phys. formulae – August 28, 2014
Cylindrical coords
dV =rdrdφdz
•
b
~
∇t=∂
r + 1 ∂φ [t]φ+∂
b
r [t]b
z [t]z
r
b
d~
l=dr r
b+rdφφ+dz
z
b
SI Base Units
2 [t]+∂ 2 [t]
∇2 t= 1 ∂r [r∂r [t]]+ 12 ∂φ
2
z
r
r
Length L⁄metre: path length of light in vacuum during
• ∂z for polar/planar
1 ∂ [rv ]+ 1 ∂ [v ]+∂ [v ]
~
∇·~
v
=
r
z z
dt = 1/299792458s. Mass M⁄kilogram: mass of an interr r
r φ φ
b
φ=atan(y/x)
b
r −sφφ
b
y
b
national prototype. Time T⁄second: dt of 9192631770 x=rcφ • z=z • x=cφb
• r=√x·x+y·y • rbb=cφx+sφ
b z
b
133
b
y=sφb
b
r +cφφ
φ=−sφx+cφ
b
y
b
Cs (hyperfine split, @ 0 K) ground state oscilla- y=rsφ z=
Delta Function/Distrib.
tions. Elec. Current I⁄ampere: produces 2E−7 N⁄m
R
~ r d
between two L = ∞, A = 0 wires 1 m apart in vacδ d (~
r ) = 1 d eik·~
d k
(2π)
cum. Thermo. Temp. Θ⁄kelvin: 1⁄273.16th the T of
N
Transforms
triple point water. Amount of Substance ⁄mole: as
∞
∞
R
R
~ r d~
~ r d
many entities (atoms, molecules, e− , etc.) as there are f(~
k)eik·~
d k • F(~
r )e−ik·~
d~
r
r ) = 1d/2 F(~
k) = 1d/2 f(~
(2π) −∞
(2π) −∞
12
atoms in 12 g of ground state C. Luminous IntenCombinatorics
sity J⁄candela: directional intensity of a monochromatic
(remaining possible) N choose n
N (f = 5.40E12 Hz) source with directional radiant inten- N
P
N·(N−1)···(N−n+1)
N!
N =2N .
= n! (N
•
n
n =
sity 1⁄683 watt per steradian.
n! "
−n)!
n=0
(order unimportant)
Constants & Units
Stirling’s Approx.
Value(s)
Base
√
n! ≈ nn e−n √2πn (1 + 1/12n)
⇐⇒ n 1
me− 9.11E−31 kg
(5.48E−4) · u
M
n −n
2πn
(sqrt sometimes needed)
n! ≈ n e
mp− 1.67E−27 kg
(1.01)
·u m
M
m
n
−n
n! ∼ n e
=⇒ ln n! ≈ n ln(n) − n
mair (29.0) · u
≈ 43 N2 + 14 O2
M
m⊗ 5.97E24 kg
(3.60E51) · u
M
Integrals – (a,b > 0)
b√
π
R
R
1.99E30 kg
(3.33E5) · m⊗
M
m
2n
2m
1 (x−a)(b−x)dx= π (√b−√a)2
1
1
2
• 0sin (x)cos (x)dx=β(n+2,m+2)
