72834610 physics ii course for students

8/3/2019 72834610 Physics II Course for Students

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ELEMENTS OF MODERN PHYSICSDr. Ing. Valeric D. Ninulescu a


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Contents1 The experimental foundations of quantum mechanics 1.1 Thermal radiation . . .. . . . . . . . . . . . . . . . . . . . . 1.1.1 Quantitative laws for blackbodyradiation . . . . . . . 1.2 Photoelectric eect . . . . . . . . . . . . . . . . .. . . . . . . 1.3 Waveparticle duality of radiation . . . . . . . . . . . . . . .1.4 Compton eect . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Atomicspectra . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Bohr model of th

e hydrogen atom . . . . . . . . . . . . . . . 1.7 Correspondence principle . . .. . . . . . . . . . . . . . . . . 1.8 Einsteins phenomenological theory of radiation processes . . 1.8.1 Relations between Einstein coecients . . . . . . . . . 1.8.2 Spontaneous emission and stimulated emission as competing processes . . . .. . . . . . . . . . . . . . . . . 1.9 Experimental conrmation of stationary states . . . . . . . . 1.10 Heisenberg uncertainty principle . . . . . . . . . . . .. . . . 1.10.1 Uncertainty relation and the Bohr orbits . . . . . . . . 1.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Fundamental physical constants 1 2 4 6 11 12 14 15 18 20 22 23 24 26 27 28 39

B Greek letters used in mathematics, science, and engineering 43


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iv

CONTENTS


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PrefaceThis text has grown out of an introductory course in modern physics taught to second year undergraduates at the Faculty of Engineering in Foreign Languages, Politehnica University of Bucharest. It was intended from the beginning that the book would include all the essentials of an introductory course in quantum physicsand it would be illustrated whenever appropriate with thechnical applications.A set of solved problems is included at the end of each chapter. V. Ninulescu 20

11


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vi

CONTENTS


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Glossary of notationse i Im(.) R c.c. z v . A A kB CM lhs rhs 1-D TDSE TISEr

exponential function, ez = exp(z) complex unity, i2 = 1 imaginary part of a complex number the set of real numbers complex conjugate of the expression written infront of c.c. complex conjugated of z vector average integral over the whole th

ree

dimensional space operator A adjoint of A roughly similar; poorly approximates Boltzmann constant centre of mass left hand side right hand side one dimensional time dependent Schrdinger equation o time independent Schrdinger equation o


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Chapter 1

The experimental foundations of quantum mechanicsAt the end of the 19th century, it seemed that physics was able to explain all physical processes. According to the ideas of that time, the Universe was composed of matter and radiation; the matter motion could be studied by Newtons laws andthe radiation was described by Maxwells equations. Physics of that time is now c

alled classical physics. The theory of special relativity (Albert Einstein, 1905) generalized classical physics to include phenomena at high velocities. This condence began to disintegrate due to the inability of the classical theories of mechanics and electromagnetism to provide a satisfactory explanation of some phenomena related to the electromagnetic radiation and the atomic structure. This chapter presents some physical phenomena that led to new ideas such us the quantization of the physical quantities, the particle properties of radiation, and the wave properties of matter. Certain physical quantities, such as energy and angular momentum, were found to take on a discrete, or quantized, set of values in some conditions, contrary to the predictions of classical physics. This discreteness property gave rise to the name of quantum mechanics for the new theory in physics. Quantum mechanics reveals the existence of a universal physical constant, P

lancks constant, whose present day accepted value in SI units is h = 6.626 068 96(33) 1034 J s . Its small value when measured in the macroscopic units of the SIshows that


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2

1 The experimental foundations of quantum mechanics

quantum physics is basically used at much smaller levels, such as the atomic andsubatomic levels.

1.1

Thermal radiation

It is well known that every body with non zero absolute temperature emits electromagnetic radiation with a continuous spectrum that contains all wavelengths. Let us explain this phenomenon in the frame of classical physics. Atoms and molecules are composed of charged particles. The irregular thermal motion produces irregular oscillatory motion of the electrons inside atoms, characterized by a continuous spectrum of frequencies. Since an oscillation is accelerated motion, eachoscillation at a particular frequency leads to the emission of electromagneticradiation of the same frequency. Examples of thermal radiation include the solar

radiation, the visible radiation emitted by a light bulb, or the infrared radiation of a household radiator. The radiation incident on the surface of a body ispartially reected and the other part is absorbed. Dark bodies absorb most of theincident radiation and light bodies reect most of the radiation. For example, asurface covered with lampblack absorbs about 97 % of the incident light and polished metal surfaces, on the other hand, absorb only about 6 % of the incident radiation, reecting the rest. The absorptance (absorption factor ) of surf ce isdened s the r tio of the r di nt energy bsorbed to th t incident upon it; thisqu ntity is dependent on w velength, temper ture, nd the n ture of the surf ce.Suppose body t therm l equilibrium with its surroundings. Such body emits

nd bsorbs the s me energy in unit time, otherwise its temper ture c n not rem

in const nt. The r di tion emitted by body t therm l equilibrium is termed therm l r di tion. A bl ckbody is n object th t bsorbs ll electrom gnetic r di

tion f

lling on it.1 This property refers to r

di

tion of

ll w

welengths

nd

ll

ngles of incidence. However,

bl

ckbody is

n

bstr

ction th

t does not exist in the re

l world. The best pr

ctic

l bl

ckbody is the surf

ce of

sm

ll holein

cont

iner m

int

ined

t

const

nt temper

ture

nd h

ving bl

ckened interior, bec

use

ny r

di

tion entering through the hole suers multiple reections, lot getting

bsorbed on e

ch reection, nd ultim tely is completely bsorbed inside(Fig. 1.1). Consider now the reverse process in whichThe term

rises bec

use such bodies do not reect incident visible r di tion, ndtherefore

ppe

r bl

ck.1


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1.1 Therm l r di tiony } U % T = const

3

Fig. 1.1 A c vity with sm ll hole beh ves s perfect bsorber. The inner w lls

re bl

ckened

nd m

int

ined

t temper

ture T = const.

r di tion emitted by the interior c vity w lls esc pes through the hole. The interior w lls re const ntly r di ting nd re bsorbing their own r di tion nd fter

while the st

te of therm

l equilibrium is

tt

ined.2 A sm

ll fr

ction of this r di tion is incident upon the hole nd p ss through it. Since the surf ce beh ves s bl ckbody, the r di tion th t esc pes the c vity h s the properties of bl ckbody r di tor. The study of the c vity r di tion led Gust v Robert Kirchho to the conclusion th t bl ckbody r di tion is homogeneous, isotropic, nonpol

rized, independent of the n ture of the w lls or the form of the c vity. The bility of surf ce to emit therm l r di tion is ch r cterized by its emitt nce , d n d as th ratio of th radiant n rgy mitt d by a surfac to that mitt d by ablackbody at th

sam

t

mp

ratur

. As th

absorptanc

, th

mittanc

d

p

nds on

wav

l

ngth, t

mp

ratur

, and th

natur

of th

surfac

. Th

mittanc

of a surfac

can b

r

lat

d to its absorptanc

as follows. Suppos

th

surfac

of th

int

rior cavity in Fig. 1.1. L

t us d

not

by L the spectra

radiance inside the cavity (power per unit surface per unit so

id ang

e per unit frequency interva

). Taking into account that at equi

ibrium the wa

s emit as much as they absorb, wehave L = L , so that = . (1.1)

This resu

t is known as Kirchho s radiation

aw (1859). According to Eq. (1.1), objects that are good emitters are a

so good absorbers. For examp

e, a b

ackened surface is an intense emitter surface as we

as an intense absorber one.The sma

size of the ho

e ensures that neither the radiation which enters the cavity from outside, nor that which escapes outside, can a

ter the therma

equi

ibrium in the cavity.

2


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4

1 The experimenta

foundations of quantum mechanics

A grey body is dened as a body with constant emittance/absorptance over a

wave

engths and temperatures. Such an idea

body does not exist in practice but thisassumption is a good approximation for many objects. The

aws governing the emis

sion or absorption of therma

radiation by b

ackbodies have a universa

character and is therefore of specia

interest.

