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Home > Courses > Physics > Quantum Physics I > Syllabus Syllabus

Course Meeting

Lectures: 2 sessions / week, 1.5 / session Recitations: 2 sessions / week, 1 / session

Course Description

Quantum Physics I explores the experimental basis of , including:

 Photoelectric effect  Compton scattering  Photons  Franck- experiment  The Bohr atom, electron diffraction  deBroglie waves  Wave-particle duality of matter and 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  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 htt p://oc w. mit. edu/courses/physics/8-04-quantu m-physics-i-spri ng-20... 2012/11/14 MI T OpenCourse Ware | Physics | 8. 04 Quantu m Physics I, Spri ng 200... 2 3 第 頁 ︐ 共 頁

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.

Exams

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

ACTIVITIES PERCENTAGES

Exam 1 20%

Exam 2 20%

Final exam 40%

Problem sets 20%

Calendar

LEC # TOPICS

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

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

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

Wavefunctions and wavepackets, probability and probability amplitude, probability 4

5 Thomson atom, Rutherford scattering

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

7 , hydrogen spectral lines

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

9 Schrödinger equation in one dimension, infinite 1D well

In-class exam 1

10 Eigenfunctions as basis, interpretation of expansion coefficients,

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

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

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

Motion of wavepackets, group and stationary phase, 1D scattering off 14 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 htt p://oc w. mit. edu/courses/physics/8-04-quantu m-physics-i-spri ng-20... 2012/11/14 MI T OpenCourse Ware | Physics | 8. 04 Quantu m Physics I, Spri ng 200... 3 3 第 頁 ︐ 共 頁

18 Harmonic oscillator: Operator method, Dirac notation

19 Schrödinger equation in 3D: Cartesian, spherical coordinates

20 , simultaneous eigenfunctions

21 Spherical harmonics

22 : Radial equation

23 Hydrogen atom: 3D eigenfunctions and spectrum

24 Entanglement, Einstein-Podolsky Rosen paradox

Final exam

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htt p://oc w. mit. edu/courses/physics/8-04-quantu m-physics-i-spri ng-20... 2012/11/14 MI T OpenCourse Ware | Physics | 8. 04 Quantu m Physics I, Spri ng 200... 1 1 第 頁 ︐ 共 頁

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

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

LEC # TOPICS

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

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

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

Wavefunctions and wavepackets, probability and probability amplitude, probability 4 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)

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

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

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

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

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

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

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

Boundary conditions, 1D problems: finite square well, delta function potential 15 (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. htt p://oc w. mit. edu/courses/physics/8-04-quantu m-physics-i-spri ng-20... 2012/11/14 ∠θ‘tuFε z

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Patrick Diamond, George M. Fuller, Tom Murphy Department of Physics, University of California, San Diego, 2012 Richard Feynman :

“use sloppy thinking”

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

In this system of units there is only one fundamental dimension, . This is accomplished by setting Planck’s constant, the , 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 of of the fundamental constants above It helps to remember the dimensions of these quantities . . .

for example, picking convenient units (for me!) units

Figure of merit for typical visible light wavelength and corresponding energy Boltzmann’s constant –from now on measure in energy units

for example . . .

but I like

with Examples:

Number Density

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

e.g., , , shear stress, etc. another example . . .

A quantum mechanics text gives the as

But I see this as . . . or whatever units you prefer . . .

or maybe even . . .

OK, why not use ergs or and centimeters or meters ? You can if you want but . . . better to be like Hans Bethe and use units scaled to the problem at hand

of a nucleon/nucleus ~ 1 fm energy levels in a nucleus ~ 1 MeV

supernova explosion energy and potentials/energies one

One falling through a potential difference of 1 = 1 = 107 erg

or fine structure constant

SI

cgs particle , atomic dimensions, etc. electron rest rest mass neutron‐proton mass difference atomic mass unit

Avogadro’s number Handy Facts: Solar System

radius of earth’s orbit around We can do all this for too !

