2. Quantum theory: techniques and applications

2.1.Translational motion

2.1.1 Particle in a box

2.1.2 Tunnelling

2.2. Vibrational motion

2.2.1 The energy levels

2.2.2 The wavefunctions

2.3. Rotational motion

2.3.1 Rotation in 2 dimensions

2.3.2 Rotation in 3 dimensions

2.3.3 Spin

translation

vibration

rotation

The energy in a molecule is stored as molecular vibration, rotation and translation.

2.1 The translational motion

2.1.1 Particle in a box

For a free particle (V=0) travelling in one dimension, the Schrödinger equation has a general solution k, where k is a value characteristic of the energy (eigenvalue) of the particle Ek.

Exm

2

22

2

k= Aeikx + Be-ikx

m

kEk 2

22

For a free particle, all the values of k, i.e. all the energies are possible: there is no quantization

Particle of mass m is confined in an infinite square well. Between the walls: V=0 and the solution of the SE is the same as for a free particle.

k= C sinkx + D coskx NB: with D= (A+B) C= i(A-B)

A. boundary condition (BC): The difference with the free particle is that the wavefunction of a confined particle must satisfy certain constraints, called boundary conditions, at certain locations.

BC1: k(0)=0 → k (0) = C 0 + D 1=0 → D=0

→ after BC1: k= C sinkx

BC2: k(L)=0 → k (L) = C sinkL =0

→ absurd solution: C=0, it gives k(x)=0 and |k(x)|2=0… the particle is not in the box!

→ physical solution: kL= n with n=1,2,… (n0 is also absurd)

The wavefunction n(x) of a particle in an infinite square well is now labeled with “n” instead of k. Because of the boundary conditions, the particle can only have particular energies En:

,...2,1;

82

/2

2222

nmL

hn

m

LnEn

,...2,1;sin)( nL

xnCxn

B. Normalization: Let’s find the value of the constant C such that the wavefunction is normalized.

L LL

n

LC

L

xn

n

LxCdx

L

xnCdxx

0

2

0

222

0

21

22sin

42

1sin)(

2

1

2

L

C

C. Properties of the solutions

,...2,1;8 2

22

nmL

hnEn

LxforL

xn

Lxn

0;sin

2)(

2

1

The solutions are labeled with n, called “quantum number”. This is an integer that specifies the energetic state of the system. In order to fit into the cavity, n(x) must have specific wavelength characterized by the quantum number.

With an increase of n, n(x) has a shorter wavelength (more nodes) and a higher average curvature → the kinetic energy of the particle increases.

The probability density to find the particle at a position x in the box is

L

xn

Lxn

22 sin2

)(

The larger n , the more uniform 2n(x): the situation is close

to the example of a ball bouncing between two walls, for which there is no preferred position between the two walls.

The classical mechanics emerges from quantum mechanics as high quantum numbers are reached.

The zero-point energy: because n>0, the lowest energy is not zero but E1=h2/(8mL2).

That follows the Uncertainty Principle: if the location of the particle is not completely indefinite (in the well), then the momentum p cannot be precisely zero and E >0.

The energy level separation E= increases with n.

E decreases with the size L of the cavity for a molecule in gas phase free to move in a laboratory-sized vessel, L is huge and E is negligible: the translational energy of a molecule in gas phase is not quantized and can be described in classical physics.

2

2

1 8)12(

mL

hnEE nn

2.1.2 Tunnelling

If the energy E of the particle is below a finite barrier of potential V, the wavefunction of the particle is non-zero inside the barrier and outside the barrier.

there is certain probability to find the particle outside the barrier, even though according to classical mechanics the particle has insufficient energy to escape: this effect is called “tunnelling”.

Transmission probability of the particle through the barrier.

For x<0: the wavefunction is that of a free particle: (x<0)= Aeikx + Be-ikx with kħ=(2mE)1/2.

Aeikx represents the incident wave, Be-ikx corresponds to the reflected wave bouncing on the wall.

For x>L: V=0, it’s like for a free particle: (x>L)= A’eikx + B’e-ikx with kħ=(2mE)1/2. But, the direction of the transmitted wave is (Left Right), hence B’=0 since B’e-ikx is a wave travelling in the (Right Left) direction. A’eikx represents the transmitted wave.

