PC Chapter 41

8/14/2019 PC Chapter 41

1/57

Chapter 41

Quantum Mechanics

8/14/2019 PC Chapter 41

2/57

Probability A Particle

Interpretation From the particle point of view, the

probability per unit volume of finding a

photon in a given region of space at aninstant of time is proportional to the

numberNof photons per unit volume at

that time and to the intensityProbability

IN

V V

8/14/2019 PC Chapter 41

3/57

Probability A Wave

Interpretation From the point of view of a wave, the

intensity of electromagnetic radiation is

proportional to the square of the electricfield amplitude, E

Combining the points of view gives

2Probability EV

2I E

8/14/2019 PC Chapter 41

4/57

Probability Interpretation

Summary For electromagnetic radiation, the probability

per unit volume of finding a particle

associated with this radiation is proportionalto the square of the amplitude of theassociated em wave The particle is the photon

The amplitude of the wave associated withthe particle is called the probabilityamplitudeor the wave function The symbol is

8/14/2019 PC Chapter 41

5/57

8/14/2019 PC Chapter 41

6/57

Wave Function, cont. The wave function is often complex-valued The absolute square ||2 = is always real

and positive * is the complete conjugate of It is proportional to the probability per unit volume

of finding a particle at a given point at someinstant

The wave function contains within it all theinformation that can be known about theparticle

8/14/2019 PC Chapter 41

7/57

Wave Function, General

Comments, Final The probabilistic interpretation of the

wave function was first suggested by

Max Born Erwin Schrdinger proposed a wave

equation that describes the manner inwhich the wave function changes inspace and time This Schrdinger wave equation represents a

key element in quantum mechanics

8/14/2019 PC Chapter 41

8/57

Wave Function of a Free

Particle The wave function of a free particle moving

along the x-axis can be written as (x) =Aeikx

k= 2/ is the angular wave number of the waverepresenting the particle

A is the constant amplitude

If represents a single particle, ||2 is the

relative probability per unit volume that theparticle will be found at any given point in thevolume ||2 is called the probability density

8/14/2019 PC Chapter 41

9/57

8/14/2019 PC Chapter 41

10/57

Wave Function of a Free

Particle, Final Because the particle must be

somewhere along thexaxis, the sum of

all the probabilities over all values ofxmust be 1

Any wave function satisfying this equationis said to be normalized

Normalization is simply a statement that

the particle exists at some point in space

21abP dx

8/14/2019 PC Chapter 41

11/57

Expectation Values is not a measurable quantity

Measurable quantities of a particle can

be derived from

The average position is called the

expectation value ofxand is defined

as* x xdx

8/14/2019 PC Chapter 41

12/57

Expectation Values, cont. The expectation value of any function of

xcan also be found

The expectation values are analogous to

weighted averages

*f x f x dx

8/14/2019 PC Chapter 41

13/57

Summary of Mathematical

Features of a Wave Function (x) may be a complex function or a

real function, depending on the system

(x) must be defined at all points inspace and be single-valued

(x) must be normalized

(x) must be continuous in space There must be no discontinuous jumps in

the value of the wave function at any point

8/14/2019 PC Chapter 41

14/57

Particle in a Box A particle is confined

to a one-dimensionalregion of space The box is one-

dimensional

The particle isbouncing elasticallyback and forthbetween twoimpenetrable wallsseparated by L

8/14/2019 PC Chapter 41

15/57

Potential Energy for a Particle

in a Box As long as the particle

is inside the box, thepotential energy does

not depend on itslocation We can choose this

energy value to be zero

The energy is infinitely

large if the particle isoutside the box This ensures that the

wave function is zerooutside the box

8/14/2019 PC Chapter 41

16/57

Wave Function for the Particle

in a Box Since the walls are impenetrable, there

is zero probability of finding the particle

outside the box (x) = 0 forx< 0 andx> L

The wave function must also be 0 at thewalls The function must be continuous (0) = 0 and (L) = 0

8/14/2019 PC Chapter 41

17/57

Wave Function of a Particle in

a Box Mathematical The wave function can be expressed as

a real, sinusoidal function

Applying the boundary conditions and

using the de Broglie wavelength

2( ) sin

x x A

( ) sinnx

x AL

8/14/2019 PC Chapter 41

18/57

Graphical Representations for

a Particle in a Box

8/14/2019 PC Chapter 41

19/57

Active Figure 41.4

(SLIDESHOW MODE ONLY)

