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Course Meeting Times

Lectures: 2 sessions / week, 1.5 hours / session

Recitations: 2 sessions / week, 1 hour / session

Course Description

Quantum Physics I explores the experimental basis of quantum mechanics, including:

Photoelectric effect

Compton scattering

Photons

Franck-Hertz experiment

The Bohr atom, electron diffraction

deBroglie waves

Wave-particle duality of matter and light

This class also provides an introduction to wave mechanics, via:

Schrödinger's equation

Wave functions

Wave packets

Probability amplitudes

Stationary states

The Heisenberg uncertainty principle

Zero-point energies

Solutions to Schrödinger's equation in one dimension

Transmission and reflection at a barrier

Barrier penetration

Potential wells

The simple harmonic oscillator

Schrödinger's equation in three dimensions

Central potentials

Introduction to hydrogenic systems

Prerequisites

In order to register for 8.04, students must have previously completed Vibrations and Waves (8.03) or Electrodynamics (6.014), and Differential Equations (18.03 or 18.034) with a grade of C or higher.

Textbooks

Required

Gasiorowicz, Stephen. Quantum Physics. 3rd ed. Hoboken, NJ: Wiley, 2003. ISBN: 9780471057000.

Strongly Recommended

French, A. P., and Edwin F. Taylor. Introduction to Quantum Physics. New York, NY: Norton, 1978. ISBN: 9780393090154.

Read Again and Again

Syllabus

第 1 頁 ︐ 共 3 頁MIT OpenCourseWare | Physics | 8.04 Quantum Physics I, Spring 200...

2012/11/14http://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-20...

Feynman, Richard P., Robert B. Leighton, and Matthew L. Sands. The Feynman Lectures on Physics: Commemorative Issue. Vol. 3. Redwood City, CA: Addison-Wesley, 1989. ISBN: 9780201510058.

References

Liboff, Richard L. Introductory Quantum Mechanics. 4th ed. San Francisco, CA: Addison Wesley, 2003. ISBN: 9780805387148.

Eisberg, Robert Martin, and Robert Resnick. Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles. New York, NY: Wiley, 1974. ISBN: 9780471873730.

Problem Sets

The weekly problem sets are an essential part of the course. Working through these problems is crucial to understanding the material deeply. After attempting each problem by yourself, we encourage you to discuss the problems with the teaching staff and with each other--this is an excellent way to learn physics! However, you must write-up your solutions by yourself. Your solutions should not be transcriptions or reproductions of someone else's work.

There will be two in-class exams. There will also be a comprehensive final exam, scheduled by the registrar and held during the final exam period.

Grading Policy

Calendar

ACTIVITIES PERCENTAGES

Exam 1 20%

Exam 2 20%

Final exam 40%

Problem sets 20%

LEC # TOPICS

1Overview, scale of quantum mechanics, boundary between classical and quantum phenomena

2Planck's constant, interference, Fermat's principle of least time, deBroglie wavelength

3Double slit experiment with electrons and photons, wave particle duality, Heisenberg uncertainty

4Wavefunctions and wavepackets, probability and probability amplitude, probability density

5 Thomson atom, Rutherford scattering

6 Photoelectric effect, X-rays, Compton scattering, Franck Hertz experiment

7 Bohr model, hydrogen spectral lines

8Bohr correspondence principle, shortcomings of Bohr model, Wilson-Sommerfeld quantization rules

9 Schrödinger equation in one dimension, infinite 1D well

In-class exam 1

10 Eigenfunctions as basis, interpretation of expansion coefficients, measurement

11Operators and expectation values, time evolution of eigenstates, classical limit, Ehrenfest's theorem

12 Eigenfunctions of p and x, Dirac delta function, Fourier transform

13Wavefunctions and operators in position and momentum space, commutators and uncertainty

14Motion of wavepackets, group velocity and stationary phase, 1D scattering off potential step

15 Boundary conditions, 1D problems: Finite square well, delta function potential

16 More 1D problems, tunneling

17 Harmonic oscillator: Series method

In-class exam 2

18 Harmonic oscillator: Operator method, Dirac notation

19 Schrödinger equation in 3D: Cartesian, spherical coordinates

20 Angular momentum, simultaneous eigenfunctions

21 Spherical harmonics

22 Hydrogen atom: Radial equation

23 Hydrogen atom: 3D eigenfunctions and spectrum

24 Entanglement, Einstein-Podolsky Rosen paradox

Final exam

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Home > Courses > Physics > Quantum Physics I > Lecture Notes

The lecture notes were typed by Morris Green, an MIT student, from Prof. Vuletic's handwritten notes.

Lecture Notes

LEC # TOPICS

1Overview, scale of quantum mechanics, boundary between classical and quantum phenomena (PDF)

2Planck's constant, interference, Fermat's principle of least time, deBroglie wavelength (PDF)

3Double slit experiment with electrons and photons, wave particle duality, Heisenberg uncertainty (PDF)

4Wavefunctions and wavepackets, probability and probability amplitude, probability density (PDF)

5 Thomson atom, Rutherford scattering (PDF)

6 Photoelectric effect, X-rays, Compton scattering, Franck Hertz experiment (PDF)

7 Bohr model, hydrogen spectral lines (PDF)

8Bohr correspondence principle, shortcomings of Bohr model, Wilson-Sommerfeld quantization rules (PDF)

9 Schrödinger equation in one dimension, infinite 1D well (PDF)

10Eigenfunctions as basis, interpretation of expansion coefficients, measurement (PDF)

11Operators and expectation values, time evolution of eigenstates, classical limit, Ehrenfest's theorem (PDF)

12 Eigenfunctions of p and x, Dirac delta function, Fourier transform (PDF)

13Wavefunctions and operators in position and momentum space, commutators and uncertainty (PDF)

14Motion of wavepackets, group velocity and stationary phase, 1D scattering off potential step (PDF)

15Boundary conditions, 1D problems: finite square well, delta function potential (PDF)

16 More 1D problems, tunneling (PDF)

17 Harmonic oscillator: series method (PDF)

18 Harmonic oscillator: operator method, Dirac notation (PDF)

19 Schrödinger equation in 3D: cartesian, spherical coordinates (PDF)

20 Angular momentum, simultaneous eigenfunctions (PDF)

21 Spherical harmonics (PDF)

22 Hydrogen atom: radial equation (PDF)

23 Hydrogen atom: 3D eigenfunctions and spectrum (PDF)

24 Entanglement, Einstein-Podolsky Rosen paradox (PDF)

Your use of the MIT OpenCourseWare site and course materials is subject to our Creative Commons License and other terms of use.