Define the Planck Mass

. . . and now the is just . . . The essence of :

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

The Einstein equations have as their solutions global coordinate systems which cover big patches of spacetime A convenient coordinate system for weak & static (no time dependence) gravitational fields is given by the following coordinate system/metric:

This would be a decent description of the spacetime geometry and gravitational effects around the earth, the sun, and white dwarf stars, but not near the surfaces of neutron stars. It turns out that in a weak the time-time component of the metric is related to the Newtonian gravitational potential by . . .

Where the Newtonian gravitational potential is

dimensionless ! Characteristic Metric Deviation

OBJECT MASS RADIUS Newtonian OBJECT MASS RADIUS Gravitational (solar masses) (cm) Potential earth 3 x 10-6 6.4 x 108 ~10-9

sun 1 6.9 x1010 ~10-6 white ~1 5 x 108 ~10-4 dwarf neutron ~0.1 ~1 106 star to 0.2 Handy Facts: the Rates and Cross Sections Eddington Luminosity Photon scattering-induced momentum transfer rate to electrons/ must be less than gravitational on proton

Electrons tied to protons via Coulomb force proton At radius r where interior mass is M( r ) and luminosity -1 (e.g., in ergs s ) is L( r ) the are equal when Sir Arthur Eddington electron www.sil.si.edu

Flux of photon Gravitational force on proton with mass mp momentum Natural units 1 Natural units

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

Introduction Natural units are intended to elegantly simplify particular algebraic expressions appearing in the laws of physics or to normalize some chosen physical quantities that are properties of universal elementary particles and are reasonably believed to be constant. However there is a choice of the set of natural units chosen, and quantities which are set to unity 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 and not from any human construct. are often, without qualification, called "natural units", although they constitute only one of several systems of natural units, albeit the best known such system. Planck units (up to a simple multiplier for each unit) might be considered one of the most "natural" systems in that the set of units is not based on properties of any , 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 ). Some physicists do not recognize temperature as a fundamental , since it expresses the energy per of freedom of a particle, which can be expressed in terms of energy (or mass, length, and time). Virtually every system of natural units normalizes Boltzmann's constant k to 1, which can be thought of as simply a way of defining the unit temperature. B In the SI unit system, electric charge is a separate fundamental dimension of physical quantity, but in natural unit systems charge is expressed in terms of the mechanical units of mass, length, and time, similarly to cgs. There are two common ways to relate charge to mass, length, and time: In Lorentz–Heaviside units (also called "rationalized"), Coulomb's law is F=q q /(4πr2), and in (also called "non-rationalized"), Coulomb's law is 1 2 F=q q /r2.[1] Both possibilities are incorporated into different natural unit systems. 1 2

Notation and use Natural units are most commonly used by setting the units to one. For example, many natural unit systems include the 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⁄ c and c = 1, hence v = 1⁄ . The equation v = 1⁄ means "the velocity v has the value one-half when 2 2 2 measured 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 in Planck units as E = m. This equation means "The rest-energy of a particle, measured in Planck , equals the rest-mass of a particle, measured in Planck units of mass." Natural units 2

Advantages and disadvantages Compared 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 equation E2 = p2c2 + m2c4 appears somewhat complicated, but the natural units version, E2 = p2 + m2, appears simpler. • Physical interpretation: Natural unit systems automatically subsume . For example, in Planck units, the units are defined by properties of quantum mechanics and . Not coincidentally, the Planck is approximately the distance at which effects become important. Likewise, atomic units are based on the mass and charge of an electron, and not coincidentally the atomic unit of length is the Bohr radius describing the orbit of the electron in a hydrogen atom. • No : A prototype is a physical object that defines a unit, such as the International Prototype , a physical cylinder of metal whose mass is by definition exactly one kilogram. A prototype definition always has imperfect reproducibility between different places and between different times, and it is an advantage of natural unit systems that they use no prototypes. (They share this advantage with other non-natural unit systems, such as conventional electrical units.) • Less precise : SI units are designed to be used in precision measurements. For example, the is defined by an atomic transition in cesium atoms, because this transition frequency can be precisely reproduced with atomic clock technology. Natural unit systems are generally not based on quantities that can be precisely reproduced in a lab. Therefore, in order to retain the same degree of precision, the fundamental constants used still have to be measured in a laboratory in terms of physical objects that can be directly observed. If this is not possible, then a quantity expressed in natural units can be less precise than the same quantity expressed in SI units. For example, Planck units use the gravitational constant G, which is measurable 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 equation means a = 16 × 10−25 m. If a represents a mass, then the equation means a = 220 kg. Therefore, if the variable a was not clearly defined, then the equation a = 1010 might be misinterpreted. By contrast, in SI units, the equation would 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 relativity space and time are so closely related that it can be useful not to have to specify whether a variable represents a distance or a time.