M L2/T 2 Θ
kB 1.38E−23 J⁄K
8.62E−5 eV⁄K
ax
∞
2
R
M L2/T 2 ΘN
R 8.31 J⁄mol K
NA kB
Γ((n+1)/2)
∞
R n −bx
xne−bx dx= 1
n≥0
x e
dx=n!/bn+1 n≥0
2 b(n+1)/2
1⁄N
NA 6.02E23 1⁄mol
≈ 279 1⁄mol
•
0
0
∞
M L2/T
h 6.23E−34 J·s
4.14E−15 eV·s
R xn
R
1
dx=Γ(n+1)ζ(n+1) n≥1
dx=−ln(1+e−x)
1 eV
IT
e 1.60E−19 C
ex −1
ex +1
•
0
e2/4π◦ ~c
α ≈ 1/137.036
–
q
∞
∞
R
2
R
n
x
e~/2me
dx=(1−1/2n)Γ(n+1)ζ(n+1) n≥1
e−ax dx= 1 π
µB 5.79E−5 eV/T
IL2
2 a
ex +1
• −∞
0
L3/T 2 M
G 6.67E−11Nm2/kg2 –
q
q
∞
∞
2
R −ax2−2bx
R
2/a
σB 5.67E−8J/sm2K4 π2kB/60c2~3
e
dx= 1 π eb
xe−a(x−b) dx=b π
2 a
a
•
−∞
−∞
M L/T 2
1 N 1kg m/s2
1(P a)m2
∞
q
R 2 −ax2
R 2π R π
1
π
2
2
2
2
x e
dx=
M L /T
1 J 1kg m /s
1N m,1C V
a·b
r)(~
b·b
r) sin θdθdφ=4π(~
a·~
b)/3
2 a3
0
0 (~
• −∞
M L2/T 3
1J/s,1V A,1ΩA2
1 W 1kg m2/s3
2−z 2
R
2
2
zR
cosθ
−
R
R
sinθ
cosθdθ
b√
√
=
R
2
T 4 I 2/L2 M
1 F 1s4 A2/m2 kg
1 VJ2 ,1 CJ ,1 sH
(R2+z2−2zR cosθ)3/2 z2 R2+z2−2zRcosθ • a2−x2 dx= πa
4
s
Js
a
M L2/T 3 I 2
1 Ω 1kg m2/s3 A2
1V
A ,1 C 2 ,1 F
∞
R xm dx
Γ(n−q)|b|2(q−n)
m+1
M L2/T 3 I
1 V 1kg m2/s3 A
1ΩA,1J/C ,1W/A
2 2 n = π sin(πq)Γ(1−q)Γ(n) q≡ 2 , 2n−m>1, m even.
kg
−∞(x +b )
M/T 2 I
1 T 1kg/s2 A
1 mJ2 A ,1 C
s
√
R
R
2
dx
√x
√ dx
=ln[x+ z 2+x2]
=
1 AJ2 ,1Ωs,1 mAT M L2/T 2 I 2
1 H 1kg m2/s2 A2
(z 2+x2)3/2
z 2 z 2+x2
z 2 +x2
•
∞
q √
Math
R −ar
bq
R
b −1dx=b π −arcsin a − a(b−a)
e
sin(br)dr= 2 b 2
2
b
b +a
Def. As
Circumference
Area
• 0
Shape
a x
∞
R1
x2 +y 2 =r 2
Circle
2πr
πr 2
√
sin(±|b|r)dr=±π/2
2 2
2 2
Ellipse
≈π[3a+3b− (3a+b)(a+3b)]
πab
n-Vol.
n-Area
4πr 3/3
2
4πr
√
2
πhr 2/3
πr 2 [1+
√ 1+(h/r)√]
ab[2+ 1+(2h/a)2 + 1+(2h/b)2 ]
abh/3
2
π n/2 r n/Γ(1+n/2)
2π n/2 r n−1/Γ(n/2)
x /a +y /b =1
Shape
3-sphere
cone⊥
pyramid⊥
n-sphere
0
r
•
cos2(x)=(1+cos(2x))/2
cosh2 (x) − sinh2 (x) = 1
n
lim 1+ x =ex
n→∞ n n
n
lim 1+ x = x
n
n
x→∞
ez =
∞
P
zn
n!
