量子力學發展史

近代科學發展之三

物理模型 粒子模型

• Allowed us to ignore unnecessary details of an object when studying its behavior

系統與剛體• Extension of particle model

波動模型 兩種新模型

• 量子粒子• 邊界條件下的量子粒子

黑體幅射物體在任何溫度下皆會發出熱幅射 (電磁幅射 )

電磁幅射波長會隨物體表面溫度的變化而改變

黑體為一理想系統會吸收所有射入的幅射由黑體發射出的電磁幅射稱為幅射

黑體近似黑體模型可近似為開了一小孔洞的金屬空腔

離開空腔的電磁幅射其性質將只與空腔的表面溫度有關

黑體實驗結論幅射的總功率與溫度的關係滿足 Stefan 定律

• Stefan’s Law

• P = A e T4

• For a blackbody, e = 1 is the Stefan-Boltzmann constant = 5.670 x 10-8 W / m2 . K4

波長分佈曲線的峰值位置隨溫度升高兒而往短波長方向偏移， Wien位移定律• Wien’s displacement lawmax T = 2.898 x 10-3 m.K

黑體幅射強度隨波長的分佈l 幅射強度隨溫度升高而增強

l 總幅射量隨溫度升高而變大• The area under the c

urve

峰值對應的波長隨溫度升高而變短

紫外危機 古典物理的預測與實驗在短波處的結果發生極大的差異

此現象稱為紫外危機，尤其古典物理預測當幅射波的波長趨近於零時更會得到無限大的能量，此與實驗觀察完全相反

Max Planck(拯救紫外危機的英雄 ) 1858 – 1947 He introduced the

concept of “quantum of action”

In 1918 he was awarded the Nobel Prize for the discovery of the quantized nature of energy

Planck’s Theory of Blackbody Radiation

In 1900, Planck developed a structural model for blackbody radiation that leads to an equation in agreement with the experimental results

He assumed the cavity radiation came from atomic oscillations in the cavity walls

Planck made two assumptions about the nature of the oscillators in the cavity walls

Planck’s Theory of Blackbody Radiation

In 1900, Planck developed a structural model for blackbody radiation that leads to an equation in agreement with the experimental results

He assumed the cavity radiation came from atomic oscillations in the cavity walls

Planck made two assumptions about the nature of the oscillators in the cavity walls

Planck’s Assumption, 1

The energy of an oscillator can have only certain discrete values En

• En = n h ƒ

• n is a positive integer called the quantum number

• h is Planck’s constant

• ƒ is the frequency of oscillation

• This says the energy is quantized

• Each discrete energy value corresponds to a different quantum state

Planck’s Assumption, 2

The oscillators emit or absorb energy only in discrete units

They do this when making a transition from one quantum state to another• The entire energy difference between the initial and

final states in the transition is emitted or absorbed as a single quantum of radiation

• An oscillator emits or absorbs energy only when it changes quantum states

Energy-Level Diagram An energy-level

diagram shows the quantized energy levels and allowed transitions

Energy is on the vertical axis

Horizontal lines represent the allowed energy levels

The double-headed arrows indicate allowed transitions

Correspondence Principle(對應原理 ) 當量子系統的量子態總數變大時，量子現象應當會連續地轉變成古典現象• Quantum effects are not seen on an everyday basis si

nce the energy change is too small a fraction of the total energy

• Quantum effects are important and become measurable only on the submicroscopic level of atoms and molecules

Photoelectric Effect

The photoelectric effect occurs when light incident on certain metallic surfaces causes electrons to be emitted from those surfaces• The emitted electrons are called

photoelectrons

The effect was first discovered by Hertz

Photoelectric Effect Apparatus When the tube is kept in the

dark, the ammeter reads zero

When plate E is illuminated by light having an appropriate wavelength, a current is detected by the ammeter

The current arises from photoelectrons emitted from the negative plate (E) and collected at the positive plate (C)

Photoelectric Effect, Results At large values of V, the

current reaches a maximum value• All the electrons emitted a

t E are collected at C The maximum current incr

eases as the intensity of the incident light increases

When V is negative, the current drops

When V is equal to or more negative than Vs, the current is zero

Photoelectric Effect Feature 1

Dependence of photoelectron kinetic energy on light intensity• Classical Prediction

• Electrons should absorb energy continually from the electromagnetic waves

• As the light intensity incident on the metal is increased, the electrons should be ejected with more kinetic energy

• Experimental Result• The maximum kinetic energy is independent of light

intensity

• The current goes to zero at the same negative voltage for all intensity curves

Photoelectric Effect Feature 2

Time interval between incidence of light and ejection of photoelectrons• Classical Prediction