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The Art of Estimation PhysicsThe Art of Estimation Physics

Patrick Diamond, George M. Fuller, Tom MurphyDepartment of Physics, University of California, San Diego, 2012

I. “Natural Units”

Ph 239also known as

“use sloppy thinking”

Richard Feynman :

“never attempt a physics problem until you know the answer”

“Natural Units”

In this system of units there is only one fundamental dimension, energy.This is accomplished by setting Planck’s constant, the speed of light,and Boltzmann’s constant to unity, i.e.,

By doing this most any quantity can be expressed as powers of energy,because now we easily can arrange for

To restore “normal” units we need only insert appropriate powers ofof the fundamental constants above

It helps to remember the dimensions of these quantities . . .

for example, picking convenient units (for me!)

length units

Figure of merit for typical visible light wavelength and corresponding energy

Boltzmann’s constant – from now on measure temperature in energy units

for example . . .

but I like

Examples:

Number Density

e.g., number density of photons in thermal equilibrium at temperature T= 1 MeV

stressesstresses

e.g., energy density, pressure, shear stress, etc.

A quantum mechanics text gives the Bohr radius as

But I see this as . . .

another example . . .

or whatever units you prefer . . .

or maybe even . . .

OK, why not use ergs or Joules and centimeters or meters ?You can if you want but . . .

better to be like Hans Betheand use units scaled to the problem at hand

size of a nucleon/nucleus ~ 1 fmenergy levels in a nucleus ~ 1 MeV

supernova explosion energy

electric charge and potentials/energies

one elementary charge

One Coulomb falling through a potential difference of 1 Volt= 1 Joule= 107 erg

fine structure constant

particle masses, atomic dimensions, etc.

electron rest mass

proton rest mass

neutron‐proton mass difference

atomic mass unit

Avogadro’s number

Handy Facts: Solar System

radius of earth’s orbit around sun

We can do all this for spacetime too !

Define the Planck Mass

. . . and now the Gravitational constant is just . . .

The essence of General Relativity:

There is no gravitation: in locally inertial coordinate systems,which the Equivalence Principle guarantees are always there,the effects of gravitation are absent!

The Einstein Field equations have as their solutions global coordinate systems which cover big patches of spacetime

A convenient coordinate system for A convenient coordinate system for weakweak & & staticstatic (no time dependence) gravitational fields(no time dependence) gravitational fieldsis given by the following coordinate system/metricis given by the following coordinate system/metric:

This would be a decent description of the spacetimegeometry and gravitational effects around the earth,the sun, and white dwarf stars, but not near the surfacesof neutron stars.

It turns out that in a weak gravitational field the time-timecomponent of the metric is related to the Newtonian gravitationalpotential by . . .

Where the Newtonian gravitational potential is

dimensionless !

OBJECTOBJECT MASSMASS(solar masses)

RADIUSRADIUS(cm)

Newtonian GravitationalPotential

earth 3 x 10-6 6.4 x 108 ~10-9

sun 1 6.9 x1010 ~10-6

whitedwarf

~1 5 x 108 ~10-4

neutronstar

~1 106 ~0.1 to 0.2

Characteristic Metric Deviation

Handy Facts: the Universe

Rates and Cross Sections

Eddington LuminosityEddington Luminosity

Sir Arthur Eddingtonwww.sil.si.edu

proton

electron

Photon scattering-induced momentum transfer rate to electrons/protons must be less than gravitational force on proton

Electrons tied to protons via Coulomb force

At radius r where interior mass is M( r ) and photon energy luminosity(e.g., in ergs s-1) is L( r ) the forces are equal when

Flux of photonmomentum

Gravitational force on proton with mass mp

Natural units 1

Natural unitsIn physics, natural units are physical units of measurement based only on universal physical constants. For examplethe elementary charge e is a natural unit of electric charge, and the speed of light c is a natural unit of speed. Apurely natural system of units is defined in such a way that some set of selected universal physical constants are eachnormalized to unity; that is, their numerical values in terms of these units are exactly 1. While this has the advantageof simplicity, there is a potential disadvantage in terms of loss of clarity and understanding, as these constants arethen omitted from mathematical expressions of physical laws.

IntroductionNatural units are intended to elegantly simplify particular algebraic expressions appearing in the laws of physics orto normalize some chosen physical quantities that are properties of universal elementary particles and are reasonablybelieved to be constant. However there is a choice of the set of natural units chosen, and quantities which are set tounity in one system may take a different value or even assumed to vary in another natural unit system.Natural units are "natural" because the origin of their definition comes only from properties of nature and not fromany human construct. Planck units are often, without qualification, called "natural units", although they constituteonly one of several systems of natural units, albeit the best known such system. Planck units (up to a simplemultiplier for each unit) might be considered one of the most "natural" systems in that the set of units is not based onproperties of any prototype, object, or particle but are solely derived from the properties of free space.As with other systems of units, the base units of a set of natural units will include definitions and values for length,mass, time, temperature, and electric charge (in lieu of electric current). Some physicists do not recognizetemperature as a fundamental physical quantity, since it expresses the energy per degree of freedom of a particle,which can be expressed in terms of energy (or mass, length, and time). Virtually every system of natural unitsnormalizes Boltzmann's constant kB to 1, which can be thought of as simply a way of defining the unit temperature.In the SI unit system, electric charge is a separate fundamental dimension of physical quantity, but in natural unitsystems charge is expressed in terms of the mechanical units of mass, length, and time, similarly to cgs. There aretwo common ways to relate charge to mass, length, and time: In Lorentz–Heaviside units (also called "rationalized"),Coulomb's law is F=q1q2/(4πr2), and in Gaussian units (also called "non-rationalized"), Coulomb's law isF=q1q2/r2.[1] Both possibilities are incorporated into different natural unit systems.

Notation and useNatural units are most commonly used by setting the units to one. For example, many natural unit systems includethe equation c = 1 in the unit-system definition, where c is the speed of light. If a velocity v is half the speed of light,then as v = 1⁄2c and c = 1, hence v = 1⁄2. The equation v = 1⁄2 means "the velocity v has the value one-half whenmeasured in Planck units", or "the velocity v is one-half the Planck unit of velocity".The equation c = 1 can be plugged in anywhere else. For example, Einstein's equation E = mc2 can be rewritten inPlanck units as E = m. This equation means "The rest-energy of a particle, measured in Planck units of energy,equals the rest-mass of a particle, measured in Planck units of mass."

