Quantum Mechanics

2006 10 15

1

Motivation Nanoscience2001 US$423 US$396 Million

NSF 21 20

H. Rohrer (1986)

IBM 2000 8 (NTT) NEC 2001 2

MagiQ Technologies, Inc. 1999 Quantum Information Technology

1 2010 90 (2002.03.20 )

0.25 0.18 0.13 0.09 (90)

1 Nanometer (nm) = 109 m = 10 Atomic Size of Hydrogen: 1.58 (Atomic Radius)

References

David J. Griffiths, Introduction to Quantum Mechanics, 2nd ed., Pearson Education, Inc., 2005.

Paul C.W. Davies and David S. Betts, Quantum Mechanics, 2nd ed., Chapman & Hall, 1994.

Richard Turton, The Quantum Dot, Oxford University Press, 1996.

2

Lecture 1

Schrdingers Equation

1.0 Origins

Plancks Energy Quantization Formula (1900)A natural fact.

The energy of any system that absorbs or emits electromagnetic radiation of frequency is an integer multiple of an energy quantum

E = h = (1.1)

Fig. 1.1: Picture of the 1913 Bohr model of the hydrogen atom showing the

Balmer transition from n=3 to n=2. The electronic orbitals (shown as dashed black circles) are drawn to scale, with 1 inch = 1 Angstrom; note that the radius of the orbital increases quadratically with n. The electron is shown in blue, the nucleus in green, and the photon in red. The frequency of the photon can be determined from Planck's constant h and the change in energy E between the two orbitals. For the 3-2 Balmer transition depicted here, the wavelength of the emitted photon is 656 nm. Photon http://en.wikipedia.org/wiki/Photon

E : energy , =2 angular frequency h : Plancks constant (6.63 1034 Js) = h/2 (1.05 1034 Js) Discrete electromagnetic energies (Quanta, Photons). Light : 1015 Hz , E = 1018 J.

3

http://en.wikipedia.org/wiki/Photon

The photon has zero rest mass and, in empty space, travels at a constant

speed c; in the presence of matter, it can be slowed or even absorbed, transferring energy and momentum proportional to its frequency. The photon has both wave and particle properties; it exhibits wave-particle duality. Max Planck (1918 Nobel Laureate in Physics)

"In recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta."

Niels Bohr (1922 Nobel Laureate in Physics) "for his services in the investigation of the structure of atoms and of the radiation emanating from them."

Einstein's Mass-Energy Equivalence Formula (1905) E=mc2

The relativistic expressions for the energy E of a particle with the rest mass m moving

with a velocity v, i.e., with a momentum p=mv can be manipulated into the fundamental

relativistic energy-momentum equation:

Note that there is no relativistic mass in this equation; the m stands for the rest mass. This

equation is a more general version of Einstein's famous equation "E=mc2".

The equation is also valid for photons, which are massless (have no rest mass):

Therefore a photon's momentum is a function of its energy; it is not analogous to the

momentum in Newtonian mechanics.

Considering an object at rest, the momentum p, in the first equation above, is zero, and we

obtain

4

http://en.wikipedia.org/wiki/Nobel_Prize_in_Physics

which reduces to

suggesting that this last well-known relation is only valid when the object is at rest, giving

what is known as the rest energy. If the object is in motion, we have

From this we see that the total energy of the object E depends on its rest energy and

momentum; as the momentum increases with the increase of the velocity v, so does the

total energy. Mass in special relativity http://en.wikipedia.org/wiki/Relativistic_mass

The fundamental relativistic energy-momentum equation for photons is E = pc (1.2) c : speed of light

Albert Einstein (1921 Nobel Laureate in Physics) "for his services to Theoretical Physics, and especially for his explanation of the photoelectric effect."

De Broglies Particle Wave Duality (1923) :

The wave-particle duality holds that light and matter exhibit

properties of both waves and of particles.

,22

====

hkhc

hp p=h, p= (1.3) k

: wavelength k : wave number

PhotonPhotoeletric EffectCoconut Shy.

One photon knocks out one electron. Photon m = 0 mass-less particle. Louis de Broglie (1929 Nobel Laureate in Physics)

" for his discovery of the wave nature of electrons."

5

http://en.wikipedia.org/wiki/Relativistic_mass

1.1 Collapse of Determinism (Probability) Electromagnetic Wave

Polarized Wave : Electric or magnetic wave.

6

Fig. 1.2: A plane-polarized light encounters an obliquely oriented polarizer only a fraction cos2 of the intensity is transmitted. = 0: All the light is transmitted = 45: Half gets through = 90: No transmission Transmitted Fraction = Transmission Probability = cos2 Weird Feature: Suppose the intensity of the light is reduced so that

Only one photon at a time arrives at the polarizer. Photon cannot be chopped in half Photon either does or does

not get through the polarizer. (Who will get through?) The polarizer has no means of sorting photons into sheep and

goats.