n=0
•
cos(x±y)=cos(x) cos(y)∓sin(x) sin(y)
Series and 
Sums
P∞ (−1)k+1 xk −1<x≤1
k=1
k
• ln(1+x) =
ln(x)+P∞ (−1)k+1 |x|>1
k=1 kxk
N
P
N−1
if
1
xn = 1−x
=⇒ 1−x
|x| < 1
1−x
n=0
N→∞
Coordinates, Vector Calculus
R~
R
H
R
H
b (∇
~ f)·d~
~ AdV
~
~ a
l=f(~
b)−f(~
a)
~
~
~ ~
∇·
= A·d~
~
a
•
• ∇×A·d~a= A·dl
~ b/ 2)=4πδ (3)(~)
∇(
∇2 (1/)=−4πδ (3)(~)
~
~
~
~
~
~
~
~
~
~
~
~
~
~ f)
A·(B×C)=B·(C×A)=C·(A×B)
∇·(∇×A) = 0 = ∇×(∇
•
~×(B×
~ C)=(
~
~ C)
~ B−(
~
~ B)
~ C
~
~ ×(∇
~ ×A
~)=∇
~ (∇·
~ A
~)−∇2A
~
A
A·
A·
∇
1
b
b
~
∇t=∂
r + 1 ∂θ [t]θ+
∂φ [t]φ
r [t]b
Spherical coords
r
r
sin
θ
•
b
b
dV =r 2 sin θdrdθdφ
d~
l=dr r
b+rdθ θ+r
sin θdφφ
h
i
2 [t]
∇2 t= 12 ∂r r 2 ∂r [t] + 2 1
∂ [sin θ∂θ [t]]+ 2 1 2 ∂φ
r
r sin θ θ
r sin θ
1
1
1
2
~
∇·~
v = 2 ∂r [r vr ]+
∂ sin θvθ ]+
∂ [v ]
r sin θ θ [
r sin θ φ φ
r
x=rcφsθ
y=rsφsθ
z=rcθ
φ=atan(y/x)
b
b
x=sθcφb
b
r +cθcφθ−sφ
φ
b
b
r
b=sθcφx+sθsφ
b
y+cθ
b
z
b
b
z=cθ
b
r
b−sθ
√θ
θ=atan( x·x+y·y/z)
φ=−sφx+cφ
b
y
b
√
r= x·x+y·y+z·z
b
r +cθsφθ+cφφ θ=cθcφx+cθsφ
b
y−sθ
b
z
b
• y=sθsφb
•b
b
β(1, 1)=π
β(1, 5)= 3π
2 2
• 23 23 π8
β(1, 3)= π
β( , )=
2 2
2
2 2
8
β(x,y)=β(y,x)=Γ(x)Γ(y)/Γ(x+y)
•
•
Classical Mech.
Trigonometry Identities
sin(x)=(eix−e−ix)/2i
sin(2x)=2sin(x)cos(x)
sinh(x)=(ex−e−x)/2
•
•
cos(2x)=1−2sin2(x)
cos(x)=(eix+e−ix)/2
cosh(x)=(ex+e−x)/2
sin2(x)=(1−cos(2x))/2
sin(x±y)=sin(x) cos(y)±cos(x) sin(y)
Misc
Γ(1)=1
Γ(n)=(n−1)!
ζ(3)≈1.202
•
√ •
ζ(2)=π2/6
ζ(4)=π4/90
Γ(1)= π
2
R
Γ(x)=(x−1)Γ(x−1)= 0∞tx−1e−tdt
r1+m2 ~
r2
mm
~ Gm1 m2 b
~=m1 ~
F=
R
µ= 1 2
s2=x2+y 2
m1+m2
m1+m2
− 2
•
•~
• ds=√1+(∂xy)2 dx
~=∂ ~
~
r=~
r1−~
r2
L
r ×~
p
~
τ=~
r ×F
t τ =~
∂L = d
∂L
m
r)2−V(~
r ) T=m[(∂t ~
r)2+r 2 (∂t φ)2]
∂qi dt ∂[∂t qi] L=T−V = 2 (∂t ~
2
• Polar
Kinetic.
Conserv. F’s., • & Inertial Frm.