8/14/2019 PC Chapter 41

20/57

Wave Function of the Particle

in a Box, cont. Only certain wavelengths for the particle

are allowed

||2 is zero at the boundaries ||2 is zero at other locations as well,

depending on the values ofn

The number of zero points increases byone each time the quantum numberincreases by one

8/14/2019 PC Chapter 41

21/57

8/14/2019 PC Chapter 41

22/57

Energy of a Particle in a Box We chose the potential energy of the

particle to be zero inside the box

Therefore, the energy of the particle is

just its kinetic energy

The energy of the particle is quantized

22

2 1 2 38 , , ,

n

h

E n nmL

K

8/14/2019 PC Chapter 41

23/57

Energy Level Diagram

Particle in a Box The lowest allowed

energy correspondsto the ground state

En = n2E1 are called

excited states E= 0 is not an

allowed state The particle can

never be at rest

8/14/2019 PC Chapter 41

24/57

Active Figure 41.5


8/14/2019 PC Chapter 41

25/57

Boundary Conditions Boundary conditions are applied to determine

the allowed states of the system

In the model of a particle under boundaryconditions, an interaction of a particle with itsenvironment represents one or moreboundary conditions and, if the interactionrestricts the particle to a finite region ofspace, results in quantization of the energy ofthe system

In general, boundary conditions are related tothe coordinates describing the problem

8/14/2019 PC Chapter 41

26/57

Erwin Schrdinger 1887 1961 Best known as one of

the creators of quantum

mechanics His approach was

shown to be equivalentto Heisenbergs

Also worked with: statistical mechanics color vision general relativity

8/14/2019 PC Chapter 41

27/57

Schrdinger Equation The Schrdinger equation as it applies

to a particle of mass m confined to

moving along thexaxis and interactingwith its environment through a potentialenergy function U(x) is

This is called the time-independentSchrdinger equation

2 2

22

d

U Em dx h

8/14/2019 PC Chapter 41

28/57

Schrdinger Equation, cont. Both for a free particle and a particle in

a box, the first term in the Schrdinger

equation reduces to the kinetic energyof the particle multiplied by the wave

function

Solutions to the Schrdinger equation indifferent regions must join smoothly at

the boundaries

8/14/2019 PC Chapter 41

29/57

Schrdinger Equation, final (x) must be continuous

(x) must approach zero asx

approaches This is needed so that (x) obeys the

normalization condition

d/dxmust also be continuous for finitevalues of the potential energy

8/14/2019 PC Chapter 41

30/57

Solutions of the Schrdinger

Equation Solutions of the Schrdinger equation may be

very difficult

The Schrdinger equation has beenextremely successful in explaining thebehavior of atomic and nuclear systems Classical physics failed to explain this behavior

When quantum mechanics is applied tomacroscopic objects, the results agree withclassical physics

8/14/2019 PC Chapter 41

31/57

8/14/2019 PC Chapter 41

32/57

Schrdinger Equation Applied

to a Particle in a Box In the region 0

8/14/2019 PC Chapter 41

33/57

8/14/2019 PC Chapter 41

34/57

8/14/2019 PC Chapter 41

35/57

Finite Potential Well

Region II U= 0

The allowed wave functions are sinusoidal

The boundary conditions no longer requirethat be zero at the ends of the well

The general solution will be

II(x) = Fsin kx+ G cos kx where Fand G are constants

8/14/2019 PC Chapter 41

36/57


Regions I and III The Schrdinger equation for these

regions may be written as

The general solution of this equation is

A and B are constants

2 22 2

2m U Ed C

dx

h

Cx Cx Ae Be

8/14/2019 PC Chapter 41

37/57


Regions I and III, cont. In region I, B = 0

This is necessary to avoid an infinite value

for for large negative values ofx In region III,A = 0

This is necessary to avoid an infinite value

for for large positive values ofx

8/14/2019 PC Chapter 41

38/57


Graphical Results for The wave functions for

various states areshown

Outside the potentialwell, classical physicsforbids the presence ofthe particle

Quantum mechanicsshows the wavefunction decaysexponentially toapproach zero

8/14/2019 PC Chapter 41

39/57


Graphical Results for2

The probability

densities for the

lowest three statesare shown

The functions are

smooth at the

boundaries

8/14/2019 PC Chapter 41

40/57

Active Figure 41.8


8/14/2019 PC Chapter 41

41/57


Determining the Constants The constants in the equations can be

determined by the boundary conditions

and the normalization condition The boundary conditions are

I II

I II

II IIIII III

and at 0

and at

d d

xdx dx

d d x L

dx dx

8/14/2019 PC Chapter 41

42/57

Application Nanotechnology Nanotechnology refers to the design and

application of devices having dimensions

ranging from 1 to 100 nm Nanotechnology uses the idea of trapping

particles in potential wells

One area of nanotechnology of interest to

researchers is the quantum dot A quantum dot is a small region that is grown in a