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

where k is the , e is the elementary charge, ℏ is the reduced , and c is the speed of e light. Therefore it is not possible to simultaneously normalize all four of the constants c, ℏ, e, and k . e Natural units 3

Electromagnetism units In SI units, electric charge is expressed in , a separate unit which is additional to the "mechanical" units (mass, length, time), even though the traditional definition of the refers to some of these other units. In natural unit systems, however, electric charge has units of [mass]1/2 [length]3/2 [time]−1. There are two main natural unit systems for : • 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 in Lorentz-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 ε = μ = 1. 0 0 Gaussian units can be derived from SI units by a more complicated set of transformations, such as dividing all electric 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

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 k is the B . Planck units are a system of natural units that is not defined in terms of properties of any prototype, physical object, or even . They only refer to the basic structure of the laws of physics: c and G are part of the structure of spacetime in general relativity, and ℏ captures the relationship between energy and frequency which is at the foundation of quantum mechanics. This makes Planck units particularly useful and common in theories of quantum gravity, including . Natural units 4

Some may consider Planck units to be "more natural" even than other natural unit systems discussed below. For example, some other systems use the mass of an electron as a parameter to be normalized. But the electron is just one of 15 known massive elementary particles, all with different masses, and there is no compelling reason, within fundamental 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 either Lorentz–Heaviside units or Gaussian units. The unit of charge is different in each.

"Natural units" ()

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 electric 5.29×10−19 C charge (L–H)

1 unit of electric 1.88×10−19 C charge (G)

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

where is the reduced Planck constant, c is the speed of light, and k is the Boltzmann constant. B Like the other systems (see above), the electromagnetism units in Planck units can be based on either Lorentz–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 the anthropogenic unit of , 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 cm can 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, k is the Boltzmann constant, B is the reduced Planck constant, and α is the fine-structure constant. was the first physicist to introduce the concept of natural units. He presented the idea in a lecture entitled "On the Physical Units of Nature" delivered to the British Association in 1874.[6] Stoney units differ from Planck units by fixing the elementary charge at 1, instead of Planck's constant (only discovered after Stoney's proposal). Stoney units are rarely used in modern physics for calculations, but they are of historical interest.

Atomic units

Quantity Expression Metric value ( 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, and are widely used in these fields. The Hartree units were first proposed by , and are more common than the Rydberg units. The units are designed especially to characterize the behavior of an electron in the of a hydrogen atom. For example, using the Hartree convention, in the Bohr model of the hydrogen atom, an electron in the ground state has orbital velocity = 1, orbital radius = 1, angular momentum = 1, = 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. The gravitational constant is extremely small in atomic units (around 10−45), which comes from the fact that the gravitational force between two electrons is far weaker than the Coulomb force. The unit length, m , is the Bohr A radius, a . 0 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 inconsistent conventions 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 nuclear physics, where quantum mechanics and relativity are omnipresent and the proton is an object of central interest".[10] Natural units 7

Geometrized units

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

Summary table

Quantity / Symbol Planck Stoney Hartree Rydberg "Natural" "Natural" (with Gaussian) (with L-H) (with Gaussian)

Speed of light in

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, • α is the gravitational coupling constant, , G

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 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 article for further discussion.