Pendulum
~g along Tb to cancel Tb ⊥ θb . Now Fnet =
Project F
−mg sin θ ≈ −mgθ. So ma =p−mgθ. Since sp= Lθ,
g
solve θ̈=− L
θ → θ(t) = A sin( g/Lt) + B cos( g/Lt).
p
p
ω
1
g/L and T = 2π
L/g .
Then, f = 2π
= 2ω
Light Interference: A cos(kr+ωt)
Double slit:
∼ 2 point sources
•
Single slit:
d = slit width
•
(I=2A)max: sin θ=nλ/d⇔n=0,±1,±2,..
(I=0)min: sin θ=nλ/2d⇔n=±1,±2,..
(I=0)min: sin θ=nλ/d⇔n=±1,±2,..
General: I(θ)=A(j0(dπ sin(θ)/λ))2
Special Relativity
p
T = [γ(v)−1]mc2
•
γ(v) = 1/ 1−v 2/c2
b
b
fobs = fstat/(γ(v)(1−~
v · d/c)) with v
b · d = cos θ.
Scattering
0
Compton: (γ+{→γ +|) Inelastic, but ∼elastic.
q
hc
1) E-cons.: mc2 + hc
m2 c4 + p2m0 c2 .
λ = λ0 +
2) P -cons.: p
~γ +
p
~
~γ 0 +~
pm0 . Isolate p
~m0 , square.
m =p
h
Substitute for p
~m0 : λ0−λ = 2 mc
sin2 θ2 θ ← p
~γ·~
pγ 0 .
Thompson: Compton, Eγ mc2 . (Elastic: {→{).
2
2R
dσ
= R4 ⇒ σ = R4 sinθdθdφ = πR2.
Hard-sphere: dΩ
Rutherford: (|+{→|+{). Elastic, Heavy target,
Coulomb.
dσ
dΩ
=
(q1 q2 )2
(16π◦ T sin2 θ )2
2
T = mc2 (γ −1).
(incoming kinetic)
Thermo./Stat. Mech.
work
R V compression (other)
1. ∆U = Q + W
• W = V f P (V )dV + Wo
heat
i
∂S
2. S ≡ kB ln Ω grows. Equilib. ⇒ ∂...
= 0 {N,U,V,...}.
3. T→0 ⇒ CV →0; dS
0
(S6=0 b/c of ∆E=0 config.s!)
P
Thermo. Identity.
dU = T dS −P dV + i µi dNi .
µi
1
∂S
P
∂S
∂S
=
•
=
•
T
∂U V,N
T
∂V U,N
T = − ∂N
i
i U,V
i
H = U + P V. Enthalpy: Eassemble & put in enviro.
Reversible dV ⇒ quasistatic. (Note: H
⇐⇒
H !) •
Q 6= 0 ⇒ irreversible. But if ∆T → 0 then ∆S ≈ 0
so “infinitesimal” Q is ∼reversible. • Quasistatic
RT C
∆S = T f TV dT .
⇒ dS = Q
T ; if dV = dN = 0 then:
i
RT C
If dN = 0 and W = −P dV then:
∆S = T f TP dT .
i
Degrees of Freedom f
[Total at STP]=(trans, rot, spring{-vib, -pot}). Spring DOF
are ‘frozen out’ in gases at STP. Monatomic gas [He, Ar]
3 =(3,0,{0,0}).
Diatomic gas [O2 , N2 , CO] 5 =(3,2,{1,1}).
Polyatomic (>2) gases: linear [CO2 ] 5 =(3,2,{4,4}), nonlinear [NO2 , H2 O] 6 =(3,3,{3,3}).
Elemental solids [Al, Pb]
6 =(0,0,{3,3}). Einstein solids 2 =(0,0,{1,1})/osc.. Adiabatic
exponent: γ ≡ 1 + 2/f .
Equipartition Theorem
At temp. T the avg. energy of any quadratic DOF is
1
2 kB T . For N entities with only f quadratic DOF
each: Uthermal = N f 12 kB T .