1.1.1

Quantitative

aws for b

ackbody radiation

StefanBo

tzmann

aw The radiant exitance (energy radiated from a body per unit area per unit time), M , of a b

ackbody at temperature T grows as T 4 : M (T ) = T4 , (1.2)

where 5.67 108 W m2 K4 i StefanBoltzmann con tant. The law wa r t formulated

9 by Jo

ef Stefan a

a re

ult of hi

experimental mea

urement

; the

ame law wa

derived in 1884 by Ludwig Boltzmann from thermodynamic conideration

. StephanBo

ltzmann law can be extended to a grey body of emittance as M (T ) = T 4 . (1.3) Wien di pacement law Wilhelm Wien experimental tudie on the pectral di tribution of blackbody radiation revealed: the radiative power per wavelength interval ha

a maximum at a certain wavelength max ; the maximum shifts to shorter wave

engths as the temperature T is increased. Wiens dispacement

aw (1893) states that the wave

ength for maximum emissive power from a b

ackbody is inverse

y proportiona

to its abso

ute temperature, max T = b . (1.4) The constant b is ca

ed Wiensdisp

acement constant and its va

ue is b 2.898 103 m K . For examp

e, when iron is heated up in a re, the rst visib

e radiation is red. Further increase in temperature causes the co

our to change to orange,


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1.1 Therma

radiation

5

then ye

ow, and white at very high temperatures, signifying that a

the visib

e frequencies are being emitted equa

y. Wiens disp

acement

aw can be used to determine the temperature of a b

ackbody spectroscopica

y by measuring the wave

e

ngth at which the intensity of the radiation is maximum. This method has been used extensive

y in determining ste

ar temperatures. Ray

eigh Jeans

aw. U

travio

et catastrofe The spectra

distribution of b

ackbody radiation is specied by thespectra

radiant exitance E that is, emitted power per unit of area and unit of wave

ength. In an attempt to furnish a formu

a for E , it proves convenient to focus our attention on the radiation inside a cavity. Let denote the spect a

adiant ene gy density, which is dened as the ene gy pe unit of vo

ume and unit of wave

ength. It can be shown that (see P ob

em 1.4) E = (1/4)c , (1.5)

whe e c is the speed of

ight in vacuum. It is a

so usefu

to conside the spect a

quantities as functions of f equency instead of wave

ength. Based on the e

ationship = c/ , the radiatio

with the wave

e

gth i

side the i

terva

(, + d) has

the freque

cy betwee

d a

d , where d = d/d d = (c/)2 d . Now, the spectra

ita ce i terms of freque cy E a d the spectra

radia t exita ce i terms of wave

e gth E are re

ated by E () d = E () d . Simi

ar

y, () d = () d . The c

t fo is (, T ) = 8 2 kB T , c3 (1.8) (1.7) (1.6)

k ow as RayleighJea s formula. The com ariso with the ex erime tal results shows a good agreeme t for small freque cies. However, do ot exhibite a maximum, bei g a i creasi g fu ctio of freque cy. More tha that, the radia t e ergy de sity is

(T ) =0

(, T ) d =

8kB T c3

0

2 d = ,

a

u

acce

table result, k

ow

as the ultraviolet catastro

he.


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6

1 The ex

erime

tal fou

datio

s of qua

tum mecha

ics

Pla cks radiatio law I 1900 Max Pla ck determi ed a formula which agrees with the ex erime t at whatever freque cy based o a ew a d revolutio ary idea: the excha

ge of e

ergy betwee

a body a

d its surrou

di

gs ca

be

erformed o

ly i

d

iscrete

ortio

s, the mi

imum e

ergy im

lied i

the excha

ge bei

g

ro

ortio

alto the freque cy, h. Pla ck radiatio law reads (, T ) = h 8 2 . c3 ex (h/kB T ))

The co sta t h is called Pla cks co sta t a d has the value h 6.626 1034 J s . Byuse of Eq. (1.7), we get (, T ) = 8hc 1 . 5 exp(hc/kB T ) 1 (1.10)

To express the radia

t exita

ce E , Eq. (1.5) is used: E (, T ) = 2hc2 1 . 5 exp(hc/T ) 1 (1.11)

P

a cks radiatio formu

a co tai s a

the i formatio previous

y obtai ed. Figure 1.2 prese

ts the graph of the spectra

e

ergy de

sity (, T ) given by Eq. (1.10)

. App

ications Py

omete

: device fo

measu

ing

e

ative

y high tempe

atu

es by measu ing adiation f om the body whose tempe atu e is to be measu ed. Inf a ed the mog aphy: a fast non-dist uctive inspection method that maps the tempe atu edie ences of any object in a ange f om 50 C to 1500 C.

1.2

Photoelectric eect

The hotoelectric eect co sists i the ejectio of electro s from a solid by a i cide t electromag etic radiatio of sucie tly high freque cy. This freque cy isi the visible ra ge for alkali metals, ear ultraviolet for other metals, a d far ultraviolet for o metals. The ejected electro s are called hotoelectro s.


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1.2 Photoelectric eect6 5 (,T), 103J m4 4 3 2 1500 K 1 1000 K 0 0 1 2 , m 3 4 5 1750 K 2000 K

7

Fig. 1.2 Spectra

e ergy de sity (, T ) of a b

ackbody at a few tempe atu es. UV adiation ' qua tz window A Vacuum chambe A U Fig. 1.3 Expe imenta

a angement f

o the study of the photoe

ect ic eect.

C

The phenomenon was st obse ved by Hein ich He tz in 1887; he noticed that the minimum vo

tage equi ed to d aw spa ks f om a pai of meta

ic e

ect odes was educed when they we e i

uminated by u

t avio

et adiation. Wi

he

m Ha

wachs (1888) found that when u

t avio

et adiation was incident on a negative

y cha ged zinc su face, the cha ge

eaked away quick

y; if the su face was positive

y cha ged, the e was no

oss of cha ge due to the incident adiation. Mo e than that, aneut a

su face became positive

y cha ged when i

uminated by u

t avio

et adiation. It is evident that on

y negative cha ges a e emitted by the su face unde i

ts exposing to u

t

avio

et

adiation. Joseph John Thomson (1899) estab

ished that the u

t avio

et adiation caused e

ect ons to be emitted, the same pa tic

es found in cathode ays.


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8i T

1 The expe imenta

foundations of quantum mechanics3I 2I I EU Fig. 1.4 Photocu ent (i) ve sus potentia

die ence (U ) between cathode and anode fo die ent va

ues of the intensity (I, 2I, and 3I) of the monoch omatic beam of

ight.

U0

0

A ty ical arra geme t for the study of the hotoelectric eect is de icted i Fig.1.3. A vacuumed glass tube co tai s two electrodes (A = a ode = collecti g late, C = cathode) a

d a quartz wi

dow allowi

g light to shi

e o

the cathode surface. The electro s are emitted at the cathode, fall o the a ode, a d com lete acircuit. The resulti g curre t is measured by a se sitive ammeter. It is very im orta t that the surface of the cathode is as clea as ossible. A adjustable

ote

tial diere ce U of a few volts is a lied across the ga betwee the cathode

a

d the a

ode to allow the

hotocurre

t co

trol. If A is made

ositive with res

ect to C, the electro s are accelerated toward A. Whe all electro s are collected, the curre t saturates. If A is made egative with res ect to C, the electriceld te

ds to re

el the electro

s back towards the cathode. For a

electro

to reach the collecti g late, it must have a ki etic e ergy of at least eU , where eis the mag itude of the electro charge a d U is the retardi g voltage. As there elli g voltage is i creased, fewer electro s have sucie t e ergy to reach the late, a d the curre t measured by the ammeter decreases. Let U0 de ote the voltage where the curre t just va ishes; the maximum ki etic e ergy of the emitted electro s is2 (1/2)mvmax = eU0 ,

(1.12)

where m de

otes the mass of the electro

. U0 is called sto i

g

ote

tial (voltage). The ex

erime

tal results for a give

emitti

g cathode are sketched i

Figs.1.4 a

d 1.5. From Fig. 1.4 it is clear that for a give

metal a

d freque

cy ofi

cide

t radiatio

, the rate at which

hotoelectro

s are ejected is

ro

ortio

alto the i

te

sity of the radiatio

. Figure 1.5 reveals other two results. First,a give

surface o

ly emits electro

s whe

the freque

cy of the radiatio

with which it is illumi

ated exceeds a certai

value; this


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1.2 Photoelectric eect2 (1/2)mvmax

9

T Fig. 1.5 De e de ce of the maximum ki etic e ergy of hotoelectro s o the freque

cy of the i

cide

t radiatio

. 0 0 E

0

freque

cy, 0 i

Fig. 1.5, is called threshold freque

cy. It is ex

erime

tally

rove that 0 de e ds o the cathode material. Seco dly, the e ergy of the hotoelectro s is i de e de t of the radiatio i te sity, but varies li early with the radiatio freque cy. The ex erime t also shows that there is o sig ica t delay betwee

irradiatio

a

d electro

emissio

(less tha

109 s). The mi

imum e

ergy needed to extract the electron from the material i called work function. If the electron ab orb a larger energy then , the energy in exce

i kinetic energy of the

photoelectron. The experimental reult

di

agree with the Maxwell wave theory o

f light: According to thi theory, a the electromagnetic wave reache the urfa

ce, the electron

tart to ab

orb energy from the wave. The more inten

e the incident wave, the greater the kinetic energy with which the electron

hould be eje

cted from theurface. In contra

t, the experiment

how

that the increa

e of th

e wave inten ity produce an increa e in the number of photoelectron , but not their energy. The wave picture predict the occurence of the photoelectric eect for any radiation, regardle

of frequency. Therehould be a

ignicant delay betwe

en theurface irradiation and the relea

e of photoelectron

(

ee Problem 1.6).