[10] Wilczek, Frank, 2007, " Fundamental Constants, (http:/ / frankwilczek. com/ Wilczek_Easy_Pieces/ 416_Fundamental_Constants. pdf)" 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 of natural units having historical use.

• Pedagogic Aides to (http:/ / www. quantumfieldtheory. info) Click on the link for Chap. 2 to find an extensive, simplified introduction to natural units. Article Sources and Contributors 9 Article Sources and Contributors

Natural 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, Michael Hardy, 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 License

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Naturalunitsdefinedby: 充=c=1(and4π εU=1).Rem血 血ngu㎡ tkchUUsentUbe Ene理乎伈V).

Quantiㄅ SymbUl naturalunits MKSA Length 〞 l/e、ㄏ 1.97327U5.1U‾ 7m出 U.2um Mass η leV 1.7826627.1U‾ 36kg 16s寫 Time t 1/e、ㄏ 6.582122U.1U‾ .66fS Frequency V leV 1.5192669.lU15Hz Speed V 1 2.99792458.1U8m/s MUmentum p leV 5.3442883.1U‾ 28kg.In/s FUrce Γ leV2 8.1194UU3.lU‾ 13N PUwer P leV2 U.2434135Um、 ㄏ Energy E leV 1.6U21773.1U‾ 、19J Charge σ 1 1.8755468.1U‾ 18C Chargedensity p leV3 244.1UU13C/m3 Current r leV 2.8494561mA Currentdensity J leV3 7.3179379.lU1UA/m2 Electhcfield E leV2 432.9U844V/mm PUtential Φ leV 85.424546m、ㄏ PUlarizatiUn P leV2 4.816756U.1U‾ 5C/m2 CUnduc七 iV北 y σ leV 1.69U4124.1U5S/m Re由 stance 見 1 29979246Ω Capacitance U l/e╮ √ 2.1955596.1U‾ 17F ⅣIagneticnux ψ 1 5.6227478.lU一 17、 ㄏb N,IagneticinductiUn B leV2 1.444U271nlT 、 ⅣIagneti2atiUn nr leV2 1.444U271.1U4A/m Induetance 乙 l/e、ㄏ 1.97327U5.lU‾ 14H sUmeconstants:

Planck’ squantum 充 1 1.U5457266.1U‾ 34J.s 几=2π 充 九 2π 6.626U755.lU‾ 34J.s ChargeUfelectrUn C 85424546.lU‾ 2 1.6U217733.1U‾ 19C 2/γ BUhrradius, 充ne2 aU 26817268.lU‾ 4/e、ㄏ5.29177249.1U一 l1m EnergyleleetrUnⅣ blt eV leV 1.6U217733.1U‾ 19J 2/2aU Γ Rydbergenergy,ε 工9Ryd 136U5698eⅣ 2.1798741.lU‾ 18J Hartreeenergy, ′/aU 既 27.211396e、 ㄏ 4.3597482.1U‾ 18J SpeedUflight ε l 2.99792458.lU8m/s Γ Permeabili七 yUf、 acuum ′tU 4π 4π .1U‾ 7H/m

PerInittivityUfvacuum εU 1/4π 8.854187817.lU一 12F/m 8/e、 BUhrmagnetUn uB 8.3585815.1U‾ ㄏ9.274Ul54.1U‾ 24J/T ㄏ A4assUfelec七 rUn r.9,e 51U.999U6ke、 9.lU93897.1U‾ 31kg

hIassUfprUtUn η甩p 938.27234〝 江e、ㄏ 1.6726231.lU‾ 27kg 27kg ⅣIassUfneutrUn η甩η 939.56563h,Ie、ㄏ 1.6749286.1U‾ GravitatiUncUnstant σ 67U711.1U‾ 57/e、ㄏ2 6.67259.1U‾ 11N.m2/kg2 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㎞ fUmuLwUfks,thoughldUes nttheexF【 tnen““cuⅣett lσ frequenε ︳es 伊七.1.2).theVyleighJmns㏑ wmmUt.UngeneralgEUunds,be sinεe 山etU“Ⅱenefgyddnsity(二ntegfatedUve【 allffequencies)“ 〔o be