Ideal Gas
P V = nRT =N kBT (Hard indist. spheres; low ρN ; elastic coll’s) • U = Nf 21kBT • CP = CV +kBN.
3 V 4πmU 2
S(N,V,U) = N kB ln N
+ 52 monatomic.
3N h2


3
2+mgz
µ=−kBT ln V 4πmU
2
N 3h N
•
3/2
=V
= V
Ztr(T,N=1)=V 2πm
vQ
h2 β
l3
Q
Z(T,V,N)= 1 [Z1(T,V)]N= 1 [Zint(T)Ztr(T,N=1)]N
N!
N!
mixing gases
b&c N
N
Nc monatomic•Pb=Pc
c
∆S=−NkB Nb ln Nb + N
N ln N
T =T •N +N =N
c
b
c
b
Free Expansion (Quasistatic+irreversible+vacuum)
∆U =
Q+
W = 0. But ∆S = N kB ln
Vf
Vi
(6= 0, 6=
Q
T ).
Isothermal (Quasistatic+slow+thermal equil.) ∆T =
= Q+W = 0 • ∂(PNV ) = 0 • ∆S = Q
0 ⇒
∆U
T
If ∆N = 0 then Wcomp. = N kB T ln Vi/Vf .
h
(
(h ∆U =
h
(h
Adiabatic (Fast+(
quasistatic)
Q+W • ∆S 6= 0
If ∆N = 0 then ∂(V T f/2 ) = 0, ∂(V γ P ) = 0, and
Wcomp. = f2 (Pf Vf − Pi Vi ).
Isentropic (Adiabatic+Quasistatic) ∆S = Q
= 0. Reversible.
T
p
p
3kB T /m from toy piston/P
vrms =
v2 =
wall .
q
p
−V ∂P
γkB T /m [Adiabatic; expand
csound =
=
ρm ∂V
∂(P V γ ) = 0, rearrange for bulk modulus.]
Heat (Energy) Capacity
C≡
Q
∆T
CP =
=
∆U −W
∆T
•
c≡
C
m
•
∂Wo
∂H
∂U
∂V
∂T P = ∂T P+P ∂T P− ∂T
P
CV =
•
∂U
∂T V,N,W
L≡
Q
m
P,W
Virial Expansion / Van der Waals
P V = nRT (1 + B(T)/(V/n) + C(T)/(V/n)2 + . . .)
Qual. OK for gases, dense fluids. a : intermolecular
attr., b : occupied V •(P + an2/V2) (V −nb) = nRT
2-State Paramagnet ∼ Spin 1/2
(symmetric!) N • Ω
N
~ || z
∼ Coins. • B
b • Ω(N, N↑ ) = N
total = 2
↑
N N
N (N↑ N↓ ok)
1+N↓/N↑ ↑ 1+N↑/N↓ ↓
eN ↑
√
N↓ N↑ 1
⇒ Ω≈ N
2πN
↑
h
i
k
S(U,B,N)=−N kB 1−z ln 1−z +1+z ln 1+z • 1 = B ln 1−z
2
2
2
2
T
2µB
1+z
Ω≈
U =µB(N↓−N↑)
U(T,B)=−N µB tanh(µB/kBT)
•z=U/(N µB) •
M =µ(N↑−N↓)
µB
µB 2
/cosh2
CB =N kB
kB T
kB T
U(T,B)=−BM(T,B)
Einstein Solid
N indep. oscil.’s, q quanta total: N bins & q stars.
)(
= bin dividers,
QQQQQ → (1, 1, 1, 2)
QQQQQ → (1, 3, 1, 0) q +(N −1) symbols,
choose q to be quanta.
.........
1
V = 21q
k s x2
−1 • E = ~ω(n+ 2 ) •
Ω(N, q) = q +N
ks
N
q
U = ~ω(q + )
ω
=
2
m
q
N high-T
q N
ln Ω ≈ ln 1+ N
Ω ≈ eq
q
N
1
q +ln 1+ N +...