Natural units 2

Advantages and disadvantagesCompared to SI or other unit systems, natural units have both advantages and disadvantages:• Simplified equations: By setting constants to 1, equations containing those constants appear more compact and

in some cases may be simpler to understand. For example, the special relativity equation E2 = p2c2 + m2c4 appearssomewhat complicated, but the natural units version, E2 = p2 + m2, appears simpler.

• Physical interpretation: Natural unit systems automatically subsume dimensional analysis. For example, inPlanck units, the units are defined by properties of quantum mechanics and gravity. Not coincidentally, the Planckunit of length is approximately the distance at which quantum gravity effects become important. Likewise, atomicunits are based on the mass and charge of an electron, and not coincidentally the atomic unit of length is the Bohrradius describing the orbit of the electron in a hydrogen atom.

• No prototypes: A prototype is a physical object that defines a unit, such as the International Prototype Kilogram,a physical cylinder of metal whose mass is by definition exactly one kilogram. A prototype definition always hasimperfect reproducibility between different places and between different times, and it is an advantage of naturalunit systems that they use no prototypes. (They share this advantage with other non-natural unit systems, such asconventional electrical units.)

• Less precise measurements: SI units are designed to be used in precision measurements. For example, thesecond is defined by an atomic transition frequency in cesium atoms, because this transition frequency can beprecisely reproduced with atomic clock technology. Natural unit systems are generally not based on quantitiesthat can be precisely reproduced in a lab. Therefore, in order to retain the same degree of precision, thefundamental constants used still have to be measured in a laboratory in terms of physical objects that can bedirectly observed. If this is not possible, then a quantity expressed in natural units can be less precise than thesame quantity expressed in SI units. For example, Planck units use the gravitational constant G, which ismeasurable in a laboratory only to four significant digits.

• Greater ambiguity: Consider the equation a = 1010 in Planck units. If a represents a length, then the equationmeans a = 16 × 10−25 m. If a represents a mass, then the equation means a = 220 kg. Therefore, if the variable awas not clearly defined, then the equation a = 1010 might be misinterpreted. By contrast, in SI units, the equationwould be (for example) a = 220 kg, and it would be clear that a represents a mass, not a length or anything else.In fact, natural units are especially useful when this ambiguity is deliberate: For example, in special relativityspace and time are so closely related that it can be useful not to have to specify whether a variable represents adistance or a time.

Choosing constants to normalizeOut of the many physical constants, the designer of a system of natural unit systems must choose a few of theseconstants to normalize (set equal to 1). It is not possible to normalize just any set of constants. For example, the massof a proton and the mass of an electron cannot both be normalized: if the mass of an electron is defined to be 1, thenthe mass of a proton has to be ≈1836. In a less trivial example, the fine-structure constant, α≈1/137, cannot be set to1, because it is a dimensionless number. The fine-structure constant is related to other fundamental constants

where ke is the Coulomb constant, e is the elementary charge, ℏ is the reduced Planck constant, and c is the speed oflight. Therefore it is not possible to simultaneously normalize all four of the constants c, ℏ, e, and ke.

Natural units 3

Electromagnetism unitsIn SI units, electric charge is expressed in coulombs, a separate unit which is additional to the "mechanical" units(mass, length, time), even though the traditional definition of the ampere refers to some of these other units. Innatural unit systems, however, electric charge has units of [mass]1/2 [length]3/2 [time]−1.There are two main natural unit systems for electromagnetism:• Lorentz–Heaviside units (classified as a rationalized system of electromagnetism units).• Gaussian units (classified as a non-rationalized system of electromagnetism units).Of these, Heaviside-Lorentz is somewhat more common,[2] mainly because Maxwell's equations are simpler inLorentz-Heaviside units than they are in Gaussian units.In the two unit systems, the elementary charge e satisfies:

• (Lorentz–Heaviside),• (Gaussian)where ħ is the reduced Planck constant, c is the speed of light, and α≈1/137 is the fine-structure constant.In a natural unit system where c=1, Lorentz-Heaviside units can be derived from SI units by setting ε0 = μ0 = 1.Gaussian units can be derived from SI units by a more complicated set of transformations, such as dividing allelectric fields by , multiplying all magnetic susceptibilities by 4π, and so on.[3]

Systems of natural units

Planck units

Quantity Expression Metric value Name

Length (L) 1.616×10−35 m Planck length

Mass (M) 2.176×10−8 kg Planck mass

Time (T) 5.3912×10−44 s Planck time

Temperature (Θ) 1.417×1032 K Planck temperature

Electric charge (Q) (L–H) 5.291×10−18 C

(G) 1.876×10−18 C

Planck units are defined by

where c is the speed of light, G is the gravitational constant, is the reduced Planck constant, and kB is theBoltzmann constant.Planck units are a system of natural units that is not defined in terms of properties of any prototype, physical object,or even elementary particle. They only refer to the basic structure of the laws of physics: c and G are part of thestructure of spacetime in general relativity, and ℏ captures the relationship between energy and frequency which is atthe foundation of quantum mechanics. This makes Planck units particularly useful and common in theories ofquantum gravity, including string theory.

Natural units 4

Some may consider Planck units to be "more natural" even than other natural unit systems discussed below. Forexample, some other systems use the mass of an electron as a parameter to be normalized. But the electron is just oneof 15 known massive elementary particles, all with different masses, and there is no compelling reason, withinfundamental physics, to emphasize the electron mass over some other elementary particle's mass.Like the other systems (see above), the electromagnetism units in Planck units can be based on eitherLorentz–Heaviside units or Gaussian units. The unit of charge is different in each.

"Natural units" (particle physics)

Unit Metric value Derivation

1 eV−1 of length 1.97×10−7 m

1 eV of mass 1.78×10−36 kg

1 eV−1 of time 6.58×10−16 s

1 eV of temperature 1.16×104 K

1 unit of electriccharge(L–H)

5.29×10−19 C

1 unit of electriccharge(G)

1.88×10−19 C

In particle physics, the phrase "natural units" generally means:[4][5]

where is the reduced Planck constant, c is the speed of light, and kB is the Boltzmann constant.Like the other systems (see above), the electromagnetism units in Planck units can be based on eitherLorentz–Heaviside units or Gaussian units. The unit of charge is different in each.Finally, one more unit is needed. Most commonly, electron-volt (eV) is used, despite the fact that this is not a"natural" unit in the sense discussed above – it is defined by a natural property, the elementary charge, and theanthropogenic unit of electric potential, the volt. (The SI prefixed multiples of eV are used as well: keV, MeV, GeV,etc.)With the addition of eV (or any other auxiliary unit), any quantity can be expressed. For example, a distance of 1 cmcan be expressed in terms of eV, in natural units, as:[5]

Stoney units

Natural units 5

Quantity Expression Metric value

Length (L) 1.381×10−36 m

Mass (M) 1.859×10−9 kg

Time (T) 4.605×10−45 s

Temperature (Θ) 1.210×1031 K

Electric charge (Q) 1.602×10−19 C

Stoney units are defined by:

where c is the speed of light, G is the gravitational constant, e is the elementary charge, kB is the Boltzmann constant,is the reduced Planck constant, and α is the fine-structure constant.