山已‘n二te! ＿ In l9一站 M6* P︳ anCk fUund a rUEmul6 by an ingen二 Uus betwee● thehigh＿ fEequen甲 wienfUfmu㏑ and the RayⅡ gh- Jmns比Ⅱ〃.The免 m山 “

z(.,η 工 (l名) 寧θ分9′ 19 ＿＿ l

,.hIe必 ,PJz”祕tr,膨 〞6,6inadI‘ mblePaEametei,.hoSe Ⅵluewas lU‾ 2te‘ S㏑w色pPEmChesthe yle七hJmns 元undtUbe● =6.6多 × B′ sec.Th二 fUm,.henv-→ ,色ndEeduces to 。 加ra‾ -1 如,η =.響 ↗什Vtη 望響〞. 〣tr (l-9) whenthefrequenεyis㏑哇●,Uf,m。 reacCurate9,whm加 》 rwere,,【 ite thefU.n山 sa9UductUfthenumbe6UfmUdeslweUbta二 n fi°m(1.9)by as d出︻d㏒ theeneEJdenskyby泛 η andan。theffaεto【 thatε6n 6nteiP【 etC● theaverageener‘ yPe【 deJeeUfffeedom

如,η =γ ,′ 〞 ′ 一 l =罕 乃T:耑告 (l1Φ fiequencksaie wese● tl出 ttheCIassrⅡ eq㎡P色Ⅱi●°nlav.k● lteredwhenever notSmaⅢ εomPπedw二th乃 T/元 .T㏑ sal〔mtion in theeq山 ㏑wshows thatthemodeshaveanavaageenefHdmtdepends on6heL uehcy,and 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

(l-ll)

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 臼) 一 φ’ 一

Πg.l〞 . φ° Dis㎡but°nofPU.,e=md伍 tedbytblackbodyⅢ v肛buStemPa9 ° 6ures.φ )ComP‘ ㎡s。 nofdat巳 atla刃 Kw︳山 PI日 nCkfU.mulaandR.,1dghJeans fU【 mula. ev“,inaccUrdw二thsomeve呼 genefaInUtbns ofclassiaⅢ Physks.Rn,le⋯ ,h 19°°,der二 vedthe‘esult ㏕ 乃 T = 一 (1η z(〞 ,T) ′ )

‾ ns甲 nt,乃 1.eig/degtnd‘ is thevelod呼 甲hete店 isBoItzm.n可 s「° =1.多8× 。 。 ●flight,‘ =多 .∞ ×l。 ηm/set.η㏑二ngEed竹nts山“.9ent mtUthedeI二Ⅵt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 ues offfeedom)fUtelectrUmagnetiε m&釭bnWith丘 equen守 inε he㏑ 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.he poten● 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. 。

國立清華犬學物理系 (所 )研 究室紀 E玉ample1.lm‵ em,sⅢ§placeⅢ eⅡ tlaw9 (a9ShUwthatthemaximuInofthePlanckener,deIlsi呼 (1.’9UcclIrSfUrawavelengmUf thefUmλ〞研=t/Γ ,whereΓ kthetemperameandtkacUnstantWhUseValueneedstUbe esjmated. a,Usetherel㏕Unth如edin(a9tUesimatcthcsdacetempemtureofastarifthe㏕ atiUn mem你 hasamaxImumwavelenmUf〞%nm.WhatktheiⅡ ●eⅢ i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 2Vλ (a9S㏑ s cev=c/λ,wehavedI,=lJ,/σ λ)〡 ˊλ=● /兄 ;wecanthuswntePlanck’ ener,den斑呼a.ㄌ ㏑●emsUfthewavelenm鉧 ㏑ⅡUW工 .

