2 (x, t)
8 2m

E - V(x, t)(x, t)  0
2
2
x
h

Class 2 Quantum
mechanics-IB
Dr. Marc Madou Chancellor’s Professor
UC Irvine, 2012
Contents
 Time Independent SE or TISE
 Applying Schrödinger’s Equation:
 Particles in a Large (Infinite) Solid
 Particles in a Finite Solid
 Quantum Wells (1D confinement)-DOS
 Quantum Wires (2D confinement)-DOS
 Quantum Dots (3D confinement)-DOS
 Tunneling
 Harmonic Well
 Central Force
Time Independent SE or TISE
 In the case V(x, t) is independent of time, the SE can be
converted into a time independent SE (TISE). Hence we obtain
the time independent form of the Schrödinger’s equation as:
2 (x, t)
8 2m

E - V(x, t)(x, t)  0
x 2
h2
2


2
  2m  x 2  V(x)  (x)=E (x)


 Solving this equation, say for an electron acted upon by a fixed
nucleus, we will see that this results in standing waves.
 The more general Schrödinger equation does feature a time
dependent potential V=V(x,t) and must be used for example
when trying to find the wave function of say an atom in a
oscillating magnetic field or other time-dependent phenomena
such as photon emission and absorption.
Applying Schrödinger’s Equation:
Particles in a Large (Infinite) Solid
 For free electrons in an infinitely large 3D piece of
metal the allowed electron states are solutions of
an expanded version of the Schrödinger equation :
2
8

m
 2  (r) 
E - V(r) (r)  0
2
h
 For electrons swarming around freely in this
infinite metal, the potential energy V(r) is zero
inside the conductor and the solutions inside the
metal are plane waves moving in the direction of

r:
k (r)  Aexpi k  r
where r is any vector in real space and k is any wave
vector.
As with a freely moving particle,

normalization is impossible as the wave extends to
infinity.
Applying Schrödinger’s Equation:
Particles in a Large (Infinite) Solid
 Plotting the energy E versus the wave
k (r)  Aexpi k  r
number kx for a free electron gas leads to a
parabolic dispersion relation.
 From classical theory we could not
 Confining electrons by
appreciate the occurrence of long electronic
limiting their propagation
mean free paths, indeed Drude used the 
interatomic distance “a” for the mean free
in certain directions in a
path . But from experiments with very pure
crystal
introduces
a
materials and at low temperatures it is clear
varying V(r) in the
the mean free path may be much longer,
Schrödinger’s
equation
8
9
actually it may be as long as 10 or 10
and this may lead to an
interatomic spacings or more than 1 cm.
electronic band gap.
 The quantum physics answer is that the
conduction electrons are not deflected by ion
cores arranged in a periodic lattice because
matter waves propagate freely through a
2
8

