Energy is absorbed and emitted in quantum packets of energy related to the frequency of the radiation: Planck constant h= 6.63 10 −34 J·s Planck constant.

Energy is absorbed and emitted in quantum packets Energy is absorbed and emitted in quantum packets of energy related to the frequency of the radiation:of energy related to the frequency of the radiation:

h= 6.63 10−34 J·s Planck constantPlanck constant

Confining a particle to a region of space imposes conditions that lead to energy quantization.

Copyright (c) Stuart Lindsay 2008

hE

• De BroglieDe Broglie:

The position of freely propagating particles can be predicted by associating a wave of wavelength

“When is a system quantum mechanical and when is it

classical?”


mv

h

m=9.1·10-31 kg; q=1.6·10-19 C

In un potenziale di 50kV:

J.qVmvE 15192 1085000010612

1

1810312 sm.m

Ev

122831 102110311019 skgm...mvp

pm.m..

.

p

h52510525

1021

10636 1222

34

Wave-like behavior

• Waves diffract and waves interfere


The key points of QM

• Particle behavior can be predicted only in terms of probability.

Quantum mechanics provides the tools for making probabilistic predictions.

• The predicted particle distributions are wave-like.

The De Broglie wavelength associated with probability distributions for macroscopic particles is so small that quantum effects are not apparent.


The Uncertainty PrincipleThe Uncertainty Principle

4

hpx

A particle confined to a tiny volume must have an enormous momentum.

12510

34

10275104

10636

4

smkg..

x

hp

Ex. speed of an electron confined to a hydrogen atom (d≈1Å)

1531

25

10851019

10275

sm..

.

m

pv

htE ~ The uncertainty in energy of a particle observed for a very short time can be enormous.

Ex. lifetime of an electronic transition with a band gap of 4eV

fss.

E

ht 110

4

10154 1515

seV.h 1510144

WavefunctionsWavefunctions

The values of probability amplitude at all points in space and time are given by a “wavefunction”

(r, t)

(r)

Systems that do not change with time are called “stationary”:

WavefunctionsWavefunctions

Since the particle must be somewhere:

r

rrr 1)()( 3* d

1

In the shorthand invented by Dirac this equation is:

Pauli Exclusion PrinciplePauli Exclusion Principle

• Consider 2 identical particles:

particle 1 in state particle 2 in state• The state could just as well be:

particle 1 in state particle 2 in state

• Thus the two particle wavefunction is

1

2 1

2

)1()2()2()1( 2121 Atotal

.

+ for Bosons, - for Fermions+ for Bosons, - for Fermions

Bosons and fermionsBosons and fermions

• Fermions are particles with odd spins, where the quantum of spin is

Electrons have spin and are Fermions

• 3He nuclei have spin and are Fermions

• 4He nuclei have spins and are Bosons

2/

2

3

2

4


2/

Two identical Fermions cannot be found in the same Two identical Fermions cannot be found in the same state.state.

For fermions the probability amplitudes for exchange of particles must change sign.

For two fermions:

Pauli Exclusion PrinciplePauli Exclusion Principle

)r()r()r()r()r,r( 21122211212

1

Bosons are not constrainedBosons are not constrained: : an arbitrary number of boson particles can populate an arbitrary number of boson particles can populate the same state!the same state!

For bosons the probability amplitudes for all combinations of the particles are added.

For two bosons:

)r()r()r()r()r,r( 21122211212

1

This increases the probability that two particles will occupy the same state (Bose condensation).

33HeHe 44HeHe

superfluiditysuperfluidity Bose-Einsteincondensation

Photons are bosons!

spin = ±

In terms of classical optics the two states correspond to left and right circularly polarized light.

hc

hE h

kp

Photons have a spin angular momentum (s=1):

The Schrödinger EquationThe Schrödinger Equation

• “Newton’s Law” for probability amplitudes:

dt

txitxtxU

x

tx

m

),(),(),(

),(

2 2

22

Time independent Schrödinger EquationTime independent Schrödinger Equation

• If the potential does not depend upon time, the particle is in a ‘stationary state’, and the wavefunction can be written as the product

• putting this into the Schrödinger equation gives

)()(),( txtx

dt

ti

txxU

x

x

m

)(

)(

1)()(

)(

2 2

22

Time independent Schrödinger Equation

Econstdt

t

ti

.

)(

)(

1

)()(

tEdt

ti

t

Eiexp)t(

E

t

iExtx

exp)(),(

Note that the probability is NOT a function of time!

Time independent Schrödinger EquationTime independent Schrödinger Equation

)()()()(

2 2

22

xExxUx

x

m

)()( xExH

Solutions of the TISE: Solutions of the TISE: 1. Constant potential 1. Constant potential

)()()(

2 2

22

xExVx

x

m

)()(2)(

22

2

xVEm

x

x

ikxAx exp)( 2

2 )(2

VEm

k

For a free particle (V=0):

m

kE

2

22

tkxiAtx exp),(

Note the quantum expression for momentum:

kp

Including the time dependence:

2

kh

p

2.Tunneling through a barrier

0

V

E

V(x)

X=0

Classically, the electron would just bounce off the barrier but……

• To the right of the barrier

22

)(2)(2

EVm

iVEm

k

x

x

(just to the left of a boundary) = (just to the right of a boundary).