1.2 When is a wave a particle?

Thomas Young's double-slit experiment in 1805 showed that light could act as a

wave, helping to defeat early particle theories of light.

7

http://en.wikipedia.org/wiki/Thomas_Younghttp://en.wikipedia.org/wiki/Double-slit_experimenthttp://en.wikipedia.org/wiki/Wavehttp://en.wikipedia.org/wiki/Elementary_particle

Physicists still regard an electron as a point-like entity but the precise location of that point may not be well-defined.

Matter waves are abstract waves (crime waves, fashions,

unemployment)

Matter waves are also probability waves.

Waves vs. Particles

Fig. 1.3: Particle double-slit experiment. Particle intensities add.

Fig. 1.4: Wave double-slit experiment. Amplitudes add.

),( tr = : wave function

= ),( tr = || e i : phase

I = || 2 : Intensity Superimposition:

8

= 1 + 2

I = ||2 = |1 + 2|2 = ))(( 2121 ++

= |1| 2 + |2| 2 +|12| ][)()( 2121 + ii ee

= I1 + I2 + )(2 2121 CosII

Interference

The wave of each individual particle passes through both slits but the particle passes through only one.

1.3 Schrdingers Wave Equation

Peter Debye (1926): If matter is a wave, there should be a wave equation to describe a matter wave.

A Traveling Sine Wave: )(2sin),( vtxAtx =

A Matter Wave:

=

= )(2exp)(2exp),( txiActxiAtx

=

= )(exp)(2exp EtpxiAt

hEx

hpiA

ipx=

xip

= (1.4)

2

2

2

2 px

=

22

2

xp

= (1.5)

iEt

=

tiE

= (1.6)

VKVmpVmvE +=+=+=22

1 22

9

= Kinetic Energy + Potential Energy Erwin Schrdinger (1926):

),(),(),(2

),(2

22

txtxVx

txmt

txi +

=

Vmt

i += 2

2

2 (1.7)

Key equation of the quantum theory. Must be accepted as a fundamental postulate.

),(),(2yxyx

===

2

2

2

2

yx

+

= or 22

2

2

2

2

zyx

+

+

= (Laplace operator)

V = ),( trV , = ),( tr

How to interpret the wave function (a complex function)?

Observables (position, speed, energy,) are real.

Max Born Postulate:

*),( 2 =tr is the probability density for

a particle to be located at point r at time t.

rd2 is the probability it will be in the infinitesimal

volume at time t. rd is not an observable quantity the phase of is arbitrary (changing) without changing the observable quantity || 2.

Normalization Condition: 1),(32 = rdtr

1D : I = (x1, x2)

10

+

= 2

1

2

1

)**(),( 2x

x

x

xdt

ttdxtx

t

dxxxm

i xx

)**(2

2

12

2

2

2

=

Rate of change of prob. for a particle to be in I.

1

2)**(2 x

xxxm

i

= (1.8)

Define *)*(2),( =

mitrj the probability current

density. (1.8) rate of increase of probability that particle is in I

= j(x1, t) j(x2, t) = flow into I from left + flow into I from right = total flow in I.

Average (or expectation) value of the particles position:

>=< rdtrrr2),(

Similarly, expectation value of other quantity )(rf

>=< rdtrrfrf2),()()(

Erwin Schrdinger (1933 Nobel Laureate in Physics)

" for the discovery of new productive forms of atomic theory." Max Born (1954 Nobel Laureate in Physics)

"for his fundamental research in quantum mechanics, especially for his statistical interpretation of the wavefunction."

11

Lecture 2

Solutions of Schrdingers Equation

2.1 Time-Independent Schrdingers Equation

)(rVV = static (time-independent) potential.

The energy of a particle moving in a static potential is conserved.

Separation of variable:

)()(),( tfrutr =

(1.10)

tf

fiurVu

mu

=

+ )(

21 22

(2.1)

EuVuum

=+ 22

2 (2.2)

)(tEftfi =

(2.3)

(2.3) iEt

etf

=)(

iEt

erutr

= )(),( (2.4) iEt

e

: a pure phase factor

The probability density is independent of the phase, i.e.,

22 )(),( rutr = (2.5)

12

What is E? Consider a free particle by putting V = 0 in (2.2), i.e.,

umEu 22 2=

Which has solutions of the form rkieu (2.6)

2

22 2mEkk ==

2222

21

22mv

mp

mkE === by (1.3)

E is the kinetic energy of the particle.

When V 0, E = Kinetic Energy + Potential Energy = Total Energy

Hence (2.2) is an energy equation.

1D Time-Independent Schrdingers Equation:

)()()()(2 2

22

xEuxuxVdx

xudm

=+ (2.7)

An eigenvalue problem : Given V, find E and u. To be physically acceptable, a wave function must satisfy the following important conditions : (1) is a single-valued function of position and time. (2) is normalizable. (3) and will be continuous everywhere except where

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V has an infinite discontinuity.