m
2
periodic structure just as predicted by:
  (r) 
E - V(r) (r)  0
2
h
Applying Schrödinger’s Equation:
Particles in a Finite Solid
 Discrete energy levels inevitably arise
whenever a small particle such as a photon or
electron is confined to a region in space.
 Sommerfeld, in 1928, was the first to show
this. He adopted Drude’s free electron gas
(FEG) and added the restriction that the
electrons must behave in accordance with the
rules of quantum mechanics (e.g., only 2
electrons per energy level) or he defined a
Fermi gas.
 In his Fermi gas, electrons are free, except for
their confinement within a cubic piece of
crystalline conductor with a finite volume of V
=L3 and they follow Fermi-Dirac statistics
instead of Maxwell-Boltzmann rules.
Applying Schrödinger’s Equation:
Particles in a Finite Periodic Solid
 Outside the 3D cube of solid the potential V(x)= and
the wave function  is zero anywhere outside the solid.
This situation applies, for example, to totally free
electrons in a metal where the ion cores do not influence
their movement. Sommerfeld actually assumed that V(x)
outside the conductor equaled the work function .
 Now let’s introduce periodicity e.g. a repeating cube with
side L. The choice of a cube shape with side L is a
matter of mathematical convenience. The value L is set
by the Born and von Karman’s periodic boundary
condition, i.e., that the wave functions must obey the
following rule: (x+L, y+L, z+L)=(x,y,z)
 Setting such a periodic boundary condition ensures that
the free-electron form of the wave function is NOT
modified by the shape of the conductor or its boundary.
This can be interpreted as follows: an electron coming to
the surface is not reflected back in, but reenters the metal
piece from the opposite surface. This excludes the
surfaces from playing any role in transport phenomena.
Applying Schrödinger’s Equation:
Particles in a Finite Periodic Solid with
Varying Vr
 Besides periodicity we also introduce V( r
) ≠ 0. The electrons are not totally free
anymore ! They feel the ion cores.
 Energy versus wave number for motion of
an electron in a one-dimensional periodic
potential. The range of allowed k values
goes from –π/a to + π/a corresponding to
the first Brillouin zone for this system.
Similarly, the second Brillouin zone
consists of two parts; on extending from
π/a to 2π/a, and another part extending
between
-π/a
and
-2π/a.
This
representation is called the extended zone
scheme. Deviations from free electrons
parabola are easily identified. Where a is
the lattice constant.
Applying Schrödinger’s Equation:
Quantum Wells (1D confinement)
 Outside the well V  and  = 0 for x  0 and x L
For the Schrödinger equation inside the
material (0 <x < L) we write:
d 2 x

 E  x
2
2me dx
2

2 nx 
n (x,t) =
sin 
L  L 
2

E n (k) =
n 2
)
2me L
(

Applying Schrödinger’s Equation:
Quantum Wells (1D confinement)
 At the lowest energy (n=1),
the ground state, the energy
remains finite despite the fact
that V=0 inside the region.
According to quantum
mechanics an electron cannot
be inside the box and have
zero energy. This is called the
zero-point energy an
important consequence of the
Heisenberg principle.
Applying Schrödinger’s Equation:
Quantum Wells (1D confinement)
 For the same value of quantum number n,
the energy is inversely proportional to the
mass of the particle and to the square of
the length of the box. For a heavier
particle and a larger box, the energy levels
become more closely spaced. Only
when
2
mL2 is of the same order as , do
quantized energy levels become important
in experimental measurements (with L =
h
1nm, E  8m L  0.36eV ). With a 1 cm3 piece
of metal (instead 
of 1nm3), the energy
levels become so closely spaced that they

seem
to be continuous E  8mh L  3.610 eV ,
in other words the quantum mechanics
formula gives the classical result for
2
 m L2 >>
dimension such that
e
2
n
E n (k) =
( )2
2me L
2
1
2
e
2
1
15
2
e
Zhores Alferov (Left) and
Herbert Kroemer (Right).
Nobel Prize in Physics 2000.
Applying Schrödinger’s Equation:
Quantum Wells (1D confinement)
 QWs developed in the early 1970s and constituted the
first lower dimensional hetero-structures. The
foremost advantage of such a design involves their
improved optical properties.
 In a quantum well there are no allowed electron states
at the very lowest energies (an electron in a box with
energy = 0 does not exist) but there are many more
available states (higher DOS) in the lowest conduction
state so that many more electrons can be
accommodated. Similarly, the top of the valence band
has plenty more states available for holes. This means
that it is possible for many more holes and electrons to
combine and produce photons with identical energy
for enhanced probability of stimulated emission
(lasing)
DOS-Bulk Materials
We calculate for the density of states for a parabolic
band in a bulk material (3 degrees of freedom) as:
3/2 1
1 2m*e  2
G(E)3D dE  2  2  E dE
2 


DOS-Quantum Well
 The density of states function (DOS) for a quantum well is different from that of a 3D
solid. The solid black curve is that for free electrons in all 3 dimensions. The bottom of the
quantum well is at energy Eg but the first level is at E0. This causes a blue shift.
 There are many more states at E0 than at Eg.This makes, for example, for a better laser.
 To calculate the current density for a 2D electron gas at a particular temperature we first
calculate the value for n(E)2D at T > 0.The function n(E)2D for a given temperature T (>0)
is shown as a red line. In the same graph we also show the Fermi-Dirac function and the
DOS function (blue).
n(E)2D dE = G(E) 2D f(E)dE
J = -n(E)2D evavg