N q
i
(1+N/q)q (1+q/N)N
eN
√
Ω≈
q, N 1
N q 1
Ω≈ q
2πq(q+N )/N
low-T
µ ≈ −kBT log(1+ q/N).
High-T : U ≈ N kB T • CV ≈ N kB
2
• CV ≈ N kB k~ωT e−~ω/kBT
Low-T : U ≈ ~ωN e−~ω/kBT
Barometric Eqn.
Air slab: dz thick, m avg. molecule mass:
dP
dz
=
N mg
−V
ideal gas
=⇒
dP
dz
=
−mg
kB T P
B
mgz
⇒ N(z) = No e−kB T
Heat Eqn.
∂t T = K∇2 T
T (~
r ; t) =
mk
r ; 0) = g(~
r ).
K = ρm T
C & T (~
R
0
0
− 2/4Kt g ~
e
r
d~
r
n/2 Rn
1
(4πKt)
Partition Functions
Bltzmn/Can’cl Gibbs/Grnd can’cl
(r,N)→rth state of
•
• sys. of N particles.
P(s)= 1 e−E(s)β
P(r,N)= 1 e−(E(r,N)−µN)β
Z
Z
P
Z(T) = s e−βE(s) • F(T) = −kBT ln(Z(T)) • hEi = −∂β[ln Z]
P
P
−βE(s)
2
2
1
•
σE
= ∂β
[ln Z]
hQi = s PsQ(s)= Z
s Q(s)e
P βµN
P
−(E(r,N)−µN)β
=
Z(T,V,N)
Z(T,µ) =
N e
(r,N) e
3/2
2 mβ
−βmv2/2
Maxwell speed: D(v)dv
dv
q = (4πv ) 2π e
p
√
2kBT
Most prob’bl: vp =
= 2vp/ π •vrms = vp 3/2
m •hvi
3/2
2
Maxwell velocity: D(3)(v)d3~
v = mβ
e−βmv /2d3~
v
2π
P(i,n)=
Ideal quantum gas Zi(T,µ)=Z(i,T,µ)
•
• P (n)= e−β(i−µ)n
Z(T,µ)=Π∞ Zi(T,V,µ) Z( ,T,µ)=P e−β(i−µ)n
i
i=1
Z(i,T,µ)
i
n
1
Fermi-Dirac: n=0,1→Z(,T,µ)=1+e−β(−µ) →hni(,T,µ)=
e+β(−µ)+1
1
1
Bose-Ein: n=0,1,...→Z(,T,µ)=
→hni(,T,µ)=
1−e−β(−µ)
e+β(−µ)−1
R
P
P
2k2
nx
L 3 3~
E(~
k)= ~2m
⇒ kx= 2πL
•
... = ... → 2π
d k...
nx,ny,nz
~
One part’cl in a box
k
R∞
P
P
P
~
D.O.S’s:
f((k)) → 0 g()f()d
~
i f(i) =
spin
k
√
√
2 3/2
BE: g() = αV /2•FD: g() = αV •α = 4π(2m/h )
R
R
N = 0∞ hni()g()d•U = 0∞ hni()g()d
Fermi gas@T=0:•F = µ(T=0)•kF3 = 3π2n
.