In order to account for the photoelectric eect experimental re ult , Albert Ein tein (1905) propo ed a new theory of light. According to thi theory, light of xedfrequency co sists of a collectio of i divisible discrete ackages, called qua

ta, whose e ergy is E = h . (1.13)

The word hoto for these qua tized ackets of light e ergy came later, give by

Gilbert Newto

Lewis (1926). Ei

stei

assumed that whe

light strikes a metal,a si

gle

hoto

i

teracts with a

electro

a

d the

hoto

e

ergy is tra

sferredto the electro

. If


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10

1 The ex

erime

tal fou

datio

s of qua

tum mecha

ics

h > , then the electron i emitted from the urface with re idual kinetic energy 2(1/2)mvmax = h . (1.14) Otherwi

e, the electron remain

trapped in the metal, re

gardle

of the inten ity of the radiation; the thre hold frequency of the photo

electric eect i 0 = /h . (1.15) Ein tein equation of the photoelectric eect [Eq. (1.14)] exactly explain the ob erved dependence of the maximum kinetic energy onthe frequency of the radiation (

ig. 1.5). Let ucontinue the explanation of th

e experimental photoelectric eect nding . Increa ing the inten ity of the radiation n time mean an n time increa e of the number of photon , while the energy of each individual photon remain the ame.

or uciently low radiation inten itie, it can be a

umed that the probability of an electron to aborb two or more p

hoton i negligibly mall compared to the probability of ab orbing a ingle photon. It follow that the number of electron emitted increa e al o by the factor n .

inally, no time i nece ary for the atom to be heated to a critical temp

erature and therefore the releae of the electron

i

nearly in

tantaneou

upon

ab orption of the light. Combining Eq . (1.12) and (1.14) we get U0 = (h/e) /e . (

1.16)

Thetopping voltage a

a function of the radiation frequency i

a

traight line

who e lope i h/e and who e intercept with the frequency axi i 0 = /h. Thi experimental dependence can be u ed to calculate Planck con tant and the work function of the metal. (The value of e i

con

idered a

known.) The re

ult for the P

lanckcon

tant i

the

ame a

that found for blackbody radiation; thi

agreement

can be con idered a a further ju tication of the portion of energy h i troducedby Pla ck. From the observed value of 0 , the workfu ctio is calculated as = h0 .Table 1.1 fur ishes the work fu ctio of some chemical eleme ts. The hotoelectric eect is erha s the most direct a d co vi ci g evide ce of the cor uscular ature of light. That is, it rovides u de iable evide ce of the qua tizatio of the electromag etic eld a d the limitatio s of the classical eld equatio s of Maxwe

ll. A

other evide

ce i

su

ort of the existe

ce of

hoto

s is

rovided by the Com

to

eect.


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1.3 Wave article duality of radiatio

11

Table 1.1 Electro work fu ctio of ome element [10]. In general, dier for each face of a monocry

talline

ample. In

ide the table, polycr i

an abbreviation fo

r polycry talline ample. Element Ag ( ilver) Plane 100 110 111 100 110 111 100 11

0 111 polycr polycr polycr /eV 4.64 4.52 4.74 4.20 4.06 4.26 5.47 5.37 5.31 2.874.5 1.95 Element Cu (copper) Plane 100 110 111 100 111 polycr polycr polycr polycr polycr polycr polycr /eV 5.10 4.48 4.94 4.67 4.81 2.29 2.93 3.66 2.36 2.261 4.33 3.63

Al (aluminum)

e (iron) K (pota ium) Li (lithium) Mg (magne ium) Na ( odium) Rb (rubidium) Ti

(titanium) Zn (zinc)

Au (gold)

Ca (calcium) Cr (chromium) C

(ce

ium)

1.3

Waveparticle duality of radiation

The phenomena of the interference and diraction can be explained only on the hypothe i that radiant energy i propagated a a wave. The re ult of the experiment on the photoelectric eect leave no doubt that in it interaction with matter,radiant energy behave

a

though it were compo

ed of particle

; it will be

hown

later thatimilar behaviour i

ob

erved in the Compton eect and the proce e o

f emi ion and ab orption of radiation. The two picture hould be under tood a

complementary view of the ame phy ical entity. Thi i called the waveparticle

duality of radiation. A particle i characterized by it energy and momentum; awave i characterized by it wave vector and angular frequency. We perform herethe link between the two picture . The energy of a photon i given by Eq. (1.13). We complete the characterization of the photon with a formula for it momentum. Cla

ical electromagnetic theory how that a wave carrie momentum in additi

on to energy; the relation between the energy E contained in a region and the corre ponding momentum p i E = cp . (1.17)

It i natural to extend thi formula for the quantum of radiation, i.e., the photon.

or a plane wave of wave vector k, the angular frequency of the


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12

1 The experimental foundation of quantum mechanic

wave i = k/c and the photon momentum can be ritten p= E = = k. c c (1.18)

This formula can be ritten ith vectors, p = k. (1.19)

To sum up, the t o behaviours of radiation are linked by Planck-Einstein relations E = and p = k . (1.20)

1.4

Compton eect

The existence of photons is also demonstrated by experiments in hich x-rays or -rays are scattered by a tar et. The experiments performed in 1922 by Arthur Holly Compton ith x-rays and a raphite tar et sho that, in addition to the incident radiation, there is another radiation present, of hi

her avelenth. The radia

tion scatterin

ith chan

e in

a

elen

th is called Compton eect. Given that is the wave

e gth of the i cide t radiatio a d that of the scattered radiatio , Compto fou d that does ot depe d upo the wave

e gth of the i cide t rays or upo the target materia

; it depe

ds o

y o

the directio

of scatterin

(an

le bet een the incident ave and direction in hich scattered aves are detected) accordin to the experimental relation = C (1 cos ) . (1.21)

The constant C 2.426 1012 m is owadays ca

ed Compto wave

e gth of the e

ectro .The scattered radiatio is i terpreted as radiatio scattered by free or

oose

y bou d e

ectro s i the atoms of matter. The c

assica

treatme t of the x raysor -rays as aves implies that an electron is forced to oscillate at a fre uencye ual to that of the ave. The oscillatin electron no acts as a source of ne electroma netic aves havin the same fre uency. Therefore, the avelen ths of t

he incident and scattered x-rays should be identical. We must therefore concludethat the classical theory fails in explainin

the Compton eect. Treatin the radiation as a beam of photons, a photon can transfer part of its ener

y and linearmomentum to a loosely bound electron in a collision


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1.4 Compton eectTar et electron at rest Scattered B photo

13

E I cide t photo

Recoi

e

ectro

Fig. 1.6 I Compto scatteri g a x ray photo of wave

e gth co

ides with a e

ectro i itia

y at rest. The photo scattered i the directio has the avelen

th > .

(Fig. 1.6). Si ce the e ergy of a photo is proportio a

to its freque cy, afterthe co

isio the photo has a

ower freque cy a d thus a

o ger wave

e gth. Tobe more specic,

et p = h/c = h/ a d p = h /c = h/ be the mome ta of the photo ba

d after the co

isio

. The e

ectro

ca

be co

sidered i

itia

y at rest. After

the co

isio

it acquires a ki

etic e

ergy T = h h = hc/ hc/ a

d a mome

tum h22 + 2 2 cos , (1.23) 2 as required by the co servatio of e ergy of mome tum, T a d pe are re

ated by the re

ativistic equatio pe = p p , p2 = (p p )2 = e T=c p2 + m2 c2 mc2 . e (1.24) I

serti

g Eqs. (1.22) a

d (1.23) i

to Eq. (1.24), rearra gi g a d squari g, we get hc hc + mc2 2

(1.22)

=

hc hc h2 c2 h2 c2 + 2 2 cos + m2 c4 . 2 C = h/mc . (1.25)

After performi

g the e

eme

tary ca

cu

atio

s it is obtai

ed Eq. (1.21), where The prese

ce of the radiatio

with the same wave

e

gth as the i

cide

t radiatio

is

ot predicted by Eq. (1.21). This radiatio

is exp

ai

ed by the scatteri

g ofthe i

cide

t radiatio

by tight

y bou

d e

ectro

s. I

this case, the photo

co

ides with the e

tire atom, whose mass is co

siderab

y greater tha

the mass of asi

g

e e

ectro

. The photo

a

d atom excha

ge mome

tum, but the amou

t of excha

ged e

ergy is much

ess tha

i

case of the photo

e

ectro

co

isio

; the wave

e

gth shift is

ot sig

ica t. Compto scatteri g is of prime importa ce to radiobio

ogy, as it happe

s to be the most probab

e i

teractio

of high e

ergy x rays with atomic

uc

ei i

ivi

g bei

gs a

d is app

ied i

radiatio

therapy.


22/56

14

1 The experime

ta

fou

datio

s of qua

tum mecha

ics

1.5

Atomic spectra

By the ear

y 1900s, the fo

owi g observatio s co cer i g the atomic emissio spectra had bee do e: Whe a gas of a e

eme t at

ow pressure is subjected to a

i

put of e

ergy, such as from a

e

ectric discharge, the gas emits e

ectromag

etic radiatio . O passi g through a very thi s

it a d the through a prism thee

ectromag etic radiatio ca be separated i to its compo e t freque cies. It was fou d that a gas at

ow pressure emits o

y discrete

i es o the freque cy sca

e. The emissio

spectrum is made up of spectra

series i

the diere t regio s (i frared, visib

e, a d u

travio

et) of the spectrum of e

ectromag etic radiatio

; the spectra

i es i a series get c

oser together with i creasi g freque cy. Each e

eme t has its ow u ique emissio spectrum. The simp

iest

i e spectrum is that of the hydroge

atom. Its four spectra

i

es i

the visib

e have wave

e

gths that ca

be represe

ted accurate

y by the Joha

Ba

mer formu

a (1885)

= C 2 2 4 = 3, 4, . . . , (1.26)

where C 364.6

m . This formu

a was writte

by Joha es Rydberg (1888) for wave

umbers, = 1 1 1 =R 2 2 2 , = 3, 4, . . . ,

where the ew co sta t R is owadays ca

ed Rydberg co sta t, a d ge era

ized i

the form R R , 1 = 2, 3, . . . , 2 = 3, 4, . . . , 2 2 1 2 (1.27) a d 1 < 2 . Here, 1 de es the spectra

series. For a give series, with i creasi g va

ues of 2 , the wave umber approaches the

imit R/ 2 . The 1 separatio of co secutive wave umbers of a give series decreases so that 1 2 = = 1 1 1 =R 2 2 1 2 1 2


