schrödinger’s Equation
 Erwin Schrödinger (1887-1961), in
1926, encouraged by Debye who
remarked that there should be a wave
equation to describe the de Broglie
waves, proposed a wave equation that
can be applied to any physical system
in which it is possible to describe the
energy
mathematically.
In
one
dimension he postulated:
 (x, t)
8 m

E - V(x, t)(x, t)  0
2
2
x
h
2
2
Schrödinger’s Equation
 The first term is the rate of change of the rate of change of the wave function with
distance x. The energy of the particle is E and the potential energy function to
describe the forces acting upon the particle is represented by V(x, t).
 The Schrödinger equation has the same central role in quantum mechanics that
Newton’s laws have in mechanics and Maxwell’s equations in electromagnetism.
 Solutions to Newton’s equations are of the form v=f(x , t), while solutions to the
wave are called wave functions (x, t).
 Like Newton’s equation, it describes the relation between kinetic energy, potential
energy, and total energy. If one knows the forces involved, one can calculate the
potential energy V and solve the equation to find .
 Solving the Schrödinger equation specifies (x,t) completely, except for a constant;
if (x,t) is a solution then A(x,t) is a solution as well.
 (x, t)
8 m

E - V(x, t)(x, t)  0
2
2
x
h
2
2
Schrödinger’s Equation
 The so-called “Copenhagen Interpretation” of Schrödinger’s equation is that
the (x,t) function is not some physical representation of a physical substance
as in classical physics (e.g., the amplitude of a water wave) but a “probability
amplitude” of the particle which, when squared, gives the probability of
finding the particle at a given place at a given time: |(x, t)|2dx = probability
the particle will be found between x and x + dx at time t and the wavefunction
itself has no physical meaning.
 Since the probability that the particle is somewhere must equal one, it holds
that one can normalize this probability function as:

 | (x,t)| dx =1
2

 The Copenhagen interpretation also holds that an unmeasured particle in a
certain sense is not real: its attributes are created or realized by the measuring
act. Another way of saying this is that a wavefunction collapses upon
measurement; before measurement a particle is described by a wave function
described
by the Schrödinger equation but upon measuring that particle’s
wave suddenly and discontinuously collapses.
 (x, t)
8 m

E - V(x, t)(x, t)  0
2
2
x
h
2
2
Schrödinger’s Equation
 Because Ψ(x,y,z,t) is complex and can be positive or negative, it cannot be
the probability directly. The Born interpretation of  places restrictions on
the form of the wavefunction:
(a)  must be continuous (no breaks);
(b) The gradient of  (d/dx) must be continuous (no kinks);
(c)  must have a single value at any point in space;
(d)  must be finite everywhere;
(e)  cannot be zero everywhere.
 (x, t)
8 m

E - V(x, t)(x, t)  0
2
2
x
h
2
2
Schrödinger’s Equation
 In operator form the SE is Hoperatoracting on function -an eigen
function=Efunction  multiplied by a number E-an eigen value)
 Where H is the one-dimensional Hamiltonian operator and in which the energy E of the
particles is called the eigenvalue, and  the eigenfunction.
 Expressed yet another way, kinetic and potential energies are transformed into the
Hamiltonian which acts upon the wavefunction to generate the evolution of the
wavefunction in time and space. The Schrödinger equation gives the quantized energies
of the system and gives the form of the wavefunction so that other properties may be
calculated.
h2  2
[ 2
 V(x,t)] (x,t)  E (x,t)
2
8 m  x
h2  2
H=  2
 V(x,t)
8 m  x 2
 (x, t)
8 m

E - V(x, t)(x, t)  0
2
2
x
h
2
2
Quantum Jokes