George Johnstone Stoney was the first physicist to introduce the concept of natural units. He presented the idea in alecture entitled "On the Physical Units of Nature" delivered to the British Association in 1874.[6] Stoney units differfrom Planck units by fixing the elementary charge at 1, instead of Planck's constant (only discovered after Stoney'sproposal).Stoney units are rarely used in modern physics for calculations, but they are of historical interest.

Atomic units

Quantity Expression(Hartree atomic units)

Metric value(Hartree atomic units)

Length (L) 5.292×10−11 m

Mass (M) 9.109×10−31 kg

Time (T) 2.419×10−17 s

Electric charge (Q) 1.602×10−19 C

Temperature (Θ) 3.158×105 K

There are two types of atomic units, closely related: Hartree atomic units:

Rydberg atomic units:[7]

Natural units 6

These units are designed to simplify atomic and molecular physics and chemistry, especially the hydrogen atom, andare widely used in these fields. The Hartree units were first proposed by Douglas Hartree, and are more commonthan the Rydberg units.The units are designed especially to characterize the behavior of an electron in the ground state of a hydrogen atom.For example, using the Hartree convention, in the Bohr model of the hydrogen atom, an electron in the ground statehas orbital velocity = 1, orbital radius = 1, angular momentum = 1, ionization energy = ½, etc.The unit of energy is called the Hartree energy in the Hartree system and the Rydberg energy in the Rydberg system.They differ by a factor of 2. The speed of light is relatively large in atomic units (137 in Hartree or 274 in Rydberg),which comes from the fact that an electron in hydrogen tends to move much slower than the speed of light. Thegravitational constant is extremely small in atomic units (around 10−45), which comes from the fact that thegravitational force between two electrons is far weaker than the Coulomb force. The unit length, mA, is the Bohrradius, a0.

The values of c and e shown above imply that , as in Gaussian units, not Lorentz–Heaviside units.[8]

However, hybrids of the Gaussian and Lorentz–Heaviside units are sometimes used, leading to inconsistentconventions for magnetism-related units.[9]

Quantum chromodynamics (QCD) system of units

Quantity Expression Metric value

Length (L) 2.103 × 10−16 m

Mass (M) 1.673 × 10−27 kg

Time (T) 7.015 × 10−25 s

Temperature (Θ) 1.089 × 1013 K

Electric charge (Q) (L–H) 5.291×10−18 C

(G) 1.876×10−18 C

The electron mass is replaced with that of the proton. Strong units are "convenient for work in QCD and nuclearphysics, where quantum mechanics and relativity are omnipresent and the proton is an object of central interest".[10]

Natural units 7

Geometrized units

The geometrized unit system, used in general relativity, is not a completely defined system. In this system, the basephysical units are chosen so that the speed of light and the gravitational constant are set equal to unity. Other unitsmay be treated however desired. By normalizing appropriate other units, geometrized units become identical toPlanck units.

Summary table

Quantity / Symbol Planck(with Gaussian)

Stoney Hartree Rydberg "Natural"(with L-H)

"Natural"(with Gaussian)

Speed of light in vacuum

Planck's constant(reduced)

Elementary charge

Josephson constant

von Klitzing constant

Gravitational constant

Boltzmann constant

Electron mass

where:• α is the fine-structure constant, approximately 0.007297,• αG is the gravitational coupling constant, ,

References[1] Kowalski, Ludwik, 1986, " A Short History of the SI Units in Electricity, (http:/ / alpha. montclair. edu/ ~kowalskiL/ SI/ SI_PAGE. HTML)"

The Physics Teacher 24(2): 97-99. Alternate web link (subscription required) (http:/ / dx. doi. org/ 10. 1119/ 1. 2341955)[2] http:/ / books. google. com/ books?id=12DKsFtFTgYC& pg=PA385 Thermodynamics and statistical mechanics, by Greiner, Neise, Stöcker[3] See Gaussian units#General rules to translate a formula and references therein.[4] Gauge field theories: an introduction with applications, by Guidry, Appendix A (http:/ / books. google. com/ books?id=kLYx_ZnanW4C&

pg=PA511)[5] An introduction to cosmology and particle physics, by Domínguez-Tenreiro and Quirós, p422 (http:/ / books. google. com/

books?id=OF4TCxwpcN0C& pg=PA422)[6] Ray, T.P. (1981). "Stoney's Fundamental Units". Irish Astronomical Journal 15: 152. Bibcode 1981IrAJ...15..152R.[7] Turek, Ilja (1997). Electronic structure of disordered alloys, surfaces and interfaces (http:/ / books. google. com/

books?id=15wz64DPVqAC& pg=PA3) (illustrated ed.). Springer. pp. 3. ISBN 978-0-7923-9798-4.[8] Relativistic Quantum Chemistry: The Fundamental Theory of Molecular Science, by Markus Reiher, Alexander Wolf, p7

[books.google.com/books?id=YwSpxCfsNsEC&pg=PA7 link]

Natural units 8

[9] A note on units lecture notes (http:/ / info. phys. unm. edu/ ~ideutsch/ Classes/ Phys531F11/ Atomic Units. pdf). See the atomic units articlefor further discussion.

[10] Wilczek, Frank, 2007, " Fundamental Constants, (http:/ / frankwilczek. com/ Wilczek_Easy_Pieces/ 416_Fundamental_Constants. pdf)"Frank Wilczek web site.

External links• The NIST website (http:/ / physics. nist. gov/ cuu/ ) (National Institute of Standards and Technology) is a

convenient source of data on the commonly recognized constants.• K.A. Tomilin: NATURAL SYSTEMS OF UNITS; To the Centenary Anniversary of the Planck System (http:/ /

www. ihst. ru/ personal/ tomilin/ papers/ tomil. pdf) A comparative overview/tutorial of various systems ofnatural units having historical use.