xAx exp)(

But QM requires:

(just to the left of a boundary) = (just to the right of a boundary).

xAx 2exp)( 22

Decay length for electrons that “leak” out of a metal is ca. 0.04 nm

Real part of

)exp()exp()( 2 ikxikxx

Is constant here

Decays exponentially here

xAx 2exp)( 22

220 A)( e

A)(

22

2

1

The distance over which the probability falls to 1/e of its value at the boundary is 1/2k.Per V-E=5eV (Au ionization energy):

1o

216218

3

2A141

1066103

51051122

.).()(

)VE(mk

22

511

c

keV

c

Em seV. 161066

o

A4402

1.

k

3. Particle in a box3. Particle in a box

• Infinite walls so must go to zero at edges

• This requirement is satisfied with

• The energy is

• And the normalized wave function is

kxB sin and kL=n i.e. k=n/L n=1,2,3....

2

222

2mL

nEn

L

xn

Ln

sin2

2

22

1 2

12

mL

nE n,n

The energy gap of semiconductor crystals that are just a few nm in diameter (quantum dots) is controlled primarily by their size!

4. Density of states for a free particle4. Density of states for a free particle

• The energy spacing of states may be infinitesimal, but the system is still quantized.

• Periodic boundary conditions: wavefunction repeats after a distance L (we can let L → )

1)0exp()exp()exp()exp( ,, zyxzyx ikLikLikLik

L

nk

L

nk

L

nk z

zy

yx

x

2

2

2


Density of statesDensity of states = number of quantum states available per unit energy o per unit wave vector.

• For a free particle:

222222

22 zyx kkkmm

E k

k-spacek-space: the allowed states are points in a space with coordinates kx, ky and kz.

The “volume” of k-space occupied by each allowed point is

32

L

L

nk

L

nk

L

nk z

zy

yx

x

2

2

2

k-space is filled with an uniform grid of points each separated in units of 2π/L along any axis.

r-space: k-spacek-space:

V

drr 2 43

232

8

4 4

dkkL

V

dkk

k

• Number of states in shell dk (V=L3):

2

2

3

2

28

4

dkVkdkVk

dn

2

2

2Vk

dk

dn

The number of states per unit wave vector increases as the square of the wave vector.

5. A tunnel junction5. A tunnel junction

Electrode 1

The gapElectrode 2

ikxrikxx L expexp)( xBxAx M expexp)(

ikxtx R exp)(

Electrode 1

The gap

Electrode 2

Real part of

• Imposing the two boundary conditions on

and continuous :

)(x

x

x

)(

LEVE

VT

2

0

20 sinh

)(41

1

Or with 2L>>1:

LiLi 02.1exp)( 0

= V0-E workfunctionworkfunction [Φ(gold)=5eV]

L in Å, in eV


TransmissionTransmissioncoefficientcoefficient

The scanning tunneling microscope

LiLi 02.1exp)( 0

The current decays a factor 10 for each Å of gap.

L=5Å V=1 Volt

i ≈ 1nA

Approximate Methods for solving the Approximate Methods for solving the SchrSchrödinger equationödinger equation

• Perturbation theoryPerturbation theory works when a small perturbing term can be added to a known Hamiltonian to set up the unknown problem:

• Then the eigenfunctions and eigenvalues can be approximated by a power series in :

HHH ˆˆˆ0

)2(2)1()0(nnnn EEEE

)2(2)1()0(nnnn


• Plugging these expansions into the SE and equating each term in each order in so the SE to first order becomes:

• The (infinite series) of eigenstates for the Schrödinger equation form a complete basis set for expanding any other function:

)1()0(0

)0()0()1( )ˆ(ˆnnnnn EHHE


)0()1(m

mnmn a

Substitute this into the first order SE and multiply from the

left by and integrate, gives, after using

The new energy is corrected by the perturbation Hamiltonian

evaluated between the unperturbed wavefunctions.


*)0(n

nmmn

)0()0()1( ˆnnn HE

• The new wave functions are mixed in using the degree to which the overlap with the perturbation Hamiltonian is significant and by the closeness in energy of the states.

• If some states are very close in energy, a perturbation generally results in a new state that is a linear combination of the originally degenerate unperturbed states.


nm mn

mmnnn EE

H)0()0(

)0()0(

)0()0( ˆnmmn HH

Time Dependent Perturbation TheoryTime Dependent Perturbation TheoryTurning on a perturbing potential at t=0 and applying the previous procedure to the time dependent Schrödinger equation:

2

0

expˆ1),(

tkm

km tdtEE

iHi

tkP

For a cosinusoidal perturbation:

P peaks at

km EE

Conservation of energy Conservation of energy in the transitionin the transition


)cos()( ttH

For a system with many levels that satisfy energy conservation

)(ˆ2),( 2

kkm EHdt

kmdP

Density of StatesDensity of States

leading to Fermi’s Golden RuleFermi’s Golden Rule, that the probability per unit time, dP/dt is

)(ˆ2),( 2

kmkm EEH

dt

kmdP

Energy is absorbed and emitted in quantum packets of energy related to the frequency of the radiation: Planck constant h= 6.63 10 −34 J·s Planck constant.

Documents

evcopyright c stuart

state particle

particle wavefunction

probability amplitudes

quantum of spin

identical particles

probability distributions

terms of probability