2.2 Infinite Square-Well Potential

Example 2.1.

=axax

xV0

)(

Eudx

udm

= 222

2 |x| < a

Solutions :

xBxAxu cossin)( += |x| < a (2.8)

= 0 |x| > 0 A, B : constants

2/1

2

2

=

mE

Continuity Condition on Boundary

Since u must be continuous at x = a, we have

=+=++

0cossin0cossin

aBaAaBaA

(2.9)

Hence, either

====

====

6,4,2,2

0sin

5,3,1,2

0cos

na

naB

na

naA

or

(2.10)

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E En = n2 2 2/ 8ma2 (2.11)

Note: (2.8) Continuity Condition on Boundary (2.9) Restriction on (2.10)

En Discrete energy levels (2.11) Energy is Quantized!

Classical limit 0 En is continuous Ema 2 m : electron mass Atom : a 1010 m En, n=1, 2 will be spaced out

by several electron volts (eV).

Ground state energy (n = 1) :

08 2

22

1 = maE

= 0 (could be classically)

Heisenbergs Principle :

p /a x = a

12

22

22)( E

mamp

a E1 0

Normalization Condition

===

a

a

a

adxaxnAdxudx 1)2/(sin 2222 n : even

a

A 1=

15

Similarly, a

B 1= . Thus

=oddnaxn

a

evennaxnaxu

:)2/cos(1

:)2/sin(1

)(

(2.12)

Fig. 2.1. Position probability densities for the ground state (n = 1) and first excited state (n = 2) of a particle trapped in a 1D impenetrable box of length 2a. Note the symmetry about the origin.

Classical standing waves with discrete frequencies of vibration

==a

adxaxnx

ax 0)2/(sin1 2 (by symmetry)

2.3 Finite Square Well

Example 2.2.

>

=axaxV

xV0

0)( 0 (2.7)

16

Eudx

udm

= 222

2 |x| < a

EuuVdx

udm

=+ 0222

2 |x| > a

2

2mE= , 2

0 )(2 EVm =

For E < V0:

udx

ud 22

2

= |x| < a

udx

ud 22

2

= |x| > a

Solution :

>+

Fig. 2.2. Graphs of cot and 12

are shown. As

is increased so the latter curves reach out farther to the right. The points of intersection determine the values of E, which correspond to the allowed energy levels

( )202332022220211 /,/,/ VEVEVE === .

Fig. 2.3. Wave functions for the first three energy levels of the finite square-well system.

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2.4 Tunneling

Fig. 2.4. A steady stream of particles from the left encounters a square-hill potential barrier with energy E < V0. Quantum penetration of the barrier allows the wave to emerge weakened on the remote side of the barrier, representing a finite probability that a given particle will quantum mechanically tunnel through the barrier. There will thus be a transmitted fraction T and a reflected fraction R. The wave function shown decays rapidly inside the barrier so the tunnel effect is small except for low thin barriers.

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2.5 Finite Difference Approximation

2

* ( ) ( ) ( ) ( ) ( , )2u x V x u x u x x a b

m + = (2.14)

( ) 0 ,u a = ( ) 0u b = Central Difference Formulas:

Partition: : mesh size, , 0,1, 1.1 i

b ah x a ih iN N= = + = ++

1 12 2

1 1 1 12 2

1 12 2( ) : , 1, , .

2 2( )

( ) , i.e., the unknown scalar is an approximate value of the unknown wave function ( ) at

i i i i

i

i i i i i ii

i i i i

u uu x h u x hu x i N

h hu u u U U Uu x

h hu x u U U

u x

+

+ +

+ = =

+ +

= .ix

(2.14)

21 1

* 2

2 2

1 1 2 1* 2 * 2

2 2 2

1 2 2 1 2* 2 * 2 * 2

2 2

1* 2 * 2

2 , 1, .2

12

22 2

2

i i ii i i

N N N N

U U U V U U i Nm h

i V U U Um h m h

i U V U U Um h m h m h

i N U V U Um h m h

+

+ + = =

= + + =

= + + + =

= + + =

20

2 2

1* 2 * 2

2 2 2

2* 2 * 2 * 2

2 2 2

3* 2 * 2 * 2

2 2 2

1* 2 * 2 * 2

2 2

* 2 * 2

0 0 0 0 02

0 0 0 02 2

0 02 2

0

00

0 0 0 0 0 02 2

0 0 0 0 0 0 02

N

N

Vm h m h

Vm h m h m h

Vm h m h m h

Vm h m h m h

Vm h m h

+

+

+

+

+

1 1

2 2

3 3

2 2

1 1

N N

N N

N N

U UU UU U

U UU UU U

=

Eigenvalue Problem:

UAU = (2.15)

21

20061015Motivation Continuity Condition on BoundaryClassical standing waves with discrete frequencies of vibrat