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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.
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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
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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:
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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)
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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
,
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(1.41)
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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.
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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.
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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)
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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)
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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 ,
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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
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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
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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
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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.
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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.
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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 =
.
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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 .
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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
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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.
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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/
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1 The experime
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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.
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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 )
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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.
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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
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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
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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
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1 The experime
ta
fou
datio
s of qua
tum mecha
ics
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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.
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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.
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42
A Fu
dame
ta
physica
co
sta
ts
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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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