X=0 X=L

xX=0 X=L

For 0<x<L: the wavefunction must be solution of the SE for a particle in a constant potential V.

The general solutions are (0<x<L)= Ceqx + De-qx with qħ=[2m(V-E)]1/2. NB: here, the two exponentials are real!

The probability to find the particle in the barrier decreases exponentionally with the distance x.

EVxm

2

22

2

The probability to find a particle in the region x<0, which travels LR, is proportional to |A|2

The probability to find a particle in the region x<0, which travels RL, is proportional to |B|2

The probability to find a particle in the region x>L, which travels LR, is proportional to |A’|2

The probability that the particle crosses the potential barrier from x<0 to x>L is given by the transmission probability: T=|A’/A|2

The probability to be reflected on the barrier is characterized by the reflection probability: R= |B/A|2

Since if the particle is not reflected, it is transmitted: T+R=1

with =E/V and q=(1/ħ) [2m(V-E)]1/2

)1(161

2

qLqL eeT

Considering that the wave function must be continuous at the edges of the barrier (for x=0 and L), as well as the derivative of the wave function; it is possible to extract the transmission probability:

For a thick barrier qL>>1: T≅ 16(1- )e-2qL

For a thick barrier qL>>1: T≃ 16(1- )e-2qL

The transmission probability decreases exponentially with the thickness of the barrier and with m1/2.

T is increased also when the energy of the particle E is higher.

Tunnelling is important for electrons, moderately important for protons (quick acid-base equilibrium reaction), and negligible for heavier particles.

J=4

J=10

J=2

A large value of J corresponds to a heavy particle or a wide barrier L

EVmL

LqJ 2*

In 1965, after he assisted in the design of Intel’s 8088 processor, Gordon Moore proposed that transistor density per die would double every year after that. “Moore’s Law”, as it was coined, led computer manufacturers to reduce the size of transistors at a rapid rate. The benefits from smaller transistors are threefold:1. Smaller transistors switch faster which leads to faster processing speeds. 2. Smaller transistors allow more complex processors to be built in the same space.3. Smaller transistors allow for a greater number of processors to be built within the same space. As a result of these economic and technical factors Intel’s first PC chip, the 8088, had 29,000 transistors with a critical dimension of 3 microns (micrometers). The Intel Pentium II processors has 7.5 million transistors with a critical dimension of .25 microns. For thirty years Intel and other chip makers have spent billions in research and development to continue product maturation at the rate explained by Moore.

Moore’s Law

Example 5: Resonant tunneling diodes

The use of a barrier to control the flow of electrons from one lead to the other is the basis of transistors. The miniaturization of solid-state devices can’t continue forever. That is, eventually the barriers that are the key to transistor function will be too small to control quantum effects and the electrons will tunnel when the transistor should be off. This is a consequence of the particle-wave duality of electrons, and the single electron characterization of Schrodinger’s equation. At the quantum level the wave nature of the electron will allow the electrons to tunnel through the barriers and create a current. Quantum effects are seen at dimensions less then a micron, but the tunneling effect is expected to be dominant when the critical dimensions approach the wavelength of an electron (approx. 10nm).Ingenious devices exploit the quantum effects of miniature structures to control electrical current. These devices operate by single electron control, and they require that electron movement be confined to two (quantum well), one (quantum wire), or zero (quantum dot) dimensions. In these devices small voltages heat electrons rapidly, inducing complex nonlinear behavior; the study of “hot” electrons, as they are termed, is central to the further development of these devices. Two such devices are the Resonant Tunneling Diode and the Resonant Tunneling Transistor. These devices create a new “switching” mechanism that requires controlled quantum tunneling to function.The Resonant Tunneling Diode (RTD) consists of an emitter and a collector separated by two barriers with a quantum well in between these barriers. The quantum well is extremely narrow (5-10nm) and is usually p doped. Resonant tunneling across the double barrier occurs when the energy of the incident electrons in the emitter match that of the unoccupied energy state in the quantum well. An illustration of the double barrier Resonant Tunneling Diode is shown in Figure 4 . When the quantum well energy level is below E0, no current may flow by the tunneling mechanism. When the bias is such that the energy level in the quantum well is aligned with a population of electrons above E0 in the emitter, the electrons may tunnel from the emitter, to the quantum well, and through to the collector. As the voltage is increased, the flow of electrons drops as the electrons are unable to tunnel above the resonant level. As the voltage bias continues to increase, the current begins to increase again, this time as a result of the electrons flowing over the top of the barriers. What results is an S shaped IV curve for the Resonant Tunneling Diode shown in Figure 5 .There are several proposed applications of the resonant tunneling diode. The interesting S shaped IV characteristic makes multistate memory and Logic circuits a possibility. Several resonant tunneling diodes can be combined to form multiple peaks. The implication is that there can be multiple operating points for a circuit. Rather then determining if the memory cell or logic state is a one or a zero, we can determine if it is any number of states. The tunneling diode has not yet been fabricated using Silicon based technology, and the operating temperature of the GaAs devices fabricated is below room temperature. Repeatable control of the size of the quantum well and other structures is not yet realizable with current technologies. These and other manufacturing issues must be resolved before the resonant tunneling diode is a widely used component.