23/56

1.6 Bohr mode

of the hydroge atom

15

the wave umber ca ot exceed the series

imit. I pri cip

e, a i ite umber of

i es

ie at the series

imit. Accordi g to Eq. (1.27) the wave umber of a y

i

e of the hydroge

spectrum is the diere ce of two spectra

terms:

1

2 = T

1 T

2 , with T

= R/

2 . (1.29) It proves empirica

y that the

i

es of other chemica

e

eme ts ca a

so be expressed as the diere ce of two terms; however, a spectra

term has a more comp

icated form. No e of the above qua titative resu

t cou

dbe exp

ai

ed satisfactori

y i

the frame of c

asica

physics. (1.28)

1.6

Bohr mode

of the hydroge

atom

Atoms have radii o the order of 1010 m . To study the i ter a

structure of atoms, Er est Rutherford, Ha s Geiger a d Er est Marsde (1909) directed -p rticles from r

dio

ctive r

dium

t thin gold foil. B

sed on the results of the -p

rticles

sc

ttering by the gold

toms, Rutherford (1911) furnished the nucle

r

tom model(pl net ry model of the tom): the tom consists of positively ch rged he vycore (nucleus) of r dius on the order of 1015 m a d a cloud of egatively chargedelectro

s. Atomic structure was

ictured as a

alogous to the solar system, withthe ucleus layi g the role of the Su a d the electro s that of the la ets bou d i their orbits by Coulomb attractio to the ucleus. About as soo as themodel was ublished it was realized that the atom model suers from two serious decie cies: 1. Orbital motio is a accellerated motio a d electro s are charged

articles. Accordi g to the electromag etic theory, the electro s should radiatee ergy i the form of electromag etic waves. Electro s lose e ergy a d they should s iral i to the ucleus i a time of the order of 108 s . This co clusio com

letely disagrees with ex erime t, si ce the atoms are stable. 2. The freque cy of the radiated e ergy is the same as the orbiti g o e. As the orbiti g freque cy

ca

take a co

ti

uous ra

ge of values, the discrete emissio

s

ectrum of atomsca

ot be ex

lai

ed. To overcome these decie cies, Niels Bohr (1913) im roved the Rutherford model of the atom by i

troduci

g the followi

g

ostulates:


24/56

16

1 The ex

erime

tal fou

datio

s of qua

tum mecha

ics For every atom there is a

i ite umber of statio ary states i which the atom ca exist without emitti g radiatio . The e ergies of the statio ary states, E1 , E2 , E3 , . . . , form a discrete set of values. Emissio a d absor tio of radiatio are always associatedto a tra

sitio

of the atom from o

e statio

ary state to a

other. The freque

cy

of the radiatio

emitted or absorbed res

ectively duri

g such a tra

sitio

is give by the equatio h = E2 E1 , (1.30)

where E1 a

d E2 de

ote the e

ergy of the atom i

the two statio

ary states. Bohrapp

ied these postu

ates for the hydroge atom. The e

ectro of mass m a d e

ectric charge e is assumed to move arou d the uc

eus of e

ectric charge e i a circu

ar orbit. The se

ectio of the a

owed statio ary states is performed by thefo

owi

g qua

tizatio

ru

e for the a

gu

ar mome

tum: the a

gu

ar mome

tum of the e

ectro i a statio ary state is a i teger mu

tip

e of : mvr = , where =1, 2, 3, . . . . (1.31)

We are

ow i

a positio

to so

ve the prob

em. Newto

s seco

d

aw app

ied for the

e

ectro

motio

o

a circu

ar orbit of radius r with the ve

ocity v u

der the attractive Cou

omb force exerted by the uc

eus yie

ds m v2 e2 = . r 40 r2 (1.32)

Combi

i

g Eqs. (1.31) a

d (1.32), we

d r

= a

d v

= The radius a0 = 40 2 2

= a0 2 me2 e2 1 . 40 (1.33)

(1.34)

40 2 5.29 1011 m me2

(1.35)


25/56

1.6 Bohr model of the hydroge atom

17

is called the Bohr radius. This value sets the scale for atomic dime sio . The

ote tial e ergy of the electro i the hydroge atom is give by V (r) = The e ergy of the atom i

state

is 1 e2 2 E

= T

+ V

= mv

2 40 r

that gives E

= m

2 2 e2 402

e2 . 40 r

(1.36)

1 ,

2

= 1, 2, 3, . . . .

(1.37)

The i teger determi es the e ergy of the bou d state of the atom a d it is called ri ci al qua tum umber. Figure 1.7 rese ts the diagram of e ergy levels for the hydroge

atom. The state of mi

imum e

ergy, called grou

d state, has thee ergy m E1 = 2 2 e2 402

13.6 eV .

(1.38)

This value sets the scale for atomic e ergy. The states of higher e ergy are called excited states. All e ergies of the bou d atom are egative; states of osit

ive e

ergies refers to the io

ized atom. The io

izatio

e

ergy of the hydroge

atom, de ed as the amou t of e ergy required to force the electro from its loweste

ergy level e

tirely out of the atom is (1.39) EI = E1 13.6 eV . Let us

ow calculate the freque

cies of the hydroge

s

ectrum. I

the tra

sitio

betwee

the states

1 a

d

2 >

1 , the freque

cy of the radiatio

emitted or absorbed is

1

2 = (E

2 E

1 )/h a

d the calculus yields

1

2 m e2 = 4 3 402

1 1 2

2

1 2

.

(1.40)

The wavele

gth

1

2 = c/

1

2 of the radiatio

is give

by 1

1

2 = m 4c3

e2 40

2

1 1 2

2

1 2

= R

1 1 2

2

1 2

,


26/56

(1.41)


27/56

18 where

1 The ex

erime

tal fou

datio

s of qua

tum mecha

ics m 4c e2 402

R =

3

1.097 107 m1

(1.42)

is the Rydberg co sta t for hydroge ; the subscri t remi ds us the remise thatthe

ucleus is exceedi

gly massive com

ared with the electro

. The great successof the Bohr model had bee i ex lai i g the s ectra of hydroge like (si gle electro arou d a ositive ucleus) atoms. Eve though the Bohr theory is ow exte ded a d altered i some esse tial res ects by qua tum mecha ics, its k owledgeco

siderably hel

s the u

dersta

di

g of

ew theories. I

fact, a

umber of

he

ome

a i

s

ectrosco

y ca

be dealt with by maki

g use of Bohr theory alo

e.

1.7

Corres o de ce ri ci le

O e of the guidi g ri ci les used i the develo me t of qua tum theory was Bohrscorres o de ce ri ci le (1920) which i dicates that qua tum theory should giveresults that a roach the classical hysics results for large qua tum umbers.To illustrate this ri ci le let us discuss the hydroge atom s ectrum. The qua

tum descri tio of the atom is erformed i the Bohr theory framework. By use ofEq. (1.40) the freque cy of the radiatio i the tra sitio betwee states 1a d is 1, = m e2 4 3 40

2

e2 1 m 1 2 = (

1)2

4 3 40

2

2 (

2

1 . 1)2

We co

sider

ow large qua

tum

umbers, i.e.,

1 . The freque

cy of the qua

tumtra

sitio

becomes

1,

e2 m 3 4 4 02

e2 m 2

= 4 3 4

2 0

2

1 .

3

O

the other ha

d the motio

of the electro

o

the

th orbit has the freque

cy2 e2 m 1 v

= ,

= 3 4 2r

2

3 0 where Eqs. (1.33) a

d (1.34) were used. Accordi

gto classical theory the atom emits radiatio

at the freque

cy

; the com

ariso

of the freque

cy ex

ressio

s gives

1,

i

agreeme

t to the corres

o

de

ce

ri

ci

le.


28/56

1.7 Corres o de ce ri ci le

19

5 4

Io

ized atom

E/eV 0.00 0.54 c c 0.85 Brackett series 1.51

656.3

m 486.1

m 434.0

m 410.2

m

2

c c c c Pasche

series Balmer series

1875 m 1282 m

3

c c

3.40

121.6 m 102.6 m 97.3 m 95.0 m c c c c Lyma series

1

13.6

Fig. 1.7 E ergy level diagram for hydroge a d a few s ectral li es of the Lyma

, Balmer, a d Pasche series.


29/56

20

1 The ex

erime

tal fou

datio

s of qua

tum mecha

ics

More that that, for large values of the qua tum umber the hydroge e ergy levels lie so close together that they form almost a co ti uum; it follows that theclassical co

ti

uum descri

tio

of the s

ectrum should corres

o

d to tra

sitio

s betwee

two such states.