瓩η=爪叨 =早 牛 (1.ll) 勝｜翠,石而品πΓΓ

η℃ m鈏山 Ⅲ Ⅲ Uf9a,Γ )ο IespUndstUθ 口(λ ,Γ9/aλ =U,w㏑ ch”elds 。 功肚 一↙“ 。 (1.12) 半年牛 ← ,十 竿 =a 卜 制 ㎡ andhence -↙ (1.13) ÷=5← 灼,

’ ∞ TU山 曲 (1.ㄌ mquen血 We㏄ 西 . 呵卯 Uvef㏕ 批 蝕 才 出 =啥

國立清華犬學物理系 (所 )研 究室紀錄 T).WecansUlvet㏑ sianscendentaIequatiUneithergrap㏑ caluUrnumeh＿ a=力 ε/(竹 ‾ -ε .Inserting山isvalueIntU(1.13),weUbtain5-ε 5十 ε bywntlnga/λ =∫ =5-5θ , ‾5=U.U337andhencea/兄 lleadstUasuggestlveapprUximatesUlutiUnε ∼ 兒 = 34Jsand U.U337=4.’ a站 . Sincea =乃 ε/(竹 Γ)andusingthevaIues乃 =6.626× lU‾ 23JK一 】 I.38U7× lU‾ ,wecanwritethewave︳ engththatcUrrespUndstUthemaxⅡ numUf PIanckenergydens丘y(lθ)asfUllUws:

‘ 易ε 1 2898θ × lU一 Π1K λ羽π = (1.14) 4.9663竹 T T

relat㏑n,whIchshUwsthatλ 〞羾 deCreaseswithincreasingtemper前 叮 eUfthebUd坊 lS icalled所“怎功⋯ 兒a名〞勿 仂w.1canbeusedtUdeterInine山 ewavelengthcUπespUndlngtU

ε v〞 aX =罕肛 (1.15) —— λ羽θX

℉ 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= ∼ 65UUK. (1.16) 446× lU一 ’m

UsingStefan’slaw(1.l),andasm㏕ ngthe就artUrad㏑ ●elikeablackbU由,wecanest㎞ 飩ethe tUtalpUwerperl1nitareaemittedatthesurfaceUfthesta.

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

ThisanenUl111Uus intensitywhichwilldecreaseaSit spreadsUver space. ●)ThedUminantwaVelengthUfthera山 at沁nemittedbyaglUwingtungsten nlamentUf temperamre33UUKis

‘ 2898.9× 1U一 InK 兄 ∼ 878.45Ilm (1.18) 羽aX一 33UUK

Theintensiyradiatedbythenlament k敼 venby

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

國旺溝華人甲物理系 Υ′

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.Such tsarehUwevernUtveryapprUpriatein parjcIephysics,wherelengthsare i5mandInasseslU‾ :caⅡ ylU‾ 2’ kg. in particIe physics are usuaⅡy quUted in terms Uf the 15m),andcrUss＿ ;θ rUr′9r用 9(1fm=lU一 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 2Ⅳ mass is山 化therestenergyisΛtε 【eV FUrexample,theprUtUnhasarest FgyUf938.28MeVUrU938GeV.UΠ enmasses(mea血ngtherest-energy ⅣaIcⅡ ts9arelUUselyquUtedinⅣ 【eVUrGeV.′

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

田U=1. unitUflengthis thentheCUmptUnwavelengthUfthestandard

尤U=」生=1; 田ηC 。 UftiIneis

℅ = 告=嘉 =t thatUfenergyis

EU=〃 Uc2=1.

thesc units, it is seen that 力=ε =1. In cUnverting back, at the end the calculatiUn tU the mUre usual units, it is usefuI tU remember that 2=197MeVhaSa =197MeVfm.Thus,apart兌 leUfmasscnergy仰 Uε

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

εi2 1 a=4冗 生 ε°乃ε 137.