Resonant Tunneling Diode

http://www.mitre.org/research/nanotech/quantum_dot_cell1.html

Forms of carbon:

diamond

graphite

fullerenes

nanotubes

“ Single-electron transistors (SETs) have been proposed as a future alternative to conventional Si electronic components. However, most SETs operate at cryogenic temperatures, which strongly limits their practical application. Some examples of SETs with room-temperature operation (RTSETs) have been realized with ultrasmall grains, but their properties are extremely hard to control. The use of conducting molecules with well-defined dimensions and properties would be a natural solution for RTSETs. We report RTSETs made within an individual metallic carbon nanotube molecule. SETs consist of a conducting island connected by tunnel barriers to two metallic leads. For temperatures and bias voltages that are low relative to a characteristic energy required to add an electron to the island, electrical transport through the device is blocked. Conduction can be restored, however, by tuning a voltage on a close-by gate, rendering this three-terminal device a transistor. Recently, we found that strong bends ("buckles") within metallic carbon nanotubes act as nanometer-sized tunnel barriers for electron transport. This prompted us to fabricate single-electron transistors by inducing two buckles in series within an individual metallic single-wall carbon nanotube, achieved by manipulation with an atomic force microscope (AFM)(Fig. C and D). The two buckles define a 25-nm island within the nanotube.”in “Carbon nanotube single-electron transistors at room temperature” by Postma-HWC; Teepen-T; Zhen-Yao; Grifoni-M; Dekker-G in Science. vol.293, no.5527; 6 July 2001; p.76-9.

Carbon nanotube single-electron transistors

2.2 The vibrational motion

Ekxxm

22

22

2

1

2

kxx

VF

2

2

1kxV

A particle undergoes harmonic motion if it experiences a restoring force proportional to its displacement

Classical mechanics Quantum mechanics

Energy separation : constant = ħω

Zero-point energy : E(=0)=½ ħω

classical limit : for a huge mass m, ω is small and the energy levels form a continuum

Eigenvalues:

2/1

,...2,1,0;2

1

m

k

E

A. The form of the wavefunctions4/12

2/ ;)()(2

mkand

xyeyHNx y

N is the normalization constant

22

222

/20

20

2/0

2/00

)(

)(

x

xy

eNx

eNeNx

22

222

/22

212

1

2/12/11

4)(

22)(

x

xy

exN

x

exN

eyNx

NB: < x >= 0 the oscillator is equally likely to be found on either side of x=0, like a classical oscillator.