1.8

Ei stei s he ome ological theory of radiatio rocesses

I 1917 Ei stei ex lai ed he ome ologically the radiatio matter i teractio based o

the qua

tum ideas of that time: Pla

cks qua

tum hy

othesis a

d Bohrs

la

etary model of the hydroge atom. Ei stei s ostulates could all be justied by thelater develo ed qua tum mecha ical treatme ts of the i teractio rocesses. Su

ose N ide tical atoms i u it volume, each atom havi g a air of bou d state e ergy levels E1 a

d E2 , E2 > E1 (Fig. 1.8). The two atomic levels are allowed to

be multi

lets with dege

eracies g1 a

d g2 . The mea

umbers of atoms

er u

it volume i the two multi let states are de oted by N1 a d N2 . Assumi g that all the atoms are i these states, N1 + N2 = N . (1.43)

The atomic medium is co sidered i a radiatio eld whose s ectral radia t e ergyde sity at freque cy give by h = E2 E1 (1.44)

is () . Ei stei co siders three basic i teractio rocesses betwee radiatio a datoms (Fig. 1.8). 1. S o ta eous emissio A atom i state 2 s o ta eously erformes a tra sitio to state 1 a d a hoto of freque cy is emitted. The hoto is emitted i a ra dom directio with arbitrary olarizatio . The robability eru it time for occure ce of this rocess is de oted by A21 a d is called Ei stei

coecie t for s o ta eous emissio . The total rate of s o ta eous emissio s is N2

(t) = A21 N2 (t)


30/56

1.8 Ei stei s he ome ological theory of radiatio rocessesT T E2 , g 2 , N 2

21

h

A21

B12 ()

B21 ()

c

c S o ta eous emissio

Absor tio

c Stimulated emissio

E1 , g 1 , N 1

Fig. 1.8 Radiative tra sitio s betwee to e ergy levels. Table 1.2 Tra sitio robability A for hydroge li es i the visible ra ge [10]. Each tra sitio is ide tied by the wavele gth a d the statistica

weights gi a d gk , of the

ower (i)a d upper (k) states. / m 410.173 434.046 486.132 656.280 gi 8 8 8 8 gk 72 50 3218 A/s1 9.732 105 2.530 106 8.419 106 4.410 107

a d the i tegratio with the i itia

co ditio at t = 0 gives N2 (t) = N2 (0) exp(A21 t) . The time i which the popu

atio fa

s to 1/e of its i itia

va

ue is 21 = 1/A21 (1.45)

and is called life

ime of level 2 wi

h respec

o

he spon

aneous

ransi

ion

olevel 1 (see Problem 1.14). A few values of

he

ransi

ion probabili

y ra

e areprovided by Table 1.2. In

he absence of a radia

ion eld

he

ransi

ion 1 2 is impossible due

o viola

ion of

he energy conserva

ion law. 2. Absorp

ion In

he presence of a radia

ion eld an a

om ini

ially in s

a

e 1 can jump

o s

a

e 2 by absorp

ion of a pho

on of frequency . The

robability

er u

it time for this

rocess is assumed to be

ro

ortio

al to the s

ectral e

ergy de

sity at freque

cy ; thetotal rate of absor

tio

s is N1 = B12 ()N1 , (1.46)


31/56

22

1 The ex

erime

tal fou

datio

s of qua

tum mecha

ics

where B12 is called Ei stei coecie t for absor tio . 3. Stimulated emissio Ei stei ostulates that the rese ce of a radiatio eld ca also stimulate the tra sitio

2 1 of the atom; e

ergy co

servatio

law asks for the emissio

of a

hoto

of freque

cy . The probability per unit time for this process is assumed to be proportional to the spectral ener y density at fre uency ; the total rate of stimulated emissio s is N2 = B21 ()N2 , (1.47)

where B21 is called Ei stei coecie t for stimulated emissio . The radiatio roduced duri g the stimulated emissio rocess adds cohere tly to the existi g o e;this mea s that radiatio created through stimulated emissio has the same freque

cy, directio

of

ro

agatio

,

olarizatio

a

d

hase as the radiatio

that forces the emissio rocess. The stimulated emissio rocess was u k ow before 1917. The reaso for its i troductio by Ei stei will be made clear duri g ext sectio . Ei stei coecie ts de ed above are i de e de t of the radiatio eld ro erties a

d are treated here as

he

ome

ological

arameters. They de

e

d o

ly o

the

ro

erties of the two atomic states. Due to all radiative tra

sitio

s

rese

tedabove, the o ulatio s N1,2 of the two e ergy levels cha ge i time accordi g to the equatio s N1 = N2 = B12 () N1 + [A21 + B21 ()]N2 , N1 + N2 = N . (1.48

1.8.1

Relatio s betwee Ei stei coecie ts

Let us co sider the atomic system de ed above i equilibrium with thermal radiatio ; the s ectral radia t e ergy de sity () is give ow by Pla cks formula [Eq. (1.9)]. This s ecial case will lead us to establish relatio s betwee Ei stei coecie ts. The equilibrium co ditio for the radiatio matter i teractio ex resses as follows. For the radiatio eld the equilibrium co ditio mea s equality of emit

ted qua

ta a

d absorbed qua

ta. For the atomic system, the equilibrium co

ditio

writes N1 = N2 = 0 . These co

ditio

s are equivale

t. From Eq. (1.48) at equilibrium, B12 (, T )N1 + [A21 + B21 (, T )]N2 = 0 ,


32/56

1.8 Ei stei s he ome ological theory of radiatio rocesses the s ectral e ergyde sity of the thermal eld ca be ex ressed as (, T ) = A21 /B21 . (B12 /B21 )(N1 /N2 ) 1

23

I

thermal equilibrium, atomic e

ergy levels are occu

ied accordi

g to the Maxwe

llBoltzma

statistics. I

exact terms the ratio of the

o

ulatio

de

sities of the two levels is N1 /N2 = (g1 /g2 ) ex [(E1 E2 )/kB T ] = (g1 /g2 ) ex (h/kB T ) .The use of this ratio i to the ex ressio of (, T ) gives (, T ) = A21 /B21 . (g112 /g2 B21 ) ex

(h/kB T ) 1 (1.49)

This formula should be Pla cks radiatio formula (1.9), so g1 B12 = g2 B21 a d 8 2A21 = 3 h . B21 c (1.50b) (1.50a)

Relatio s (1.50) are k ow as Ei stei s relatio s. These relatio s ermit to ex ress the tra sitio rates betwee a air of levels i terms of a si gle Ei stei coecie t.

1.8.2

S o ta eous emissio a d stimulated emissio as com eti g rocesses

A atom i a excited state ca jum to a lower e ergy state through a s o ta eous emissio or a stimulated o e. Which rocess is most likely? To ask this questio , we co sider the ratio R of the stimulated emissio rate a d s o ta eous emissio o e. This ratio is R= B21 () A21

a d is de e de t o the radiatio eld a d freque cy. To be more s ecic, let us co

sider the thermal radiatio eld. The use of Pla cks radiatio formula a d the seco d Ei stei equatio gives R= 1 . ex (h/kB T ) 1


33/56

24100

1 The ex erime tal fou datio s of qua tum mecha ics

10 R

20

10

40

10

60

Fig. 1.9 Ratio of stimulated emissio

robability a

d s

o

ta

eous emissio

roba

bility for a two

level atom i

a thermal radiatio

eld of tem

erature T = 300 K.

10

10

10

11

10 10 (Hz)

12

13

10

14

10

15

For

umerical evaluatio

s we choose a ty

ical tem

erature T = 300 K. The ratio is R = 1 for = (kB T /h) l

2 4.331012 Hz ( 69.2 m) i

i

frared regio

. The depe

dce R versus freque

cy is prese

ted i

Fig. 1.9. I

the microwave regio

, for examp

e at = 1010 Hz, R 624, so the stimu

ated emissio

i

more

ike

y tha

the spo

ta

eous emissio

. For sma

er freque

cies spo

ta

eous emissio

becomes

eg

igib

e with respect to stimu

ated emissio

. I

the

ear i

frared a

d visib

e regio

the ratio R takes o

very sma

va

ues, so the spo

ta

eous emissio

is the domi

a

t process. Ca

stimu

ated emissio

domi

ate over spo

ta

eous emissio

for a c

assica

source of visib

e radiatio

? It ca

be prove

that the spectra

e

ergy de

sity of a spectroscopic

amp is

ot sucie t to e sure a ratio R > 1 . The stimu

ated emissio

domi

ates over spo

ta

eous for a

aser.

1.9

Experime

ta

co

rmatio

of statio

ary states

The idea of statio

ary states had bee

i

troduced to exp

ai

the discrete spectr


34/56

um of atomic systems. The rst experime ta

co rmatio of this hypothesis was provided by James Fra ck a d Gustav Hertz (1914). Their experime ta

set up is prese

ted schematica

y i

Fig. 1.10. E

ectro

s emitted from a hot cathode C are acce

erated toward the mesh grid G through a

ow pressure gas of Hg vapour by mea

s of a adjustab

e vo

tage U . Betwee the grid a d the a ode A a sma

retardi g vo

tage U 0.5 V is app

ied


35/56

1.9 Experime ta

co rmatio of statio ary statesC G A

25

Hg vapour

A U T Curre

t U

Fig. 1.10 Fra ck Hertz set up.