Applying Schrödinger’s Equation:
Quantum Wires (2D confinement)
 For a particle in a finite sized, 2-D
infinitely deep potential well, we define
a wave function similar to the 1-D
potential well, but now we obtain x,y)
solutions that are defined by 2 quantum
numbers one associated with each
confined dimension.
Applying Schrödinger’s Equation:
Quantum Wires (2D confinement)
 Compared to the fabrication of
quantum wells, the realization of
nanoscale quantum wires requires
more difficult and precise growth
control in the lateral dimension, and,
as a result, quantum wire
applications are in the development
stage only.
1
2
 4 
 n1 x   n 2 y 
 n1n 2 (x,y)= 
sin


 sin 

L
L
L
 1 2
 1   L2 
E n1n 2 (k )=E  E
x
n1
y
n2
h 2  n12 n 22 

 

8m  L21 L22 
NEC’s Sumio Iijima
DOS-Quantum Wire
 Quantum wires again feature a blue
shift.
 Also the density of states (DOS) (blue)
and occupied states (red) for a
quantum wire are different than those
of a 3D electron gas. The Fermi-Dirac
function is f(E) and n(E)1D is the
product of f(E) and G(E)1D at T>0.
n(E)1D dE = G(E) 1D f(E)dE
J = -n(E)2D evavg

DOS-Quantum Wire
 For a quantum wire the resistance or current is
found to further simplify to a very simple
expression that does not depend on voltage but
only on the number of available levels.
 The amount of current is dictated only by the
number of modes (also called sub-bands or
channels)
M
(EF),
that
are
filled
between EF(1) andF with each mode
2/h2 or:
contributing 2e2e

M(E F )
h
and :
R
h
1
2e 2 M(E F )
This quantized resistance R has a value of 12.906
DOS-Quantum Wires
 Each of the discrete peaks in the density of
states (DOS) is due to the filling of a new
lateral sub-band. The peaks in the density of
states functions at those energies where the
different sub-bands begin to fill are called
criticalities or Van Hove singularities. These
singularities (sharp peaks) in the density of
states function leads to sharp peaks in
optical spectra and can also be observed
directly with scanning tunneling microscopy
(STM).
 Singularities again emerge when dk/dE = 0
Applying Schrödinger’s Equation:
Quantum Dots (3D confinement)
 In the early 1980s, Dr. Ekimov
discovered quantum dots with his
colleague, Dr. Efros, while working
at the Ioffe Institute in St.
Petersburg (then Leningrad),
Russia. This team’s discovery of
quantum dots occurred at nearly the
same time as Dr. Louis E. Brus a
physical chemist then working at
AT&T Bell Labs found out how to
grow CdSe nano crystals in a
controlled manner.
Columbia’s Louis Brus
Applying Schrödinger’s Equation:
Quantum Dots (3D confinement)
 The solution of the SE for a
square semiconductor
quantum dot (3D
confinement) with side =L
and volume V =L3 is given
as:

1
2
k (r)  V exp(ik  r)
kx 

2n y
2n x
2n z
, ky 
, kz 
,
L
L
L
n x , n y , n z  1, 2, 3, ...
h 2  n12 n 22 n 32 
h2
2
2
2
E n1 ,n 2 ,n3 =
n