三音舞 馬::子I::I;忍:l呈 r｝ 己姆 :l::甘 Ff:胖:燙i:#｝l∵掣r: ′ i一 a = 一 生 4π 凹 as in (1.12). A sIniIar denni● Un is used tU relate charges and cUupIing cUnstants in theUther interacuUns.

研究至紀碌 Z.一,由〞停/

ShUwthat,inUne-dimensiUnaIprUbIems,theenergyspectiumofthebUundstates is

a︳ wavsnUn-degenera亡 e.

l●°and.φ2●● FUIthesakeUfargument letussuppUsethattheoppUsieis true.Letφ thenbetwoInearlVindependenteigenfuncuUnsw丘 hthesameenergyeigenvalueE.FrUm theequauUnS ’ ㎡+管σ一◤v1=U,” +管 σ一/v2=U, weUbtain 考告=考告=弓等γ一D,

Ⅱ.e. .Ψ ’ 一 古l)’ =U.＿ 2一 仍Ψl=(φ i煦)’ 仰Ψ

一〧 ＿ ＿ ＿＿＿

After integratIngthiSequatiUnwenndthat nstant φiΨ 2一 Ψtφ ⊥=ac。

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.Ψ 1=“ 炒,whjchcUntradicts the assumed︳inear independenceUfthetwUfunctiUns.

國立清華犬學物理系(所 )研 究室紀錄 J.劦 翃 ⊿口:::i;一口t,勿抸

partlcleis initiallyσ

FUr′

動=与 = (U≦ x≦ a). (lU.76) 勢房吾=㏕。 (a9Whenthewallis mUvedSlUWl坊 theadiabatictheUrem山 aatesthatthepⅢ iclewillbe fUundatⅡ me′ in thegrUundstateUfthenewpUtentialwell〔 hewellwithwalIsatx=Uand x=8a).Thus,wehave

瑚=石 ㎡ω=摴 血(等 (U≦ x ≦a). (lU.77) 話名吾F=齬 )

2力 2/φ 2方 2/φ ThewUrk neededtUmUvethewall isΔ 〃=El-El(〞 )=π 〞a2)一 π 〞“a)2)= 2/(128〞 ωπ2力 a2). (b9Ⅵ竹enthewallis mUvedrapldl坊 thepart㏑ lewill nnd北 selfinstant妙 (att≧ U)in the newpUtentialwe11;itsene:掛 leVelsandwave㎞ ctiUnarenUwgivenby

(U≦ x≦ 8a) (lU.78) =手 , =V:9蕊n(咢 矺=籍 全告弩青蛻ω )

TheprUbabilΠ yUffindingtheparticlein thegrUundstateUfthe newbUxpUtential canbe UbtainediUm(lU.73):Pll=〡 2where 〈〃i︳ 〞I〉 〡

a㎡ r九 一 ＿2Vㄎ (。 拖 血 , 〣=“ 一吻ω嵞= 僗)血 僗)蝕 =奇 (lU.79)

兩l=｜ 〈叫〡吻〉虍=(署 -2v劾 =aUU77m7%. (1U.8Φ )2‘

TheprUba㏑li妙 Ufnndingthepart始 lein山e白rstexcitedstateUfthenewbUxpUtent㏑ l isg∥en 2where ”Pl2=｜ 〈吒｜〞l)｜

三 r北 激= 拖 血 n僗 =。 碗〣=Xa溺 劬吻ω 僗)㎡ )禿 影戶

拖 =｜ (暆 ｜吻〉=(告 99一 19U/U. )2=峏 AsinⅡ IarcalculatiUnleadstU 「

°befUund㏑ ㏑gh田 ::五∵磊拌臨遣Υ撇 諡 s控 r;:∵點 ‘ 〞仰〞ˊ族翃ㄔ協 七sume山at,atIme′ =a山ewavefuna㏑ nφ←,r9Ufauㄩ比 (EprUbIemI6,CLapterIII9: 炳 芯ofthefUIm ﹉ 毗Φ=畾 銤p├ ,伊 =4矽 . 「匍 Inves位 gate山 echange㏑ 6meUfthiswaVe” ack說 氓fUrr>U,nU,gε 叩t° n與 pai㏑Ie.一