Fig. 1.11 Curre t through a tube of Hg vapor versus acce

erati g vo

tage i theFra ckHertz experime t. 0 E 5 10 15 Acce

erati g vo

tage, V

0

so that o

y those e

ectro s above a e ergy thresho

d (eU ) wi

reach it. Thea odic curre t as a fu ctio of vo

tage (Fig. 1.11) does ot i crease mo oto ica

y, as wou

d be the case for a vacuum tube, but rather disp

ays a series of pea

ks at mu

tip

es of 4.9 V. The experime

ta

resu

t is exp

ai

ed i

terms of e

ectro mercury atom co

isio s. Due to the i teractio betwee e

ectro a d atom, there is a excha ge of e ergy. Let us de ote E1 a d E2 the e ergy of the atom i its grou

d state a

d rst excited state, respective

y. The mi

imum ki

etic e

ergy of the e

ectro for the atom excitatio is practica

y E2 E1 (see Prob

em 1.12). If the e

ectro has a ki etic e ergy T < E2 E1 it is ot ab

e to excite a mercury atom. The co

isio is e

astic (i.e., the ki etic e ergy is co served) a d thee

ectro moves through the vapor

osi g e ergy very s

ow

y (see Prob

em 1.13). U ished! = h/p (1.51)

If there was o cou tervo

tage app

ied, a

e

ectro s wou

d reach the a ode a d o modu

atio of the curre t sig a

cou

d be measured.


36/56

26

1 The experime

ta

fou

datio

s of qua

tum mecha

ics

1.10

Heise

berg u

certai

ty pri

cip

e

A i teresti g i terpretatio of the wave partic

e dua

ity of a

physica

e tities has bee give by Wer er Heise berg (1927). The u certai ty pri cip

e, or i

determi

acy pri

cip

e, refers to the simu

ta

eous measureme

t of the positio

a

d the mome tum of a partic

e a d states that the u certai ty x i vo

ved i the measureme t of a coordo ate of the partic

e a d the u certai ty px i vo

ved i themeasureme t of the mome tum i the same directio are re

ated by the re

atio ship x px /2 . (1.52)

To i

ustrate the u certai ty pri cip

e, we co sider a thought experime t. Suppose a para

e

beam of mo oe ergetic e

ectro s that passes through a arrow s

ita

d is the

recorded o

a photographic p

ate (Fig. 1.12). The precisio

with whi

ch we k

ow the y

positio

of a

e

ectro

is determi

ed by the size of the s

it;if d is the width of the s

it, the u certai ty of the y positio is y d . Reduci

g the width of the s

it, a diractio patter is observed o the photographic p

ate. The wave

e

gth of the wave associated to the e

ectro

s is give

by Eq. (1.51), where p desig ate the mome tum of the e

ectro s. The u certai ty i the k ow

edge of the y compo e t of e

ectro mome tum after passi g through the s

it is determi ed by the a g

e correspondin to the central maximum of the diraction pattern; accordin to the theory of the diraction produced by a rectan ular slit, thean le is iven by d sin = . We have py p si = p and h = d d

h = h, d i agreeme t with Eq. (1.52). Note that: y py d To reduce the u certai tyi the determi atio of the coordi ate y of the partic

e, a arrower s

it is required. This s

it produces a wider ce tra

maximum i the diractio patter , i.e.

, a

arger u

certai

ty i

the determi

atio

of the y

compo

e

t of the mome

tum of the partic

e. To improve the precisio

i

the determi

atio

of the y compo

e

tof the mome

tum of the partic

e, the width of the ce

tra

maximum i

the diractio

patter

must be reduced. This requires a

arger s

it, which mea

s a

arger u

certai

ty i

the y coordi

ate of the partic

e.


37/56

1.10 Heise berg u certai ty pri cip

ey T I comi g beam E E T E d E E c

27

O

Screen ith a sin le slit

Observin screen

Fi . 1.12 Diraction throu h a sin le slit.

It is common to consider in everyday life that a measurement can be performed ithout chan in the state of the physical system. The above example clearly sho sthat in contrast to the classical situation, at the atomic level, measurement inevitably introduces a si

nicant perturbation in the system. The uncertainty prin

ciple implies that the concept of trajectory for particles of atomic dimensionsis meanin less. It follo s that for such particles the description of the motionneeds a dierent picture. For everyday macroscopic objects the uncertainty principle plays a ne

li

ible role in limitin

the accuracy of measurements, because the uncertainties implied by this principle are too small to be observed.

1.10.1

Uncertainty relation and the Bohr orbits

In Sect. 1.6 the electron of the hydro en atom is supposed in a circular motionaround the nucleus. Let us investi ate the compatibility of this picture ith the uncertainty relation [E . (1.52)]. The classical treatment of the electron mot

ion is justied for small uncertainties in the position and momentum, x r from which we i

fer a

d px p , (1.53)

x px 1. r p

(1.54)

For the motio

o

the Bohr orbit

, the qua

tizatio

ru

e gives pr =

.


38/56

28

1 The experime

ta

fou

datio

s of qua

tum mecha

ics

The use of Heise berg u certai ty re

atio yie

ds 1 x px . r p 2 It is c

ear owthat this resu

t is i co tradictio with Eq. (1.54) for sma

va

ues of the qua

tum

umber

. It fo

ows that the c

assica

motio

picture of the e

ectro

o

circu

ar orbits must be rejected.

1.11

Prob

ems

1.1 Determi e the umber of photo s emitted per seco d by a P = 2 mW He Ne

aseroperati

g o

the wave

e

gth = 632.8

m.So

utio . The e ergy of a photo is hc/ . The umber of photo s emitted per seco

d is P 6.37 1015 photo s/s . hc/

1.2 I

a te

evisio

tube, e

ectro

s are acce

erated by a pote

tia

diere ce of 25

kV. Determi

e the mi

imum wave

e

gth of the x

rays produced whe

the e

ectro

sare stopped at the scree .So

utio . The e ergy acquired by a e

ectro acce

erated by the pote tia

diere ce U = 25 kV is E = eU . This e

ergy may be radiated as a resu

t of e

ectro

stoppi g; the mi imum wave

e gth radiated is obtai ed whe a

e ergy is radiated asa si g

e photo : mi = hc/E = hc/eU 5.0 1011 m = 50 pm . A

most a

of this radiatio is b

ocked by the thick

eaded g

ass i the scree .

1.3 The temperature of a perso ski is skin = 35 C . (a) Determi e the wavele gth at which the radiatio emitted from the ski reaches its eak. (b) Estimate the et loss of ower by the body i a e viro me t of tem erature environment = 20 C . The huma ski has the emitta ce = 0.98 in infrar d and th surfac ar a ofa typical p

rson can b

stimat

d as A = 2 m2 . (c) Estimat

th

n

t loss of

n

rgy during on

day. Expr

ss th

r

sult in calori

s by us

of th

conv

rsion r

lation 1 cal = 4.184 J .


39/56

1.11 Probl

msSolution. (a) By us

of Eq. (1.4), max = b 9.5 m . Tski

29

This wave

e gth is i the i frared regio of the spectrum. (b) The power emittedby the body is [see Eq. (1.3)]

4 Pemitted = T kin A 974 W ,

while the power aborbed from the environment i

4 Pab orbed = Tenvironment A 819 W .

The net outward ow of energy i P = Pemitted Pab orbed 155 W . (c) The net lo of energy in a time t = 24 3 600

i

E = P t 3 207 kcal. The re

ult i

an overe

timation of the real net lo

of energy, becau e the clothe we wear contributeto a ignicant reduction of the kin emittance.

1.4 Prove that the relation between radiant exitance E of a blackbody and the energy den ity of the b

ackbody cavity is E = (1/4)c . (1.55)

So

ution. Let us conside a sma

su face of the body; this can be conside ed asp

ane. Let us denote by A the a ea. We choose a 3-D Ca tesian coo dinate systemwith the o igin O on the emitting su face and the Oz-axis pe pendicu

a to thesu face and di ected outwa d (see gu e be

ow). z T n We rst rite the ux of ener ythrou h the surface of area A inside the solid an le d = sin d d around the direction n determined by polar an les and . Durin the time interval t , the e ergy emitted i side the so

id a g

e d is located inside the cylinder of eneratrix paralell to n and len th c t . The e ergy i side this vo

ume is Ac t cos and only the fraction d/4 ro agates i the co sidered solid a gle.

A O

c t

The tota

e

ergy emitted through the surface of area A i

the time i

terva

t is=/2 =0 =2

Ac t cos =0

sin d d 1 = cA t. 4 4


40/56

30

1 The ex

erime

tal fou

datio

s of qua

tum mecha

ics

It follows that the e ergy emitted i u it time by u it area is give by Eq. (1.55). Remark. I case we are i terested i the relatio betwee the s ectral qua

tities E a

d , we est ict ou se

ves to the adiation in the wave

ength inte va

(,

+ d). The ene gy emitted with the wave

ength in the specied inte va

in t is=/2 =0 =2

d Ac t cos =0

1 sin d d = c d A t. 4 4

The e ergy emitted i u it time through the u it area of surface is E d = (1/4)c d f om which Eq. (1.5) is infe ed.

1.5 De ive Wiens displacemen

law from Eq. (1.10).

Solu

ion. d 1 hc exp(hc/kB T ) 40hc hc 1 + () = + exp d 6 [exp(hc/kB T ) 1]2he co ditio d ()/d = 0 yie

ds the t anscendenta

equation 1 x/5 = exp(x) , where

shor

hand no

a

ion x = hc/kB T have bee used. Besides the trivia

so

utio x =0 , a positive so

utio

exists, x 4.965 (see gure be

ow). 1 T .

1 x/5 exp(x) 0 0 1 2 3 4 5 Ex

Graphical solu

ion of

he equa

ion 1 x/5 = exp(x) .

The spec

ral energy densi

y is maximum for

he waveleng

h max give by max T hc 2.898 103 m K. 4.965 kB

1.6 A zi c p

ate is irradiated at a dista ce R = 1 m from a mercury

amp that em

its through a spectra

ter P = 1 W radiatio

power at = 250

m.