• Pedagogic Aides to Quantum Field Theory (http:/ / www. quantumfieldtheory. info) Click on the link for Chap. 2to find an extensive, simplified introduction to natural units.

Article Sources and Contributors 9

Article Sources and ContributorsNatural units Source: http://en.wikipedia.org/w/index.php?oldid=522811064 Contributors: 6birc, A. di M., A.C. Norman, Akitstika, AlexStamm, AnonMoos, Army1987, Bcharles, Bdesham,Bkl0, Bradeos Graphon, CRGreathouse, Chzz, CosineKitty, Dah31, Daniel5Ko, Dark Formal, Dhollm, Dicklyon, Douglasheggie, Dpb2104, Ehrenkater, Enenn, Exuwon, Finell, Giftlite,GrahamN, Hairy Dude, Hauntedpz, Headbomb, Henning Makholm, Heron, Hymyly, InverseHypercube, Jimp, Just granpa, Justlettersandnumbers, Kehrli, LeadSongDog, Lemmiwinks2, MichaelHardy, Montanabw, Nicholasink, Nneonneo, Nnh, Obondu, OrangeDog, Owlbuster, Peter Horn, Pomona17, Quondum, Rbj, Rmashhadi, Robert K S, Sbyrnes321, SebastianHelm, Sigmundur,Srleffler, Stephan Leeds, Stevecrye, Strait, Tamfang, Thumperward, Thunderbird2, Tibbets74, Truthnlove, Unitfreak, WISo, WarrenPlatts, Wdfarmer, Woz2, Wtshymanski, Yakiv Gluck, 老陳,164 anonymous edits

LicenseCreative Commons Attribution-Share Alike 3.0 Unported//creativecommons.org/licenses/by-sa/3.0/

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1.97327U5.1U￣ 7m出 U.2um1.7826627.1U￣ 36kg

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5.3442883.1U￣ 28kg.In/s

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1.444U271nlT

1.444U271.1U4A/m

1.97327U5.lU￣ 14H

l/e、ㄏ

leV1/e、ㄏ

leVleV2leV2leV1

leV3leVleV3leV2leVleV2leV1

l/e╮√

leV2leV2l/e、ㄏ

sUmeconstants:

Planck’squantum几=2π充

ChargeUfelectrUn

BUhrradius, 充2/γne2

EnergyleleetrUnⅣ blt

Rydbergenergy,ε 2/2aU

Hartreeenergy, ′/aUSpeedUflight

Permeabili七 yUf、ΓacuumPerInittivityUfvacuum

BUhrmagnetUnA4assUfelec七 rUn

hIassUfprUtUn

ⅣIassUfneutrUn

GravitatiUncUnstant

85424546.lU￣ 2

26817268.lU￣ 4/e、ㄏ

leV136U5698eⅣΓ

27.211396e、ㄏ

8.3585815.1U￣ 8/e、ㄏ

51U.999U6ke、ㄏ

938.27234〝江e、ㄏ

939.56563h,Ie、ㄏ

67U711.1U￣ 57/e、ㄏ2

1.U5457266.1U￣ 34J.s

6.626U755.lU￣ 34J.s

1.6U217733.1U￣ 19C

5.29177249.1U一 l1m

1.6U217733.1U￣ 19J

2.1798741.lU￣ 18J

4.3597482.1U￣ 18J

2.99792458.lU8m/s

4π .1U￣7H/m

8.854187817.lU一12F/m

9.274Ul54.1U￣ 24J/T

9.lU93897.1U￣ 31kg

1.6726231.lU￣ 27kg

1.6749286.1U￣ 27kg

6.67259.1U￣ 11N.m2/kg2

eV工9Ryd

η甩p

η甩η

TheL︳ m︳.sofC︳ass︳ ca︳

ⅢeRaylcighJeansL.,(l刁 )UφnSmadeaminUr t。 ●tS

deE二va●Un)dUes notagreeMthe*P日㏑ ent巳t⋯ghn.quencies,1 theW㎞frequenε︳es

山etU“Ⅱenefgyddnsity(二ntegfatedUve【 allffequencies)“

山已‘n二te! ＿

In l9一站 M6* P︳anCk fUund a rUEmul6 by an ingen二 Uus

θ分9′19 ＿＿ l

RayⅡ gh-

(l名)

,.hIe必 ,PJz”祕 tr,膨〞6,6inadI‘ mblePaEametei,.hoSe Ⅵluewas

元undtUbe● =6.6多 × lU￣ 2te‘B′

sec.Th二 S㏑w色pPEmChesthe yle七hJmns

fUmuLwUfks,thoughldUes nttheexF【 tnen““cuⅣett lσ

伊七.1.2).theVyleighJmns㏑ wmmUt.UngeneralgEUunds,be

betwee● thehigh＿ fEequen甲 wienfUfmu㏑ and the

Jmns比Ⅱ〃.The免m山 “

z(.,η 工寧

fUm,.henv-→。,色ndEeduces to

如,η =.響↗�加什ra￣ �Vtη-1

望響〞.�〣tr

whenthefrequenεyis㏑哇●,Uf,m。 reacCurate9,whm加》thefU.n山 sa9UductUfthenumbe6UfmUdeslweUbta二 n

d出︻d㏒ theeneEJdenskyby泛η andan。theffaεto【 thatε6n

theaverageener‘ yPe【 deJeeUfffeedom

如,η =γ′,′〞

sinεe

〔o be

rwere,,【 ite

fi°m(1.9)by6nteiP【etC● as

(l-ll)

wese● tl出ttheCIassrⅡ eq㎡P色Ⅱi●°nlav.k●lteredwhenever fiequencksaie

notSmaⅢ εomPπedw二th乃 T/元.T㏑sal〔mtion in theeq山㏑wshowsuehcy,andthatthemodeshaveanavaageenefHdmtdepends on6heL

日mttheh七 hffequen守 mUdeshavea● e呼 sm色ll6VeEage .T㏑se任eaⅣe

cut-Uffemovesthe山伍cultIoftheRavle七 hJea“ densi呼 :thetUtaI

ene‘舒㏑ aavkyUfu㎡tvUlume“ no lUnge6innake.wehave

=罕乃T:耑告

國立清華犬學物理系 (所 )研究室紀錄

一K｝

●45U° K

12盯 K

●UUU° K

。 ●

。.UUU 犯 UUU aU,∞ o 4U.UUUSU.UUU m.UUU m.UUU 6U.qUU

W●Ⅳ●｜●ng● h :n 犬

W日 Ⅳ●︳●ngth :n A

一臼) φ’