• For very weak light, a measurable time interval should pass between the instant the light is turned on and the time an electron is ejected from the metal

• This time interval is required for the electron to absorb the incident radiation before it acquires enough energy to escape from the metal

• Experimental Result• Electrons are emitted almost instantaneously, even at very

low light intensities

• Less than 10-9 s

Photoelectric Effect Feature 3

Dependence of ejection of electrons on light frequency • Classical Prediction

• Electrons should be ejected at any frequency as long as the light intensity is high enough

• Experimental Result• No electrons are emitted if the incident light falls below s

ome cutoff frequency, ƒc

• The cutoff frequency is characteristic of the material being illuminated

• No electrons are ejected below the cutoff frequency regardless of intensity

Photoelectric Effect Feature 4

Dependence of photoelectron kinetic energy on light frequency• Classical Prediction

• There should be no relationship between the frequency of the light and the electric kinetic energy

• The kinetic energy should be related to the intensity of the light

• Experimental Result

• The maximum kinetic energy of the photoelectrons increases with increasing light frequency

Photoelectric Effect Features, Summary

The experimental results contradict all four classical predictions

Einstein extended Planck’s concept of quantization to electromagnetic waves

All electromagnetic radiation can be considered a stream of quanta, now called photons

A photon of incident light gives all its energy hƒ to a single electron in the metal

Photoelectric Effect, Work Function

Electrons ejected from the surface of the metal and not making collisions with other metal atoms before escaping possess the maximum kinetic energy Kmax

Kmax = hƒ – is called the work function

• The work function represents the minimum energy with which an electron is bound in the metal

Some Work Function Values

Photon Model Explanation of the Photoelectric Effect

Dependence of photoelectron kinetic energy on light intensity• Kmax is independent of light intensity

• K depends on the light frequency and the work function

• The intensity will change the number of photoelectrons being emitted, but not the energy of an individual electron

Time interval between incidence of light and ejection of the photoelectron• Each photon can have enough energy to eject an electro

n immediately

Photon Model Explanation of the Photoelectric Effect, cont

Dependence of ejection of electrons on light frequency• There is a failure to observe photoelectric

effect below a certain cutoff frequency, which indicates the photon must have more energy than the work function in order to eject an electron

• Without enough energy, an electron cannot be ejected, regardless of the light intensity

Photon Model Explanation of the Photoelectric Effect, final

Dependence of photoelectron kinetic energy on light frequency• Since Kmax = hƒ – • As the frequency increases, the kinetic energ

y will increase• Once the energy of the work function is exceeded

• There is a linear relationship between the kinetic energy and the frequency

Cutoff Frequency

The lines show the linear relationship between K and ƒ

The slope of each line is h The absolute value of the y-

intercept is the work function

The x-intercept is the cutoff frequency• This is the frequency below

which no photoelectrons are emitted

Cutoff Frequency and Wavelength

The cutoff frequency is related to the work function through ƒc = / h

The cutoff frequency corresponds to a cutoff wavelength

Wavelengths greater than c incident on a material having a work function do not result in the emission of photoelectrons

ƒcc

c hc

Applications of the Photoelectric Effect

Detector in the light meter of a cameraPhototube

• Used in burglar alarms and soundtrack of motion picture films

• Largely replaced by semiconductor devices

Photomultiplier tubes• Used in nuclear detectors and astronomy

Arthur Holly Compton

1892 - 1962 Director at the lab of

the University of Chicago

Discovered the Compton Effect

Shared the Nobel Prize in 1927

The Compton Effect, Introduction

Compton and coworkers dealt with Einstein’s idea of photon momentum• Einstein proposed a photon with energy E carries a m

omentum of E/c = hƒ / c

Compton and others accumulated evidence of the inadequacy of the classical wave theory

The classical wave theory of light failed to explain the scattering of x-rays from electrons

Compton Effect, Classical Predictions

According to the classical theory, electromagnetic waves of frequency ƒo incident on electrons should• Accelerate in the direction of propagation of the x-rays

by radiation pressure

• Oscillate at the apparent frequency of the radiation since the oscillating electric field should set the electrons in motion

Overall, the scattered wave frequency at a given angle should be a distribution of Doppler-shifted values

Compton Effect, Observations

Compton’s experiments showed that, at any given angle, only one frequency of radiation is observed

Compton Effect, Explanation

The results could be explained by treating the photons as point-like particles having energy hƒ and momentum hƒ / c