2/122 )

2

1()()(

mkdxxxxx

B. The virial theorem

In a 1-dimensional problem with a potential V(x)= xn, the expectation values of the kinetic energy <T> and the potential energy <V> verify the following equality:

2 <T> = n <V> ; with the total energy: <E>= <T> + <V>

The harmonic oscillator, V=½kx2, is a special case of the virial theorem since n=2

)2

1(

2

1)

2

1(

2

1

2

12/1

2

m

kxkV

2

1Eand we have seen that

EV2

1

we also know that <E>= <T> + <V>

<T> = <V>

0xtp xtp

C. Quantum behavior of the oscillator

The probability to find an oscillator (in its ground state: =0) beyond the turning point xtp (the classical limit), is:

2/12

max

2

2

1

k

ExkxEVV tptp

Classical behavior

Quantum behavior

In the harmonic approximation, a diatomic molecule in the vibration state = 0 has a probability of 8% to be stretched (and 8% to be compressed) beyond its classical limit. These tunnelling probabilities are independent of the force constant and the mass of the oscillator.

Classical limit: for huge (the case of macroscopic

object), P 0

08.0)(20

tpx

dxxP

Classical behavior

Qua

ntum

be

havi

or

Qua

ntum

be

havi

or

-xtp0xtp

2.3 The rotational motion

2.3.1 Rotation in 2 dimensions

Classical mechanics:

The angular momentum |Lz|= ∓prThe moment of inertia I= mr2

I

L

mr

L

m

pETV zz

222;0

2

2

22

In quantum mechanics: not all the values of Lz are permitted, and therefore the rotational energy is quantized. Where does this quantization come from?

hr

Lh

pandprL zz

The wavelength of the wavefunction () cannot have any value. When increases beyond 2, we must have ()= (+2), such that the wavefunction is single-valued: |()|2 is then meaningful.

The wavelength should fit to the circumference 2r of the circle. The allowed wavelengths are = 2r/ml ; where ml is an integer that is the quantum number for rotation.

lll

z mhm

r

hrmhrL

22

No physical meaning

Lz

A. Schrödinger equation for rotation in 2D

2

2

2

22

2 yxmH

Go to cylindrical coordinates:

x= r cos; y= r sin

0

2

11

2 2

2

2

2

2

2

22

22

rbecause

mrrrrrmH

)()(2 2

22

EI

2/12)(

lime

Schrödinger equation:

The normalized general solutions:

have to fulfill the cyclic boundary condition ()= (+2):

)(

22)2( 2

2/1

2

2/1

)2(

l

lllmi

miimim

eeee

11sincos 2222 llll mmmimi iee

2ml = an even integer ml = 0, ∓1, 2, 3, ...∓ ∓

I

m

I

LE lz

22

222 The eigenvalues are given by

NB: With ml2, the energy does not depend on the sense of rotation

2/12)(

lime For an increasing ml, the real part of the

wavefunction has more nodes

the wavelength decreases and consequently, the momentum of the particle that travels round the ring increases (de Broglie relation): p=h/

The probability density to find the particle in is a constant: |()|2=1/2

knowing the angular momentum precisely eliminates the possibility of specifying the particle’s location: the operator position and angular momentum do not commute: uncertainty principle.

Plots of the real part of the wavefunction ()

B. The angular momentum operator Lz

ux, uy, uz are unitary vectors

Classical mechanics

Quantum mechanics

Correspondence

principles (chap 1)

x

yy

xi

Lz

cylindrical coordinates:

x= r cos; y= r sin

i

Lz

What are the eigenfunctions and eigenvalues of Lz?

Let’s apply Lz to the wavefunctions that are solutions of the Schrödinger equation:

The solutions of the Schrödinger equation, eigenfunctions of the Hamiltonian operator, are also eigenfunctions of the angular momentum operator Lz: H and Lz are commutable: the energy and the angular momentum can be known simultaneously.

ml() is an eigenfunction of the angular momentum operator Lz and corresponds to an angular momentum of mlħ.

Vector representation of angular momentum: the magnitude of the angular momentum is represented by the length of the vector, and the orientation of the motion in space by the orientation of the vector

0,0

zpbecause

ypxp

ppp

zyx

uuu

prL

z

xy

zyx

zyx

z

l

l

ll mlim

lmmz mei

imi

L

2/12

1

2.3.2 Rotation in 3 dimensions

x

y

z

r

A particle of mass m free to travel (V=0) over a sphere of radius r.