n

n


 2  2  2 =
1
2
3
8m  L1 L2 L3  8mL2
DOS Quantumdot
 A quantum dot (QD) is an atom-like
state of matter often referred to as an
“artificial atom.’
 What is so interesting about a QD is that
electrons trapped in them arrange
themselves as if they were part of an
atom although there is no nucleus for
the electrons to surround here.
 The type of atom the dot emulates
depends on the number of atoms in the
well and the geometry of the potential
well V(r) that surrounds them.
Conclusion Reduced DOS as a
Function of Dimensionality
 An important consequence of decreasing the dimensionality beyond that
of quantum wells and quantum wires is that the density of states (DOS)
for quantum dots features an even sharper and yet more discrete DOS.
 As a consequence, quantum dot lasers exhibit a yet lower threshold
current than lasers based on quantum wire and wells, and because of the
more widely separated discrete quantum states they are also less
temperature sensitive.
 However since the active lasing material volume is very small in
quantum wires and dots, a large array of them has to be made to reach a
high enough overall intensity. Making quantum wires and dot arrays with
a very narrow size distribution to reduce inhomogeneous broadening
remains a real manufacturing challenge and as a result only quantum well
lasers are commercially mature.
Applying Schrödinger’s Equation:
Tunneling
 Outside the box, in regions I and III, the
boundary condition is that V(x) = Vo.
These are regions that are “forbidden” to
classical particles with E < Vo. With E <
V0 a classical particle cannot penetrate a
barrier region: think about a particle
hitting a metal foil and only penetrating
the foil if its initial energy is greater than
the potential energy it would possess
when embedded in the foil and where
2
otherwise it will be reflected.
d 2 x

 E  V0   x  The solution inside the well is an
2
2m dx
oscillating wave just as in the case of the
well with infinite walls.
Applying Schrödinger’s Equation:
2m(V  E)
Tunneling


 Defining as :
2
d 2
2



Yields: dx 2

0
2
 In a region with E < V0 there is
an immediate effect on the

waveform for the particle
because, kx is real under these
conditions and we can write:
kx =  =
2m(V0  E)
Applying Schrödinger’s Equation:
Tunneling
 We find for the general solution in Region
I and III, a wave-function of the form ,
x  Aex +Bex
i.e., a mixture of an increasing and a
decreasing exponential function.

With a barrier that is infinitely thick we
can see that the increasing exponential
must be ruled out as it conflicts with the
Born interpretation because it would
imply an infinite amplitude. Therefore in a
barrier region the wave-function must
simply be the decaying exponential . The
important point being that a particle may
be found inside a classically forbidden
region (Region I and III).
Applying Schrödinger’s Equation:
Tunneling
exponential decay of the wave
 The
function inside the barrier is given as:
 If the barrier is narrow enough (L)
there will be a finite probability P of
finding the particle on the other side of
the barrier. The probabilityof an
electron reaching across barrier L is:
P  | (x) |2 = A2e -2L
where A is a function of energy E and
barrier height V0.
 (x) = Ae-x
Applying Schrödinger’s Equation:
Tunneling
 The tunneling current, picked up
by the sharp needle point of an
STMis given by: I = fw (E)A2e -2L
 Where fw(E) is the Fermi-Dirac
function, which contains a
weighted joint local density of
electronic states in the solid
surface that is being probed and
those states in the needle point.
Applying Schrödinger’s Equation:
Harmonic Well
 The time independent
Schrödinger
equation
(TISE) for a harmonic
oscillator is given as:

2
2

d  x
 E  1 2 k x 2  x
2
2m dx
1
2

-
 n (x)  N n H n ( x)e
x 2
2
with n  0,1,2,3...and   (
Km
2
Nn (a normalization const ant)
=
)
1
2
1
(2n n!)
 14
( )
1

2
E n  n  12  , n  0,1,2,3...
Applying Schrödinger’s Equation:
Harmonic Well
 As expected from Bohr’s
Correspondence principle the
higher the quantum numbers
the better the quantized
oscillator resembles the
classical non-quantized
oscillator.
Applying Schrödinger’s Equation:
Central Force
 An electron bound to the
hydrogen nucleus is an example
of a central force system: the
force depends on the radial
distance between the electron
and the nucleus only.
 The solutions of the
Schrödinger equation with this
potential are spherical Bessel
functions.
Summary SE
Applications
 Summarizing, the quantization
for three of the most important
potential profiles leads to the
following mathematical
solutions of the Schrödinger
equation: for the central force
we obtain spherical Bessel
functions, for an infinite
square well potential sines,
cosines and exponentials and
for an oscillator Hermite
polynomials.
Quantum Jokes