Assume the energy and momentum of the isolated system of the colliding photon-electron are conserved• Adopted a particle model for a well-known wave

This scattering phenomena is known as the Compton Effect

Compton Shift Equation

The graphs show the scattered x-ray for various angles

The shifted peak, ', is caused by the scattering of free electrons

• This is called the Compton shift equation

' 1 cosoe

h

m c

Compton Wavelength

The unshifted wavelength, o, is caused by x-rays scattered from the electrons that are tightly bound to the target atoms

The shifted peak, ', is caused by x-rays scattered from free electrons in the target

The Compton wavelength is

0.00243e

hnm

m c

Photons and Waves Revisited

Some experiments are best explained by the photon model

Some are best explained by the wave model We must accept both models and admit that

the true nature of light is not describable in terms of any single classical model

Light has a dual nature in that it exhibits both wave and particle characteristics

The particle model and the wave model of light complement each other

Louis de Broglie 1892 – 1987 Originally studied

history Was awarded the

Nobel Prize in 1929 for his prediction of the wave nature of electrons

Wave Properties of Particles

Louis de Broglie postulated that because photons have both wave and particle characteristics, perhaps all forms of matter have both properties

The de Broglie wavelength of a particle is

h h

p mv

Frequency of a Particle

In an analogy with photons, de Broglie postulated that particles would also have a frequency associated with them

These equations present the dual nature of matter• particle nature, m and v

• wave nature, and ƒ

ƒE

h

Davisson-Germer Experiment If particles have a wave nature, then und

er the correct conditions, they should exhibit diffraction effects

Davission and Germer measured the wavelength of electrons

This provided experimental confirmation of the matter waves proposed by de Broglie

Electron Microscope

The electron microscope depends on the wave characteristics of electrons

The electron microscope has a high resolving power because it has a very short wavelength

Typically, the wavelengths of the electrons are about 100 times shorter than that of visible light

Quantum Particle

The quantum particle is a new simplification model that is a result of the recognition of the dual nature of light and of material particles

In this model, entities have both particle and wave characteristics

We much choose one appropriate behavior in order to understand a particular phenomenon

Ideal Particle vs. Ideal Wave

An ideal particle has zero size• Therefore, it is localized in space

An ideal wave has a single frequency and is infinitely long• Therefore, it is unlocalized in space

A localized entity can be built from infinitely long waves

Particle as a Wave Packet

Multiple waves are superimposed so that one of its crests is at x = 0

The result is that all the waves add constructively at x = 0

There is destructive interference at every point except x = 0

The small region of constructive interference is called a wave packet• The wave packet can be identified as a particle

Wave Envelope

The blue line represents the envelope function This envelope can travel through space with a

different speed than the individual waves

Speeds Associated with Wave Packet

The phase speed of a wave in a wave packet is given by

• This is the rate of advance of a crest on a single wave

The group speed is given by

• This is the speed of the wave packet itself

phasev k

gdv dk

Speeds, cont

The group speed can also be expressed in terms of energy and momentum

This indicates that the group speed of the wave packet is identical to the speed of the particle that it is modeled to represent

2 1

22 2g

dE d pv p u

dp dp m m

Electron Diffraction, Set-Up

Electron Diffraction, Experiment

Parallel beams of mono-energetic electrons are incident on a double slit

The slit widths are small compared to the electron wavelength

An electron detector is positioned far from the slits at a distance much greater than the slit separation

Electron Diffraction, cont

If the detector collects electrons for a long enough time, a typical wave interference pattern is produced

This is distinct evidence that electrons are interfering, a wave-like behavior

The interference pattern becomes clearer as the number of electrons reaching the screen increases

Electron Diffraction, Equations

A minimum occurs when

This shows the dual nature of the electron• The electrons are detected as particles at a localized

spot at some instant of time

• The probability of arrival at that spot is determined by finding the intensity of two interfering waves

sin sin2 2 x

hd or

p d

Electron Diffraction, Closed Slits If one slit is closed, the

maximum is centered around the opening

Closing the other slit produces another maximum centered around that opening

The total effect is the blue line

It is completely different from the interference pattern (brown curve)

Electron Diffraction Explained

An electron interacts with both slits simultaneously

If an attempt is made to determine experimentally which slit the electron goes through, the act of measuring destroys the interference pattern• It is impossible to determine which slit the electron

goes through In effect, the electron goes through both slits

• The wave components of the electron are present at both slits at the same time

Werner Heisenberg 1901 – 1976 Developed matrix

mechanics Many contributions

include• Uncertainty Principle

• Rec’d Nobel Prize in 1932

• Prediction of two forms of molecular hydrogen

• Theoretical models of the nucleus

The Uncertainty Principle, Introduction

In classical mechanics, it is possible, in principle, to make measurements with arbitrarily small uncertainty