2

2

2

2

2

22

2 zyxmH

spherical coordinates:

x= r sincos; y= r sin sin; z= r cos

222

22 12

2 rrrrmH

∂r = 0 (the particle stays on the sphere)

sinsin

1

sin

12

2

22

is the Legendrian

),(2

),(1

22

2

mE

r

2

2

2

IE

The Schrödinger equation is :

withSince I = mr2, we can write:

)()(),(

22

2

sinsinsin1

d

d

d

d

d

d

22

21lm

d

d

22sinsinsin

lmd

d

d

d

At the moment, ml2 is just

introduced as an arbitrary constant

We consider that (,) can be separated in 2 independent functions:

→ the Hamiltonian can be separated in 2 parts → the SE is divided into 2 equations

2

2

2

lmd

d

Same as for the rotation in 2-D with

2/12)(

l

l

im

m

e

The solutions should also fulfill the cyclic boundary condition: ()=(+2); because of that another quantum number “l” appears and is linked to ml. Pl

m(cos ) is a polynomial called the associated Legendre functions. Nlm is the normalization constant.

)(cos)( mllmPN

mlml

imlmml YPe

Nl

,2/1, )(cos)2(

)()(

l = 0, 1, 2, 3,…

|m|≤l

+ ml2 -ml

2

The normalized functions lm(,)=Ylm(,) are called spherical Harmonics

The figure represents the amplitude of the spherical harmonics at different points on the spherical surface.

Note that the number of nodal lines (where lm(,)=0) increases as the value of l increases: a higher angular momentum implies higher kinetic energy.

From the solution of the SE, the energy is restricted to:

→ The energy is quantized and is independent of ml. Because there are (2l+1) different wavefunctions (one for each value of ml) that correspond to the same energy, the energetic level characterized by “l” is called “(2l+1)-fold degenerate”.

,...2,1,0;2

)1(2

lI

llE

http://www.sci.gu.edu.au/research/laserP/livejava/spher_harm.html

http://mathworld.wolfram.com/SphericalHarmonic.html

Spherical harmonics

ml = 0: a path around the vertical z-axis of the sphere does not cut through any nodes. For those functions, the kinetic energy arises from the motion parallel to the equator because the curvature is the greatest in that direction.

The comparison between the classical energy E=L2/2I and the previous expression for E, shows that the angular momentum L is quantized and has the values (→ length of the vector):

L={l(l+1)}1/2 ħ ; l= 0, 1, 2,...

As for the rotation in 2-D, the z-component Lz is also quantized, but with the quantum number ml (→ orientation of the vector L):

Lz= ml ħ ; ml= l, l-1, …, -l

For a particle having a certain energy (e.g. characterized by l=2), the plane of rotation can only take a discrete range of orientations (characterized by one of the 2l+1 values ml)

The orientation of a rotating body is quantized

Vector representation of the angular momentum

While L2 and Lz commute, Lz and Lx (or Ly) do not commute

Lz and Lx (or Ly) cannot be measured accurately and simultaneously

If Lz is known precisely, Lx and Ly are completely unknown: representation with a cone is more realistic than a simple vector. It means that once the orientation of the rotation plane is known, Lx and Ly can take any value.

Cone representation of the angular momentum

Notation: L is also often written J in textbooks

2.3.2 Spin of a particle

The wavefunction of the particle has to satisfy specific boundary conditions for this motion (not the same as for the 3D-rotation). It follows that this spin angular momentum is characterized by two quantum numbers:

s (in place of l) > 0 and ∈ R → the magnitude of the spin angular momentum: {s(s+1)}1/2ħ

ms ≤ |s| → the projection of the spin angular momentum on the z-axis: msħ

The spin “s” of a particle is an angular momentum characterizing the rotation (the spinning) of the particle around its own axis.

NB: In this course the spin is introduced as such. But in the Relativistic Quantum Field Theory, the spin appears naturally from the mathematics.

Electrons: s = ½ → the magnitude of the spin angular momentum is 0.8666 ħ. The spins may lie in 2s+1= 2 different orientations (see figure). The orientation for ms= +½, called and noted ; the orientation for ms= -½ is called and noted .

Photons: s = 1 → the angular momentum is 21/2 ħ

The properties of fermions are described in the statistic of Fermi-Dirac.

The properties of bosons are described in the statistic of Bose-Einstein.