Complex numbers 複數
Chapter 8
Gialih Lin, Ph. D. Professor
8.1 Concepts
• x2 = -1 are x = ± √-1
• The square root of -1 as a new number which is
usually represented by symbol I (sometimes j)
with the property
• i2 = 1
• I, 4i, and -4i are numbers called imaginary to
distinguish them from real number.
• z= x + iy
• Such numbers are called a complex number
(Discuss in Chapter 8)
Powers of I
•
•
•
•
•
•
•
i3=i2 I =-I
i4= (i2)2 = (-1)2 =1
i-1 = 1/i=i/i2=i/-1=-I
i4n=1
I4n+1 =i
I4n+2=-1
I4n+3=-i
8.2 Algebra of complex numbers
•
•
•
•
•
•
•
•
z1= x1 + iy1; z2= x2 + iy2
Equality
z1=z2 if x1=x2 and y1=y2
Addition
z1+z2 = x1 + x2 + i(y1+y2)
Multiplication
z1z2 = (x1+iy1)(x2+iy2)
=(x1 x2 - y1y2)+i(x1y2+y1x2)
The complex conjugate 共軛
z*= x - iy
• zz*= x2 + y2
• ½(z+z*)= ½(x + iy + x - iy )=x=Re(z)
• ½(z-z*)= ½(x + iy - x + iy )=iy=I Im(z)
• zz*=(x + iy)(x - iy)= x2 + y2 (real and
positive)
Division
• z1/z2 = (x1+iy1)/(x2+iy2)= z1z2*/z2z2*
• = (x1+iy1)(x2-iy2)/(x2+iy2)(x2-iy2)
• = (x1+iy1)(x2-iy2)/(x22 +y22)
• =(x1x2+y1y2)/(x22+y22)+i[(y1x2-x1y2)/(x22+y22) ]
•
•
•
•
(2+3i)/(3+4i)= (2+3i)(3-4i)/(3+4i)(3-4i)
=(6+9i-8i-12i2)/(9+16)
=(6+i+12)/(9+16)
=(18+i)/25=(18/25)+i(1/25)
8.3 Graphical representation
• r=(x2+y2)1/2
• Modulus or absolute
value of z and is
written
• r =mod z = ｜z｜
y(imaginary axis)
z=x+iy
r
y

x
x (real axis)
The polar representation
• x=r cos, y=r sin
• z=r(cos + I sin)
• Argument or angle of z, 
 =arg z
• =tan-1(y/x)
if x>0
• =tan-1(y/x)+p if x<0
8.5 Euler’s formula
•
•
•
•
The Euler number
e = 1+ 1/1! + 1/2! + 1/3!+ 1/4!+ …
=1+1+0.5+0.16667+0.041667+
=2.71828
Euler’s formula
•
•
•
•
•
•
•
ez =exp(z)= 1+ z/1! + z2/2! + z3/3!+ z4/4!+ …
ei =exp(i)
= 1+ i + (i)2/2! + (i)3/3!+ (i)4/4!+ …
=[1 - 2/2! + 4/4!- 6/6!+ …] +i[ - 3/3!+ 5/5!+…]
Real part
Imaginary part
From Section 7.6
ei=exp(i)=cos + isin
exp [± iwt] = cos (wt) ± i sin (wt)
Driven, Damped Weight on a
Spring
(Harmonic Oscillator)
f
k
m
The Driven, Damped Weight on a Spring: One of
the Most Important Problems in Physical Science
(Damped Harmonic Oscillator)
Net force = m(d2x/dt2)= -kx (simple harmonic oscillator)
Weight on a spring
Damping force (frictional force opposing the motion)
= -f(dx/dt) where f: damping constant
The equation of motion for the position, x, of a weight
of mass, m, on a spring of force constant, k, driven by
an oscillating force, F,
F = F0cos(wt) the applied driving force
and subject to a frictional (damping) force proportional
to the velocity of the moving weight, is given by:
m(d2x/dt2)= net force= -kx – f(dx/dt) + F0cos(wt)
Solution General Case real
x = x’ cos (wt) + x’’ sin(wt)
where
x’ = Fo m(wo2-w2) / (m2(wo2 - w2)2 + f2 w2)
and
x’’ = Fo f w / (m2(wo2 - w2)2 + f2 w2)
where w0 = (k/m)1/2
Natural frequency for weight-on-a-spring
Solution in terms of complex
exp [± iwt] = cos (wt) ± i sin (wt)
m(d2x/dt2)+ kx + f(dx/dt) = F0cos(wt)
x is real
m(d2x/dt2)m + kx + f(dx/dt)= F0 exp [iwt]
complex
Complex s = c exp [iwt]
c = Fo / ((k- w2)+ if w)
= Fo / (m2(wo2 - w2)2 + if w)
1/(a+ib) = (a-ib)/(a2+b2)
x is
c = Fo ((m(wo2 - w2)2 + if w)/ (m2(wo2 - w2)2 + f 2w2)
Re (complex x) = Re (c exp[iwt])
= Fo m(wo2-w2) / (m2(wo2 - w2)2 + f2 w2)cos(wt)
+ Fo f w / (m2(wo2 - w2)2 + f2 w2) sin(wt)