41/56

1.11 Prob

ems

31

The pe etratio depth of the radiatio is approximate

y equa

to the radiatio wave

e gth. I the c

assica

mode

of radiatio matter i teractio , the radiatio

e

ergy is equa

y shared by a

free e

ectro

s. Ca

cu

ate the mi

imum irradiati

o

time for a

e

ectro

to accumu

ate sucie

t e

ergy to escape from the meta

. Free e

ectro de sity i zi c is = 1029 m3 a d the work fu ctio is = 4 eV.Solu

ion. In a

ime in

erval

he pla

e of area A receives

he energy A P

4R2 a

d this is accumulated by the free electro

s i

the volume A . O

the co

ditio

that a e

ectro acquires the e ergy , A P

= nA . 4R2 The mi imum irradiatio timeis t= 4R2 2.0 105 s P

in s

rong con

radic

ion

o

he experimen

al resul

s.

1.7 Blue ligh

of waveleng

h = 456 m a d power P = 1 mW is i cide t o a photose sitive surface of cesium. The e

ectro work fu ctio of cesium is = 1.95 eV. (a) De

ermine

he maximum veloci

y of

he emi

ed elec

rons and

he s

opping vol

age. (b) If

he quan

um eciency of

he surface is = 0.5 %, determine t

e magnitude of t

e p

otocurrent. T

e quantum eciency is dened as t

e ratio of t

e number ofp

otoelectrons to t

at of incident p

otons.Solution. (a) T

e t

res

old wavelengt

of t

e p

otoelectric eect for cesium is 0 =hc/ 636 nm < , so e

ectro s are extracted from cesium. To ca

cu

ate the maximumve

ocity of the photoe

ectro s, we make use of Eq. (1.14), where = c/ . We get vmax 5.2 105 m s1 . As vmax /c 1, the o re

ativistic expressio of the ki etic e ergy proves to be justied. The stoppi g vo

tage is give by Eq. (1.16), where = c/. We get U0 0.77 V. (b) The umber of e

ectro s extracted i u it time is = P

c/


42/56

32

1 The experime

ta

fou

datio

s of qua

tum mecha

ics

formi g a curre t of mag itude I = e = eP 1.8 A . hc

1.8 Prove that a free e

ectro

ca ot absorb a photo

.

So

utio

. The co

isio

betwee

the photo

a

d the e

ectro

is i

vestigated i

their ce tre of mass refere ce frame, i.e., the frame of refere ce i which the tota

mome tum is zero. The hypothetica

process is prese ted be

ow. Here, p de ote the mag

itude of the mome

tum of the e

ectro

a

d photo

. p E p After absorpt

Before absorptio

Let us de

ote by me the e

ectro

mass. The i

itia

e

ergy of the system is pc +m2 c4 + p2 c2 , whi

e the e ergy of the a

state is me c2 . It is c

ear that e co servatio of e ergy is vio

ated, so the process ca ot occur. Remark. A e

ectro participati g i the photoe

ectric eect is ot free, but bou d to either a atom, mo

ecu

e, or a so

id. The e

ectro

a

d the heavy matter to which the e

ectr

o

is coup

ed share the e

ergy a

d mome

tum absorbed a

d it is a

ways possib

e to satisfy both mome tum a d e ergy co servatio . However, this heavy matter carries o

y a very sma

fractio of the photo e ergy, so that it is usua

y ot co

sidered at a

.

1.9 X rays of wave

e gth 70.7 pm are scattered from a graphite b

ock. (a) Determi e the e ergy of a photo . (b) Determi e the shift i the wave

e gth for radiatio

eavi g the b

ock at a a g

e of 90 from the directio of the i cide t beam.(c) Determi e the directio of maximum shift i wavele gth a d the mag itude ofthis shift. (d) Determi e the maximum shift i the wavele gth for radiatio scattered by a electro tightly bou d to its carbo atom.Solutio . (a) E = hc/ 2.81 1015 J 1.75 104 eV . Remark. This e ergy is severa

oers of mag itude

arger tha the bi di g e ergy of the outer carbo e

ectro s, s

o treati

g these e

ectro

s as free partic

es i

the Compto

eect is a good approximatio

. (b) By maki

g use of Eq. (1.21), we get = 2.43 pm. (c) The directio

aximum shift i

wave

e

gth is = , i.e., the

hoto

is scattered backwards; ()max =2C 4.85 pm.


43/56

1.11 Prob

ems

33

(d) The photo co

ides with the e tire atom whose mass is 12 u . Compto wave

e gth of the carbo atom is 1.111016 m. The maximum cha ge of the wave

e gth due toscatteri

g is 2.221016 m, too sma

to be measured.

1.10 I a Compto scatteri g experime t (see Fig. 1.6) a photo of e ergy E is scattered by a statio ary e

ectro through a a g

e . (a) Determine the an le bet

een the direction of the recoilin

electron and that of the incident photon. (b)Determine the kinetic ener y of the recoilin electron.Solution. (a) Given that p is the momentum of the incident photon and p and pe the momenta of scattered photon and electron after the collision, the conservationof momentum re

uires that pe = p p . T

is relation is projected on two perpendicular axes s

own in t

e gure: pe cos = h h cos , h pe si = sin .

Dividin side by side the t o e uations e et cot =

By use of E

. (1.21), the an

le is

iven by cot = 1 + y T p*

E C ta

= 1 + ta

. 2 mc2 2

pR e

p-

Ex

Momentum conservation in the Compton eect and the choice of a xOy coordinate system.

(b) The kinetic ener

y of the recoilin

electron is e

ual to the ener

y loss ofthe photon: T = hc hc hc C (1 cos ) (E/mc2 )(1 cos ) = = E. + C (1 c(1 cos )


44/56

34

1 The experimental foundations of

uantum mechanics

1.11 Use position-momentum uncertainty relation to estimate the dimensions and the ener y of the hydro en atom in its round state.Solution. Let the electron be conned to a re ion of linear dimension r . We estim

ate the ener

y of the atom. The potential ener

y of the electron is of the orderof V = e2 . 40 r

Accordi

g to the u

certai

ty

ri

ci

le, the u

certai

ty p i

its mome

tum is p /r. The average ki etic e ergy of the e

ectro is T = (1/2m) p2 a d assumi g a zero average mome tum we get T =2 1 (p)2 . 2m 2mr2

The tota

e ergy of the atom is2

E= T + V Let us de

ote Emi

(r) =

2mr22

e2 . 40 r

2mr2

e2 . 40 r

The atom settles to a state of lowest e

ergy. Notice that for large r values, the

ote

tial e

ergy domi

ates; it follows that ex

a

di

g the atom i

creases the total e

ergy. However, for small e

ough r, the ki

etic e

ergy is the domi

a

t co

tributio

a

d the total e

ergy lowers as the atom ex

a

ds. It follows that theremust exists a value of r for which the total e

ergy is a mi

imum. We have2 e2 e2 dEmi

= 3 + = dr mr 40 r3 40 r3

r

40 2 me2

.

The e

ergy attai

s i

deed a mi

imum for r = 40 2/me2 = a0 . The total e

ergy for this radius is exactly the e

ergy of the grou

d state of the hydroge

atom [see Eq. (1.38)]. It should be

oi

ted out that the Heise

berg u

certai

ty relatio

ca

be used to give o

ly a

order of mag

itude of the hydroge

atom size or its grou

d state e

ergy. The exact qua

titative agreeme

t is accide

tal. I

fact, we had i

te

tio

ally chose

the lower bou

d value i

the Heise

berg u

certai

ty relatio

i

such a way (i.e., ) that the resulti

g grou

d state e

ergy is exact.


45/56

1.11 Problems

35

1.12 A electro is bombardi g a atom at rest i its grou d state of e ergy E1. Prove that the threshold ki etic e ergy of the electro , required for the atomexcitatio

to the rst excited state of e ergy E2 , is a roximately equal to E2

E1 .Solutio . Let us de ote m a d M the mass of the electro a d the atom, res ectively. We have m 1 m . M MH 1836 The threshold co ditio for the excitatio of theatom as a result of the collisio

will be ex

ressed i

the ce

tre of mass (CM)frame of refere ce: all ki etic e ergy of the system is used for the atom excitatio . I other words, i the CM frame of refere ce, the electro a d the atom are both at rest after the collisio . We de ote a d P the i itial mome tum of the electro

a

d the atom i

the CM frame of refere

ce, res

ectively. The co

servatio of mome tum a d e ergy i the CM frame of refere ce gives + P = 0 a d Combi i g these equatio s we get 2 = 2M m (E2 E1 ) . M +m 1 2 1 2 + P + E 1 = E2. 2m 2M

The i

itial velocities of the

articles with res

ect to the CM frame of refere

ce are P a d V = = . m M M The i itial velocity of the electro with res ect to the atom is v= vV = The sought ki etic e ergy is T = (M + m)2 2 m 1 (E2 E1 ) E2 E1 . m(v V )2 =

= 1+ 2 2M 2 m M 1 1

. + m M

1.13 Show that the cha ge i ki etic e ergy of a article of mass m, with i itial ki etic e ergy T , whe it collides with a article of mass M i itially at rest i the laboratory frame of refere ce is T = Discuss the case M m . 4M/m T. (1 +M/m)2


46/56

36

1 The ex

erime

tal fou

datio

s of qua

tum mecha

ics

Solutio . Let us desig ate the velocity of the rojectile i the laboratory frame of refere ce by v. We choose a axis with the orie tatio of v. The co servatio

of mome

tum a

d e

ergy gives mv = mv + M V a

d 1 1 1 mv 2 = mv 2 + M V 2 , 2 2

2

where v a d V de ote the al velocity of article m a d M , res ectively. Arra gi

g the two equatio

s i

the form m(v v ) = M V a

d m(v v )(v + v ) = M V 2 ,

rst degree equatio s are obtai ed: v v = (M/m)V a d v + v = V .