silicon crystal that acts as a potential well

8/14/2019 PC Chapter 41

43/57

8/14/2019 PC Chapter 41

44/57

Tunneling, cont. Classically, the particle is reflected by the

barrier Regions II and III would be forbidden

According to quantum mechanics, all regionsare accessible to the particle The probability of the particle being in a classically

forbidden region is low, but not zero According to the uncertainty principle, the particle

can be inside the barrier as long as the timeinterval is short and consistent with the principle

8/14/2019 PC Chapter 41

45/57

Tunneling, final The curve in the diagram represents a full

solution to the Schrdinger equation

Movement of the particle to the far side of thebarrier is called tunneling orbarrier

penetration

The probability of tunneling can be described

with a transmission coefficient, T, and a

reflection coefficient, R

8/14/2019 PC Chapter 41

46/57

Tunneling Coefficients The transmission coefficient represents the

probability that the particle penetrates to theother side of the barrier

The reflection coefficient represents theprobability that the particle is reflected by thebarrier

T+ R= 1 The particle must be either transmitted or reflected T e-2CL and can be nonzero

Tunneling is observed and provides evidence

of the principles of quantum mechanics

8/14/2019 PC Chapter 41

47/57

Applications of Tunneling Alpha decay

In order for the alpha particle to escape

from the nucleus, it must penetrate abarrier whose energy is several timesgreater than the energy of the nucleus-alpha particle system

Nuclear fusion Protons can tunnel through the barrier

caused by their mutual electrostaticrepulsion

8/14/2019 PC Chapter 41

48/57

More Applications of Tunneling

Resonant Tunneling Device

Electrons travel in the gallium arsenide

semiconductor They strike the barrier of the quantum dot from

the left The electrons can tunnel through the barrier

and produce a current in the device

8/14/2019 PC Chapter 41

49/57

8/14/2019 PC Chapter 41

50/57

More Applications of Tunneling

Scanning Tunneling Microscope An electrically

conducting probe with avery sharp edge is

brought near thesurface to be studied The empty space

between the tip and thesurface represents the

barrier The tip and the surface

are two walls of thepotential well

8/14/2019 PC Chapter 41

51/57

Scanning Tunneling

Microscope The STM allows

highly detailedimages of surfaces

with resolutionscomparable to thesize of a single atom

At right is thesurface of graphiteviewed with theSTM

8/14/2019 PC Chapter 41

52/57

Scanning Tunneling

Microscope, final The STM is very sensitive to the distance

from the tip to the surface This is the thickness of the barrier

STM has one very serious limitation Its operation is dependent on the electrical

conductivity of the sample and the tip Most materials are not electrically conductive at

their surfaces The atomic force microscope overcomes this

limitation

8/14/2019 PC Chapter 41

53/57

Simple Harmonic Oscillator Reconsider black body radiation as

vibrating charges acting as simple

harmonic oscillators The potential energy is

U= kx2 = m2x2

Its total energy is

K+ U= kA2 = m2A2

8/14/2019 PC Chapter 41

54/57

Simple Harmonic Oscillator, 2 The Schrdinger equation for this

problem is

The solution of this equation gives the

wave function of the ground state as

2 22 2

2

1

2 2

dm x E

m dx h

22m x Be h

8/14/2019 PC Chapter 41

55/57

Simple Harmonic Oscillator, 3 The remaining solutions that describe the

excited states all include the exponential

function

The energy levels of the oscillator are

quantized

The energy for an arbitrary quantum number

n is En = (n + ) where n = 0, 1, 2,

2Cxe

8/14/2019 PC Chapter 41

56/57

8/14/2019 PC Chapter 41

57/57

Energy Levels in the

Harmonic Oscillator The state n = 0 corresponds to the ground

state The energy is Eo =

The state n = 1 is the first excited state The separations between adjacent levels are

equal and are given by E=

As n increases, the agreement between theclassical and the quantum-mechanical resultsimprove

Documents

PC Chapter 41