= x’ cos (wt) + x’’ sin (wt)
c = x’ + i x’’
x’’ ultimately is expressed as the amplitude of the 90o outof-phase response
lim x’’ = 0 (f→0) so that the system can respond only inphase with the driver
Scattering limit: negligible damping
and/or far from resonance
when f << ∣m(wo2-w2) /w ∣
lim (x’) = F0 / m(wo2-w2)
lim (x’’) = 0
Escatt is proportional to d2(x’cos(wt))/dt2
=-F0w2cos(wt)/m(w02-w2)
=-F0cos(wt)/m((w/w0 ) 2-1)
Iscatt = constant/(wo/w)2-1)2
Rayleigh limit: driving frequency smaller than natural frequency
When w<<∞, lim w<<w0 Iscatt = constant (w4/wo4)
Rayleigh Scattering (visible light)
Thomson limit: Driving frequency larger than natural frequency
deBroglie relation, l=h/mn, where h is Plank’s constant
lim w>>w0 Iscatt = constant
X-ray, fast moving electron
Why sky is blue
• The blue light is scattered much more strongly than red light
(w blue >> w red)
• In particular, this explains why the sky is blue, since light
reaching an observer in directions other than in a straight line
from the sun must arise from driven oscillating electrons on
molecules in the atmosphere. Furthermore, since the
wavelength of visible light is of the order of dimensions of
very large macromolecules in solution, such as synthetic or
biological polymers, the light waves scattered from different
parts of a big molecule will interfere destructively to an extent
that depends on macromolecular size and shape.
Lorentz limit: driving frequency
near natural frequency
• when ∣(wo- w) ∣<< (wo+ w)
• when wo = w
 w02-w2 = (wo-w)(w0+w) = 2w0(w0-w)
• x = x’ cos (wt) + x’’ sin(wt)
• x’ = Fo m(wo2-w2) / (m2(wo2 - w2)2 + f2 w2)
• x’’ = Fo f w / (m2(wo2 - w2)2 + f2 w2)
Lorentz Limit: Driving Frequency
Near Natural Frequency
∣wo-w∣<< wo+ w
w02-w2=(w0-w)(w0+w) ≒2w0 (w0-w)
lim ∣wo-w∣/(wo+ w) <<1(x’)
= 2mw0 (w0-w) Fo / (4m2wo2 (w0-w))2 + f2 w02)
lim ∣wo-w∣/(wo+ w) <<1(x’’)
= f w0Fo / (4m2wo2 (w0-w))2 + f2 w02)
Finally, recognizing that friction (damping) coefficient, f,
has units of mass/time, we may simplify by defining a
characteristic relation time , t,
t = 2m/f
Lorentz Limit Spectroscopy
Dispersion = lim ∣wo-w∣/(wo+ w) <<1(x’)
= Fo { (w0-w)t2 /[1 +(w0-w)2t2 ]}/2mw0
Absorption = lim ∣wo-w∣/(wo+ w) <<1(x’’)
= Fo { t2/[1 +(w0-w)2t2 ]}/2mw0
• Spectroscopy (w=w0)
• Electronic, vibration, pure rotation, ion
cyclotron resonance, electron spin
resonance, nuclear magnetic resonance
8.6 Periodicity
• Rotation in quantum mechanics
• Schrodinger equation for a rigid rotor in
plane is
• -(ħ2/2I)(dy2/d2)=Ey
• dy2/d2=-a2y
 yn()=Cein, n=0, ±1, ± 2,...
• En=ħ2n2/2I
40.4 Molecular Spectra
Example 40-4: Reduced mass.
Show that the moment of inertia of a
diatomic molecule rotating about its center
of mass can be written
I =  r 2,
where
m1m2
=
m1 + m2
.
40.4 Molecular Spectra
Figure 40-16 goes
here.
A diatomic molecule
can rotate around a
vertical axis. The
rotational energy is
quantized.
40.4 Molecular Spectra
These are some
rotational energy
levels and allowed
transitions for a
diatomic molecule.
理論計算(Gaussian 03) 聯苯(biphenyl)轉動中心C-C鍵能量變化曲線圖
旋轉聯苯中心C-C鍵而成之三構形(conformation)能階示意圖
理論計算(Gaussian 03) 4-n-butylcarbamyloxy-4’-acetyloxy-biphenyl 轉動中心C-C鍵
能量變化曲線圖
旋轉4-n-butylcarbamyloxy-4’-acetyloxy-biphenyl中心C-C鍵而成之
三構形(conformation)能階示意圖