ItknecessarytodeteⅡ linethewavefunctionφ 什,● wh㏑hsatis且 estheSchr° dinger equa位oIl 一 ∟ 磁留 =Πㄅ←,U,

andw㏑ 山,at tlmer=U,6山 eg㏑mfuncjUnⅨ x,U9.Wi山 thatend㏑ Ⅵ叩WeeXpand -independentdgenfunctiUnsφ φ°,U9㏑ temls ofthesetUΓ UrthonU了 maI位 ㏄ 口fx9, (Hw口fx9=E紳Π←》 ◆eep.羽 ,fUUtnUte9,thus:

φ什,U9=Σ 幼%(功, 偏 =Iφ X° φ←,U9激 .

Thefunc● Un干 死φK° exp├ E:兮 thensajsnestheschrδ d㏑gerequatUn,and,皯 ÷ time〞 =U,cUincideswithΨ (x,U1.Hence ‾E.t, Ψ(x,r9=Σ 而φ〞(X》 ÷

1.e. ㄍx,′ )=fC(ε ,馮 抑(ξ ,ω 遮 where ‾÷〦f . U估,x,● =Σ φ;6)1.什 )θ

Since,in thecaseoffreemUtiUn,theeigenfunctionsaΓ e ㏕=晶 。 啡⋯ theGΓ een’ sfunctionβ .Θ becUmes(withpcUntinuUu助

雊,ω =｝ 唧 一動一 〞 嘉 忙卜 制 ｝ vㄅ ├扣 器. =｛蒜｝ FrUm(8.39and(8a)ifUI1Uwsthat v2石 -ξ 岭 ,● =｝ 唧 +芻 砟, 甾紡 蒜而 毒 ｛ ｝ ├ 。 〢 whenceweUbtain nnaIly,fUrthewavefunctiUn,

叭馬 ●=勿 #十 一 +翔 ← 畾 啊 呼 帶 ｝ ｜ andfUr山eprUbabiItydens” 2=Pπ 阢叫 伊+馫 唧 . 〔 ｝1珔 ├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 〞 6t6幻 6加 ˊ Ha一m勿 托 坊 夕 a伍 田 匕 ˊ ˊ

2X al osc:llat° i,X+ω S㏑cethegeⅡ erals。lu山nUftheequatlUnUfmotl° nofaclass㏑ =U,:softhefUmx=C㎡n●甿+ψ),thetUtalener田 士 El=T十 ◤=等 +二咢′

山 岫 m㎜ 研k山 n晒 =神 S㏑ceT一 awehaveEl-匕 W㏑hme日 各 ::ihξ驟 瓏 r穿 ±咢 ir驟 驟 揣 亂 絲 迋 留i/mB=到“〞ω.田H曲由㎡poh吐玀比”印 驟眾豔 褓糮 毛荔

iSprUpUrt㏑ naltUthe6me′ rw㏑ch加 takestUpass thrUughⅢs㏑ terva1.If山 ●p田山心 Uf UsciⅡ atiUn isΓ =2π /ω .then 〞 ω 赤 — ↗ 一 t9 西 一 = 一 一 放 = 一 一 一 吼人x)汝 = Γ ∴ 犬 羽 〞 π“9CUS(ω′+ψ )

wl心 chiStheΓ equiredexpressiUn.

F︳ G.II.23.

ItcanbeSeenthatth心 prUbabilhyisg了 eateStattheturningpointsx=二 L‘ (Fi＿g.II.23). AccordingtUquantummechanicstheprobauli” Ufnnd㏑gtheparⅡde㏑ theinterval (x,x+〞 x)沁 9Vq.(x)激 -Il2有 ax2exp ˊx =坑 (-講｝

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”

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國立清華大學物理系(所 )研 究室紀錄

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