Quantum theory predicts that it is fundamentally impossible to make simultaneous measurements of a particle’s position and momentum with infinite accuracy

Heisenberg Uncertainty Principle, Statement

The Heisenberg Uncertainty Principle states if a measurement of the position of a particle is made with uncertainty x and a simultaneous measurement of its x component of momentum is made with uncertainty p, the product of the two uncertainties can never be smaller than

2px x

Heisenberg Uncertainty Principle, Explained

It is physically impossible to measure simultaneously the exact position and exact momentum of a particle

The inescapable uncertainties do not arise from imperfections in practical measuring instruments

The uncertainties arise from the quantum structure of matter

Heisenberg Uncertainty Principle, Another Form

Another form of the Uncertainty Principle can be expressed in terms of energy and time

This suggests that energy conservation can appear to be violated by an amount E as long as it is only for a short time interval t

2tE

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 an instant of time is proportional to the number N of photons per unit volume at that time and to the intensity

Probability

V

NI

V

Probability – A Wave Interpretation

From the point of view of a wave, the intensity of electromagnetic radiation is proportional to the square of the electric field amplitude, E

Combining the points of view gives

2Probability

VE

2I E

Probability – Interpretation Summary

For electromagnetic radiation, the probability per unit volume of finding a particle associated with this radiation is proportional to the square of the amplitude of the associated em wave• The particle is the photon

The amplitude of the wave associated with the particle is called the probability amplitude or the wave function• The symbol is

Wave Function

The complete wave function for a system depends on the positions of all the particles in the system and on time• The function can be written as (r1, r2, … rj…., t) = (rj)e-it

• rj is the position of the jth particle in the system

= 2 ƒ is the angular frequency

• 1i

Wave Function, con’t 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 some instant The wave function contains within it all the

information that can be known about the particle

Wave Function, General Comments, Final

The probabilistic interpretation of the wave function was first suggested by Max Born

Erwin Schrödinger proposed a wave equation that describes the manner in which the wave function changes in space and time• This Schrödinger Wave Equation represents a key

element in quantum mechanics

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 wave representing the particle

• A is the constant amplitude If represents a single particle, ||2 is the relat

ive probability per unit volume that the particle will be found at any given point in the volume• ||2 is called the probability density

Wave Function of a Free Particle, Cont In general, the probability

of finding the particle in a volume dV is ||2 dV

With one-dimensional analysis, this becomes ||2 dx

The probability of finding the particle in the arbitrary interval axb is

and is the area under the curve

b

a

2

ab dxP

Wave Function of a Free Particle, Final

Because the particle must be somewhere along the x axis, the sum of all the probabilities over all values of x must be 1

• Any wave function satisfying this equation is said to be normalized

• Normalization is simply a statement that the particle exists at some point in space

1dxP 2

ab

Expectation Values

is not a measurable quantityMeasurable quantities of a particle can

be derived from The average position is called the

expectation value of x and is defined as

dxx*x

Expectation Values, cont

The expectation value of any function of x can also be found

• The expectation values are analogous to averages

dxxf*xf

Particle in a Box

A particle is confined to a one-dimensional region of space• The “box” is one-

dimensional The particle is bouncing

elastically back and forth between two impenetrable walls separated by L

Classically, the particle’s momentum and kinetic energy remain constant

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 for x < 0 and x > L

The wave function must also be 0 at the walls• The function must be continuous(0) = 0 and (L) = 0

Potential Energy for a Particle in a Box As long as the particle is

inside the box, the potential energy does not depend on its location• We can choose this

energy value to be zero The energy is infinitely

large if the particle is outside the box• This ensures that the

wave function is zero outside the box

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

xx A

( ) sinn x

x AL

Graphical Representations for a Particle in a Box

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 of nThe number of zero points increases by

one each time the quantum number increases by one

Momentum of the Particle in a Box

Remember the wavelengths are restricted to specific values

Therefore, the momentum values are also restricted

2

h nhp

L

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

21, 2, 3

8n

hE n n

mL

Energy Level Diagram – Particle in a Box

The lowest allowed energy corresponds to the ground state

En = n2E1 are called excited states

E = 0 is not an allowed state The particle can never be at

rest The lowest energy the

particle can have, E = 1, is called the zero-point energy

Boundary Conditions

Boundary conditions are applied to determine the allowed states of the system

In the model of a particle under boundary conditions, an interaction of a particle with its environment represents one or more boundary conditions and, if the interaction restricts the particle to a finite region of space, results in quantization of the energy of the system