Πg.l〞. φ° Dis㎡but°nofPU.,e=md伍 tedbytblackbodyⅢ v肛buStemPa96ures.φ )ComP‘㎡s。nofdat巳 atla刃°Kw︳山 PI日nCkfU.mulaandR.,1dghJeansfU【mula.

ev“,inaccUrdw二thsomeve呼 genefaInUtbns ofclassiaⅢ Physks.Rn,le⋯,h19°°,der二vedthe‘esult

z(〞,T) (1η )

甲hete店 isBoItzm.n可 s「°ns甲 nt,乃 =1.多8×。。

1.eig/degtnd‘ is thevelod呼

●flight,‘ =多 .∞ × l。η m/set.η㏑二ngEed竹nts山“ .9ent mtUthedeI二Ⅵ t二on

weEe(l)thed小 S6出 Ia,Jofequ啷 tqn ofeneEH,accUId㏒ tUwⅡch山e

巳ⅣeEageeneI8yPefde胛 °ffieedomfUiadFnamimlsystem inequⅡ︳b已um︳ s,

㏑山二scontext,4泛冗andφ )them︳ cu㏑●on。fthenu平比【ofmodes〔 .e.,de召uesofffeedom)fUtelectrUmagnetiε m&釭bnWith丘 equen守 inεhe㏑ terval(,,〞 +劦),cUnnnedinacavity.°

.The.qui●“u‘。.㏑,Jp【.dic‘ 山6tth● .nc.g9P“ ●q“ee.ff.●edom k● T/2.rUi

6nU㎡ Ⅲ .U.一 nd山 .m。 des。 rtheelearUm㎎㏄血 6.id.Ees血 Ψ ㏑ haEmon︳ cU㏕血 .U“一

6con● .iibudo口 of乃t/2f,Um山.k︳.etic.ieigy:sm.tchcdb,a1:k.con山山uuon i° m.hepoten● Ⅲ●ne咕 y-3iΨ㏑gti

㏕潶糕甜掛抖戳甲咒盅 :l熬萬犍 :驟智全甘比S●

。nd.° mU.dl,ens㏑mlh臼“n°㏕CUsdⅡttors.

T乃㏕一′

國立清華犬學物理系 (所)研究室紀

E玉ample1.lm‵em,sⅢ§placeⅢ eⅡtlaw9(a9ShUwthatthemaximuInofthePlanckener,deIlsi呼 (1.’9UcclIrSfUrawavelengmUf

thefUmλ〞研=t/Γ,whereΓ kthetemperameandtkacUnstantWhUseValueneedstUbeesjmated.

a,Usetherel㏕Unth如edin(a9tUesimatcthcsdacetempemtureofastarifthe㏕ atiUn

mem你 hasamaxImumwavelenmUf〞%nm.WhatktheiⅡ ●eⅢi呼田datedby山,s如?

●)E血matethewaveI叫曲 andthe上山田6i呼 fthe㏕㎞ Uneml臼edbya」㏕ ng屾gsten

nIamentwhUsesmfacetemperameis33UUK.

SUlm田UⅢ

(a9S㏑cev=c/λ,wehavedI,=lJ,/σλ)〡

λ=● /兄2Vλ

;wecanthuswntePlanck’ s

ener,den斑呼a.ㄌ㏑●emsUfthewavelenm鉧㏑ⅡUW工 .

瓩 η=爪叨勝｜

=早翠牛,石而品πΓΓ

η℃ m鈏山ⅢⅢ Uf9a,Γ )ο。IespUndstUθ 口(λ ,Γ9/aλ =U,w㏑ ch”elds

半年牛卜 ←一↙功“肚,十制㎡

竿。 =a

(1.ll)

(1.12)

(1.13)

andhence

÷=5← -↙灼,

’TU山呵卯曲 (1.ㄌ Uvef㏕ mquen血批 We㏄蝕才∞

出西 =啥 .

a=力 ε/(竹 T).WecansUlvet㏑ sianscendentaIequatiUneithergrap㏑ caluUrnumeh＿

bywntlnga/λ =∫ -ε .Inserting山isvalueIntU(1.13),weUbtain5-ε =5-5θ 5十ε,

lleadstUasuggestlveapprUximatesUlutiUnε ∼ 兒5=U.U337andhencea/兄

=U.U337=4.’ a站 . Sincea =乃 ε/(竹 Γ)andusingthevaIues乃 =6.626× lU￣

34Jsand

I.38U7× lU￣23JK一】

,wecanwritethewave︳ engththatcUrrespUndstUthemaxⅡ numUfPIanckenergydens丘y(lθ)asfUllUws:

λ羽π =易ε 1 2898θ × lU一

‘Π1K

4.9663竹 T T(1.14)

relat㏑n,whIchshUwsthatλ 〞羾 deCreaseswithincreasingtemper前叮eUfthebUd坊 lS

icalled所“怎功⋯兒a名〞勿仂w.1canbeusedtUdeterInine山 ewavelengthcUπespUndlngtU

v〞 aX ——

=罕肛ε

λ羽θX(1.15)

℉Ⅱsrelat㏑n曲UwsthatthepeakUfthera山 atlUn speciumUccursataiequency山 at沁 prUpUi

●UnaltUthetemperature

°)IftheradiatiUnemi仇 edbythestarhasamaximumw出 KlengthUf兄羽π =44‘ nm,its

surfacetempeIattlreisgivenby

2898.9× lU一‘mK

T=446× lU一’m

∼ 65UUK. (1.16)

UsingStefan’slaw(1.l),andasm㏕ ngthe就artUrad㏑●elikeablackbU由,wecanest㎞飩ethe

tUtalpUwerperl1nitareaemittedatthesurfaceUfthesta.

E=σ /=5.67× 1U￣8Wm￣ 2K￣4× (65UUK)4生 lU1.2× 1U‘ Wm￣ 2 (1.17)

ThisanenUl111Uus intensitywhichwilldecreaseaSit spreadsUver space.