The velocity of

article M after the collisio

is V = 2 v. 1 + M/m

The cha ge i the ki etic e ergy og the rojectile m is 1 4M/m T = M V 2 = T 2 (1+ M/m)2 For M m we have 4M/m m T =4 1. T (M/m)2 M

1.14 Justify the

ame of

ifetime for the time i

terva

de

ed by = 1/A [see Eq. (1.45)].Solu

ion. We calcula

e

he average

ime spen

by

he a

om before de exci

a

ion.Suppose a

= 0

here are N (0) a

oms in

he upper energy level. During

he

ime in

erval (

d

/2,

+ d

/2), a number AN (

) d

= AN (0) exp(A

) d

of a

oms de exci

e. These a

oms have spen

a

ime

in

he upper energy level. Thus,

he probabili

y

ha

an a

om remains in

he upper energy level a

ime

is AN (

) d

=A exp(A

) d

. N (0) The average

ime spen

by an a

om in

he upper energy levelis 0

A exp(A

) d

=

exp(A

)0

+0

exp(A

) d

=

1 =. A


47/56

1.11 Problems

37

1.15 The wavepacke

for a par

icle of mass m has

he uncer

ain

y x . After what time the wavepacket wi

spread appreciab

y?So

utio

. The u

certai

ty i

the partic

e mome

tum is p /2x . The u

certai

ty i

t

he partic

e ve

ocity is the

v = p/m /2mx . After a time t the spreadi

g of the wavepacket due to the u certai ty of its ve

ocity is (v)t . This spreadi g becomes importa t whe it is of the same order of mag itude as the i itia

spreadi g, x .We de e the spreadi g time ts by (v)ts = x . It resu

ts ts = x 2m(x)2 . v

1.16 The wave

e gth associated to a partic

e is k ow up to , where . Derive amu

a for the u certai ty x i the partic

e positio .So

utio

. The u

certai

ty i

the wave

e

gth imp

ies a

u

certai

ty i

the mome

tum h h 2 . p = The u certai ty i the partic

e positio is x 2 /2 = . p 4


48/56

38

1 The experime

ta

fou

datio

s of qua

tum mecha

ics


49/56

Appe dix A

Fu

dame

ta

physica

co

sta

tsThe tab

es give va

ues of some basic physica

co

sta

ts recomme

ded by the Committee o Data for Scie ce a d Tech o

ogy (CODATA) based o the 2006 adjustme t [11, 12]. The sta dard u certai ty i the

ast two digits is give i pare thesis.


50/56

40

A Fu

dame

ta

physica

co

sta

ts

Tab

e A.1 A abbreviated

ist of the CODATA recomme ded va

ues of the fu dame ta

co sta ts of physics a d chemistry. Qua tity speed of

ight i vacuum mag eticco

sta

t e

ectric co

sta

t 1/0 c2 Newto

ia

co

sta

t of gravitatio

Avogadro co

sta

t mo

ar gas co

sta

t Bo

tzma

co

sta

t R/NA mo

ar vo

ume of idea

gas (T =273.15 K, p = 101.325 kPa) Loschmidt co sta t NA /Vm e

eme tary charge Faraday co sta t NA e P

a ck co sta t h/2 electro mass e ergy equivale t i MeV electro charge to mass quotie

t

roto

mass m

= Ar (

) u e

ergy equivale

t i

MeV

eutro mass m = Ar ( ) u e ergy equivale t i MeV roto electro mass ratio e structure co sta t e2/40 c i verse e structure co sta t Rydberg co sta t 2 me c/2h R hcin eV Wien displ cement l w const nts b = max T b = max /T Stefa Bo

tzma co sta t( 2/60)k 4/ 3 c2 Bohr radius /4R = 40 2/me e2 Com

to

wavele

gth h/me c classical eltro radius 2 0 Bohr m gneton e /2me nucle r m gneton e /2mp me me c 2 e/me mp mpc 2 mn mn c 2 mp /me 1/ R Symbol c, c0 0 0 G NA R k Value 299 792 458 m s1 (exact)107 N A2 8.854 187 817... 1012 F m1 6.674 28(67) 1011 m3 kg1 s2 6.022 141 79(30ol1 8.314 472(15) J mol1 K1 1.380 6504(24) 1023 J K1 8.617 343(15) 105 eV K1 22

(39) 103 m3 mol1 2.686 7774(47) 1025 m3 1.602 176 487(40) 1019 C 96 485.3399(24)ol1 6.626 068 96(33) 1034 J s 4.135 667 33(10) 1015 eV s 1.054 571 628(53) 10346.582 118 99(16) 1016 eV s 9.109 382 15(45) 1031 kg 0.510 998 910(13) MeV 1.758 820150(44) 1011 C kg1 1.672 621 637(83) 1027 kg 1.007 276 466 77(10) u 938.272 013(23) MeV 1.674 927 211(84) 1027 kg 1.008 664 915 97(43) u 939.565 346(23) MeV 1836.152 672 47(80) 7.297 352 5376(50) 103 137.035 999 679(94) 10 973 731.568 527(73)m1 13.605 691 93(34) eV 2.897 7685(51) 103 m K 5.878 933(10) 1010 Hz K1 5.670 400(40) 108 W m2 K4 0.529 177 208 59(36) 1010 m 2.426 310 2175(33) 1012 m 2.817 9408) 1015 m 927.400 915(23) 1026 J T1 5.788 381 7555(79) 105 eV T1 5.050 783 24(1J T1 3.152 451 2326(45) 108 eV T1

Vm 0 e F h

b b a0 C re B N


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41

Tab

e A.2 The va

ues i

SI u

its of some

o

SI u

its. Name of u

it Symbo

Va

uei

SI u

its a

gstrm o A 0.1

m = 100 pm = 1010 m a e

ectro

vo

t : (e/C) J eV 1.602 176 487(40) 1019 J b (u ied) atomic mass u it 1 u = mu = (1/12)m(12 C) = 103 kgmo

1/NA u 1.660 538 782(83) 1027 kg a The e

ectro vo

t is the ki etic e ergy acquired by a

e

ectro

i

passi

g through a pote

tia

diere ce of o e vo

t i

vacuum.

b The u

ied atomic mass u

it is equa

to 1/12 times the mass of a free carbo

12atom, at rest a d i its grou d state.


52/56

42

A Fu

dame

ta

physica

co

sta

ts


53/56


54/56

44

B Greek

etters used i

mathematics, scie

ce, a

d e

gi

eeri

g

Appe dix B

Greek

etters used i

mathematics, scie

ce, a

d e

gi

eeri

g

Name of

etter a

pha beta gamma de

ta epsi

o

zeta eta theta iota kappa

ambda mu u xi omicro pi rho sigma tau upsi

o phi chi psi omega Capita

etter A B EZ H I K M N O

T X Lo er-case letter , , ,


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B

o r p

y[1] H. H

H. C. Wo

f, T

P

ys cs of Atoms

Qu t (7t

t o ), pr r-V

r

, B

r

(2005). [2] R. E s

r

R. R

s

c

, Qu tum P

ys cs of Atoms, Mo

cu

s, o

s, Nuc

,

P

rt c

s (2

t o

), o

W

y & o

s,N w

!

or

(1985). [3] rw y [4] I. M. Pop scu,

" c, Vo

. II, E

tur D

ct c P

o c Bucur t , Bu s s cur t (1983). s [5] V.

or scu, L ct

M c c Cu t c, E

tur

#

v

rs t

Bucur

t ,

t s Bucur

t (2007). s [6] C. Co

-T

ou

j , B. D u,

. L

o, Qu

tum M

c

cs, Vo

s. I

II, o

W

y & o s, N w

!

or

(1977). [7] D. . Gr t

s, I tro

uct o to Qu tum M c

cs, Pr t c H

(1995). [8] B. H. Br s

C. . o c

, Qu tum M c

cs (2

t o

), P rso

, Lo

o

(2000). [9] R.

r, Pr

c p

s of Qu tum M

c

cs (2

t o ), K

uw r Ac

m c, N w!

or

(1994). [10] Prop rt s of so

s, CRC H

oo

of C

m stry

P

ys cs, I t r t V rs o 2005, D v

R. L

,

.,

ttp://www.

cp t

s .com, CRC Pr ss, Boc R to ,

L (2005). [11] P. . Mo

r, B. N. T

y

or,

D. B. N

w

, CODATA r

comm

v

u

s of t

fu

m t

p

ys c

co st ts: 2006, R v. Mo

. P

ys 80(2), 633730 (2008); . P

ys. C

m. R f. D

t 37(3), 11871284 (2008). T

rt c

c

so

fou

t

ttp://p

ys cs. st.

ov/cuu/Co st ts/p p rs.

tm

. [12] CODATA I t r t o

y r comm

v

u s oft

fu

m t

p

ys c

co

st ts,

ttp://p

ys cs.

st.

ov/cuu/Co

st ts.


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