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

Erwin Schrödinger 1887 – 1961 Best known as one of

the creators of quantum mechanics

His approach was shown to be equivalent to Heisenberg’s

Also worked with• statistical mechanics

• color vision

• general relativity

Schrödinger Equation

The Schrödinger equation as it applies to a particle of mass m confined to moving along the x axis and interacting with its environment through a potential energy function U(x) is

This is called the time-independent Schrödinger equation

2 2

22

h dU E

m dx

Schrödinger Equation, cont

Both for a free particle and a particle in a box, the first term in the Schrödinger equation reduces to the kinetic energy of the particle multiplied by the wave function

Solutions to the Schrödinger equation in different regions must join smoothly at the boundaries

Schrödinger Equation, final

(x) must be continuous(x) must approach zero as x approache

s ±• This is needed so that (x) obeys the normali

zation condition

d / dx must also be continuous for finite values of the potential energy

Solutions of the Schrödinger Equation

Solutions of the Schrödinger equation may be very difficult

The Schrödinger equation has been extremely successful in explaining the behavior of atomic and nuclear systems• Classical physics failed to explain this behavior

When quantum mechanics is applied to macroscopic objects, the results agree with classical physics

Potential Wells

A potential well is a graphical representation of energy

The well is the upward-facing region of the curve in a potential energy diagram

The particle in a box is sometimes said to be in a square well• Due to the shape of the potential energy

diagram

Schrödinger Equation Applied to a Particle in a Box

In the region 0 < x < L, where U = 0, the Schrödinger equation can be expressed in the form

The most general solution to the equation is (x) = A sin kx + B cos kx• A and B are constants determined by the boundary an

d normalization conditions

22

2 2

2d mEk

dx

Schrödinger Equation Applied to a Particle in a Box, cont.

Solving for the allowed energies gives

The allowed wave functions are given by

• The second expression is the normalized wave function

• These match the original results for the particle in a box

22

28n

hE n

mL

2( ) sin sin

n x n xx A

L L L

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

• Storage of binary information using quantum dots is being researched

Quantum Corral

Corrals and other structures are used to confine surface electron waves

This corral is a ring of 48 iron atoms on a copper surface

The ring has a diameter of 143 nm

Tunneling

The potential energy has a constant value U in the region of width L and zero in all other regions

This a called a square barrier

U is the called the barrier height

Tunneling, cont

Classically, the particle is reflected by the barrier• Regions II and III would be forbidden

According to quantum mechanics, all regions are 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 time interval is short and consistent with the Principle

Tunneling, final

The curve in the diagram represents a full solution to the Schrödinger equation

Movement of the particle to the far side of the barrier is called tunneling or barrier penetration

The probability of tunneling can be described with a transmission coefficient, T, and a reflection coefficient, R

Tunneling Coefficients

The transmission coefficient represents the probability that the particle penetrates to the other side of the barrier

The reflection coefficient represents the probability that the particle is reflected by the barrier

T + R = 1• The particle must be either transmitted or reflected

• T e-2CL and can be non zero Tunneling is observed and provides evidence of

the principles of quantum mechanics

Applications of Tunneling

Alpha decay• In order for the alpha particle to escape from

the nucleus, it must penetrate a barrier whose energy is several times greater than the energy of the nucleus-alpha particle system

Nuclear fusion• Protons can tunnel through the barrier caused

by their mutual electrostatic repulsion

More Applications of Tunneling – Scanning Tunneling Microscope

An electrically conducting probe with a very sharp edge is brought near the surface to be studied

The empty space between the tip and the surface represents the “barrier”

The tip and the surface are two walls of the “potential well”

The vertical motion of the probe follows the contour of the specimen’s surface and therefore an image of the surface is obtained

Cosmic Temperature

In the 1940’s, a structural model of the universe was developed which predicted the existence of thermal radiation from the Big Bang• The radiation would now have a wavelength d

istribution consistent with a black body

• The temperature would be a few kelvins

Cosmic Temperature, cont

In 1965 two workers at Bell Labs found a consistent “noise” in the radiation they were measuring• They were detecting the background radiation

from the Big Bang

• It was detected by their system regardless of direction

• It was consistent with a back body at about 3 K

Cosmic Temperature, Final

Measurements at many wavelengths were needed

The brown curve is the theoretical curve

The blue dots represent measurements from COBE and Bell Labs