●)ThedUminantwaVelengthUfthera山 at沁nemittedbyaglUwingtungsten nlamentUftemperamre33UUKis

2898.9× 1U一‘InK

兄羽aX一33UUK

∼ 878.45Ilm (1.18)

Theintensiyradiatedbythenlament k敼 venby

E=σΓ4=567× lU￣8Wm￣ 2K￣4× (33UUK)4=ε .7× lU‘ Wm￣2 =6.7WⅢn2.(1.19)

國旺溝華人甲物理系

UnIts加 H珀h】mr白 P1外比S

1.唧 TSINHIGH＿ ENERGYPHYSICS

fundamcntaluΠ i6siΠ phy∫ icsareUfIeng亡 1l,Inass,andtIme,thefan】i】iar

@呸KS)exprcssingthese㏑ meters,kⅡ Ugrams,andsecUnds.SuchtsarehUwevernUtveryapprUpriatein parjcIephysics,wherelengthsare:caⅡ ylU￣ i5mandInasseslU￣ 2’

Lengths in particIe physics are usuaⅡy quUted in terms Uf the;θrUr′9r用9(1fm=lU一 15m),andcrUss＿

sectiUnsin tel1Ⅱ sUfthetar刀b=1U￣ 28m2),millibarn(1mb=lU一 3im2),Ur micrUbarn(lub=34m2).Theun丘 UfenergyisbasedUntheelectrUnvUIt(leV=1.6×⊥’jUuleS)w丘htheIargerunitsⅣ ㏑V(=lU° eV),GeV(=lU’ eV),and(=lUlzeV).Massesareusuallymeasured㏑ MeV/ε2,meaningthatif

mass is山化therestenergyisΛtε2Ⅳ【eV FUrexample,theprUtUnhasarest

FgyUf938.28MeVUrU938GeV.UΠ enmasses(mea血ngtherest-energyⅣaIcⅡts9arelUUselyquUtedinⅣ 【eVUrGeV.′

Incalcula“ Uns,thequantijes乃 =九/2π andε Uccurfrequently,anditadvantageUustUuseasysteⅡ lUfunits inwhich乃 =ε =1 WedUthisng§ UmestandardInass沕 U(e.g.,theprUtUnmasS)astheunit:

田U=1.

unitUflengthis thentheCUmptUnwavelengthUfthestandard

尤U=」生 =1;田η

UftiIneis

thatUfenergyis

EU=〃 Uc2=1.

thesc units, it is seen that 力=ε =1. In cUnverting back, at the endthe calculatiUn tU the mUre usual units, it is usefuI tU remember that

=197MeVfm.Thus,apart兌 leUfmasscnergy仰 Uε2=197MeVhaSa

wavelengthUf馬/用 Uε =竹〃〞℅′=1fm.ThrUughUut this text we shall be dealing w丘 h the cUupling Uf

strUng,clectricandweak一 tUmediatingbUsUns. In MKS units,Hccharge,色 iSIneasuredinCUulUmbsandthenne＿ structurecUnStant is

區venby

告=嘉 =t

εi2 1

a=4冗ε°乃ε

生137.

三音舞馬::子I::I;忍:l呈己姆 :l::甘 r｝Ff:胖:燙i:#｝l∵掣r:i一

凹生

′一4π

as in (1.12). A sIniIar denni● Un is used tU relate charges and cUupIingcUnstants in theUther interacuUns.

研究至紀碌

Z.一,由〞停/

ShUwthat,inUne-dimensiUnaIprUbIems,theenergyspectiumofthebUundstates is

a︳ wavsnUn-degenera亡 e.

FUIthesakeUfargument letussuppUsethattheoppUsieis true.Letφl●°and.φ2●●

thenbetwoInearlVindependenteigenfuncuUnsw丘 hthesameenergyeigenvalueE.FrUm

theequauUnS

weUbtain

Ⅱ.e.

㎡+管σ一◤v1=U,”’+管σ一/v2=U,

考告=考告=弓等γ一D,

一〧

.Ψ2一仍

’Ψl=(φ i煦)’

一仰古Ψl)’ =U.＿

＿＿＿＿＿

After integratIngthiSequatiUnwenndthat

φiΨ2一Ψtφ⊥=ac。 nstant

Since,at innnity,吻 =竹 =U(b。 undstates),we了 nuSthaVethecUnstant=Uandhence

ηp1 Ψ2

馬〞1 勺〞2

IIltegratingUncemUreWehaVelnφ 1=Inφ 2十】nε ,i.e.Ψ1=“ 炒,whjchcUntradicts the

assumed︳inear independenceUfthetwUfunctiUns.

J.劦翃 ⊿口:::i;一口t,勿抸

partlcleis initiallyσ <U9㏑ thegrUl1ndstateUfan山區nite,Une-dimensiUnalpUtentialweIl

wallsatx=Uandx=a(a,rthewallatx=a6mUved∫ㄌ〃〞tUx=8a,nnd山ecnergyandwavemnct㏑ nUfthelein thenewwel1.CalculatethewUrkdUnein this prUcess.

°)fthewallatx=θ is nUW∫班dde刀〞mUVedlatr= U)tUx=8a,calculatethenrstexcitedstate,and(iiDtheili妙 Ufnn山ngthepartideina)thegrUundstate,(iDthe

excitedstateUfthenewpUtentialweI1.

FUr′ <Uthep盯ⅡCleWas inapUtentialwellwithwallsatx

動=与勢房吾=㏕。=

瑚=石話名吾F=齬㎡ω=摴血(等)

矺=籍 =手全告弩青, 蛻ω=V:9蕊n(咢)

TheprUbabilΠ yUffindingtheparticlein thegrUundstateUfthe

UbtainediUm(lU.73):Pll=〡〈〃i︳〞I〉〡2where

(a9Whenthewallis mUvedSlUWl坊 theadiabatictheUrem山 aatesthatthepⅢ iclewillbe

fUundatⅡme′ in thegrUundstateUfthenewpUtentialwell〔 hewellwithwalIsatx=Uandx=8a).Thus,wehave

ThewUrk neededtUmUvethewall isΔ 〃 =El-El(〞 )=π2力 2/φ

〞a2)一 π2方 2/φ

〞“a)2)=

ωπ2力 2/(128〞a2).

(b9Ⅵ竹enthewallis mUvedrapldl坊 thepart㏑ lewill nnd北 selfinstant妙 (att≧ U)in the

newpUtentialwe11;itsene:掛 leVelsandwave㎞ ctiUnarenUwgivenby

=Uandx==θ ,andhence

(U≦ x≦ a). (lU.76)

(U≦ x ≦a). (lU.77)

(U≦ x≦ 8a) (lU.78)

newbUxpUtential canbe

〣=“a㎡

一(。吻ω嵞= 血僗)血僗)蝕 =奇

兩l=｜〈叫〡吻〉虍=(署)2‘

-2v劾 =aUU77m7%.

一拖

三拖

(1U.8Φ

TheprUba㏑li妙 Ufnndingthepart始 lein山e白rstexcitedstateUfthenewbUxpUtent㏑ l isg∥en

”Pl2=｜〈吒｜〞l)｜

2where

碗〣=Xa溺劬吻ω激= 血僗)㎡n僗

)禿 =。影戶

拖 =｜ (暆｜吻〉

=(告)2=峏

99一 19U/U.

AsinⅡ IarcalculatiUnleadstU

＿2Vㄎ ,

(lU.79)

::五∵磊拌臨遣Υ撇諡 s控 r;:∵點°befUund㏑㏑gh田

‘〞仰〞族翃ㄔ協七sume山at,atIme′ =a山ewavefuna㏑nφ←,r9Ufauㄩ比炳芯ofthefUIm(EprUbIemI6,CLapterIII9:

毗Φ=畾「銤p├匍,伊 =4矽. ﹉

Inves位 gate山echange㏑ 6meUfthiswaVe” ack說氓 fUrr>U,nU,gε叩 t°n與 pai㏑Ie.一

ItknecessarytodeteⅡ一linethewavefunctionφ 什,● wh㏑hsatis且 estheSchr° dinger

equa位oIl

磁留�∟=Πㄅ←,U,

andw㏑山,at tlmer=U,6山 eg㏑mfuncjUnⅨ x,U9.Wi山 thatend㏑ Ⅵ叩 WeeXpand

φ°,U9㏑ temls ofthesetUΓ UrthonU了 maI位㏄ -independentdgenfunctiUnsφ口fx9,

(Hw口fx9=E紳Π←》 ◆eep.羽 ,fUUtnUte9,thus:

φ什,U9=Σ 幼%(功, 偏 =IφX°φ←,U9激 .

Thefunc● Un干死φK° exp├÷

E:兮 thensajsnestheschrδ d㏑gerequatUn,and,皯

time〞 =U,cUincideswithΨ (x,U1.Hence

Ψ(x,r9=Σ 而φ〞(X》 ÷E.t,

ㄍx,′)=fC(ε ,馮抑(ξ,ω遮

U估,x,● =Σ φ;6)1.什 )θ÷〦f .

Since,in thecaseoffreemUtiUn,theeigenfunctionsaΓ e

㏕。=晶啡 ⋯

theGΓeen’sfunctionβ .Θ becUmes(withpcUntinuUu助

雊 ,ω =｝嘉

唧忙卜

一動一制｝

=｛蒜｝vㄅ器├扣

FrUm(8.39and(8a)ifUI1Uwsthat

岭 ,● =｝｛甾紡｝

蒜而唧├ 毒

+芻。

〢砟,

whenceweUbtain nnaIly,fUrthewavefunctiUn,

叭馬 ●=勿

啊+翔

呼帶｝

←一畾

andfUr山eprUbabiItydens”

阢叫2=Pπ伊〔+馫

｝1珔唧

├ 可角剎

殈 express㏑nhasthesamefU一 Ⅲ asthei㎡曲 IprUbau︳” densi垀

〡岭,U9l2=仰δㄅv2exp├蟲｝

國立清華犬學物理系 (所)研究室紀錄

夕 a伍田匕ˊ 6加 ˊˊ

Ha一m勿托〞 6t6幻坊

S㏑cethegeⅡ erals。lu山nUftheequatlUnUfmotl° nofaclass㏑ al osc:llat° i,X+ω 2X

=U,:softhefUmx=C㎡n●甿+ψ),thetUtalener田

El=T十◤=等 +二咢士′

山岫 m㎜研k山n晒 =神

各 ::ihξ驟瓏 r穿±咢 ir驟S㏑ceT一 awehaveEl-匕 W㏑hme日

驟揣亂絲迋玀驟眾豔褓糮毛荔留i/mB=到“〞ω.田H曲由㎡ poh吐比”印

iSprUpUrt㏑ naltUthe6me′ rw㏑ch加 takestUpass thrUughⅢs㏑terva1.If山●p田山心 Uf

UsciⅡatiUn isΓ =2π /ω .then

吼人x)汝 =“9CUS(ω′+ψ )

西一↗

一〞一

一羽=

放赤一犬

ω一∴=

〞一Γ

wl心 chiStheΓ equiredexpressiUn.

ItcanbeSeenthatth心 prUbabilhyisg了 eateStattheturningpointsx=二 L‘ (Fi＿g.II.23).

AccordingtUquantummechanicstheprobauli” Ufnnd㏑gtheparⅡde㏑ theinterval

(x,x+〞 x)沁

9Vq.(x)激 =坑-Il2有 ax2exp

Ⅱ ShUuⅡ benU竹dthat吒.ω hasmax㏑anearthedaSs㏑ 1tum㏑gpU㏑6仙 =∼巧 9/〞ω,

↙=v伍γ〞ω｝,but,in ooniast㏕ ththedaSsicaIcase,itdoesnUtvanishbeyUndtheSepoin“ .

Thkphen°InenUn,ofthepenetrajonUfapartdeintU比」°nS㎡山“ne螂 .Ⅱnemcenergv”

(｜ x｜ >ω ,dUesnUtIeadtUanycUntradictUnbecausetheequ㏕妙 E=F+◤ ㏑ quantum

mechanicsis notaslnp︳eΓelaJUnbetweennumbers,butbetweenUpeIators;theⅡ n由沁and

山epUtenualenerglescannUt㏑ factbedetem㏑eds㏑mItan.UuSlV.

FUrh珀herleveIs,丘 kfUundthatthecurve2〞 da)becUmestheenvelUpeof山 epeak§ Uf

吒u)㏑山ecla$㏑㏕Ⅱ㎡t刀 →∞｛吐Π&Ⅱ.Z,w㏑hrepresents%.ω =〡Ψl命)尸 ,

2=V21砳司.一＿＿＿＿

(-講｝x

F︳ G.II.23.

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國立活華大學物理系 (所 )研究室紀錄

分類 :

編號 :

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分一編一總

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分類 :

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國立清華大學物理系 (所 )研究室紀錄

國立汗華大學物理系 (所 )研究室紀錄

田立活華大學物理系 (所 )研究室紀錄

(q工 /〃他幻

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一一上

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國立清華大學物理系 r所 )研究室紀錄

分類 :

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一一一一國立清華大學物理系 (所 )研究室紀錄

E兌竹fnf山九＿

— — —

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′ 〞

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田立汗華大孕物理系 (所 )研究室紀

= ′方∠

國立活華大學物理系 (所 )研究室紀

c.rθn:

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分類 :

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田立清華大學物理系 (所 )研究室紀

力 tㄌ」●￣￣

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田立活華大學物理系 (所 )研究室紀錄

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