3/2
U
= 3F/5•P
= 2U/3 R
n = 2αF /3• N
∞
R
PV P
2
L 3 3~
d k → πV
Black body • µ = 0 •
→ 2 2π
2c3 ω dω
polariz
0 3
~
∞
∞
k
R
R
1
~ω
U
V = uω(ω,T)dω = u(,T)d• = ~ω•uω(ω,T) = π 2c3 β~ω
e
−1
0
4σB
4
1
3
•
~ω
≈
2.82k
T
•
U
=
VT
max
B
c
π 2 (~c)3 eβ−1
16σB
3
S = 3c VT
• P V = U/3 • Rad.E-flux: eσB T4
ω
phon
RD
P BZ
P
k3 T3
6π2Nv3
3
Debye •
→ 9N
dωω 2 ... • ωD
= B~3D = V s
ω3
polariz~
D 0
k
ω
TR/T
RD~ω3dω 9N(kBT)4TD
R/T
9NkBT3 D
x3dx
x3exdx
U= 9N
= (k T )3
ex−1 • CV =
(ex−1)2
ω3
T3
eβ~ω−1
BD
D0
D
0
0
h
i3R
h i3
h i3
∞√
T<Tc
N
xdx Nnorm
2 2πmkBTc 2
√
BEC• N
= TTc 2• cond
=1− TTc 2
V = π
ex−1 •
N
N
h2
0
∞
∞
h i3
h i3
R 3/2 R
5γNk
x1/2dx
U = γNkBT TTc 2•CV = 2 B TTc 2•γ = xex−dx
1
ex−1 ≈0.77
0
0
0
u(,T) =
Non-Rel. Quantum Mech.
P
hf (j)i= j f (j)Prob(j)
R
hf (x)i= f (x)ρprob (x)dx
iλA
√
= σ
σj
−iλA
j σj ← std.dev. •↑ var
(iλ)2
2
Qe
=Q+iλ[A,Q]+
[A,Q] +...
2!
• e
Schrödinger
Eqn.
2
2
= −~
r , t) Ψ(~
r , t)
(= H(~
r, p
~, t)Ψ).
2m ∇ +V(~
Time-Indep. Schrödinger Eqn.
[AB,C]=A[B,C]+[A,C]B
i~ ∂Ψ
∂t
•
2≡h(j−hji)2 i=hj 2 i−hji2 "
σj
If ∂tV
=
0,
using Ψ(~
r, t) = ψ(~
r )φ(t) (‘sep of vars’)
leads
to
H(~
r, p
~)ψ = Eψ.
General
sol’n
is Ψ =
P
r, t) with Ψn
=
ψn φn
=
ψn (~
r )exp iEnt
&
n cn Ψn(~
−~
R ∗
(~
r )Ψ(~
r , t = 0)d~
r En∈R, cn∈C.
cn = hψn|Ψt=0i = ψn
1. Ψn ← stationary states; only phase evolves in t.
2. If V is symmetric ⇒ each ψn (can be) even or odd.
3. Each En>Vmin . (Else Ψ is not normalizeable.)
4. E > V(−∞) or E > V(∞) ⇒ scatter state(s).
5. ψn ← an orthonormal set. (Completeness assumed.)
R +
6. ψ continuous. ∆(∂xψ) = 2m
lim→0
V(x)ψ(x)dx.
−
~2
7. ψn+1 has one more zero-crossing than ψn .
im
[H(t),~
r]
8. i~∂tQ(t) = [Q(t),H(t)] ⇒
p
~=
~
Interpretation, Requirements, Misc.
∗
~~
~p
p
~ → ~i ∇
r→i~∇
r• ~
~ • [x,px]=i~ • [f(x),px]=i~∂xf • hg|fi=hf|gi
Eigen-funcs: x
b:
x|ψy(x)i = y|ψy(x)i ⇒ |ψy(x)i = δ(x−y).
√
p
bx :
i~∂x|ψpx(x)i = px|ψpx(x)i ⇒ |ψpx(x)i = eipx x/~/ 2π~.
So: Φ(px, t) ≡ hψpx(x)|Ψ(x, t)i and Ψ(y, t) ≡ hψ−px(y)|Φ(px, t)i
P
hQi = hΨ| [Q|Ψi] =
[hΨ|Q†] |Ψi = n qn|cn|2 R ∗ ~ 3
R
~~
~p
hQ(~
r, p
~)i = Ψ Q ~
r, i ∇
r = Φ∗Q i~∇
~ Φd3 p
~
r Ψd ~
~, p
P
Spectral decomp.: Q = nqn Pn if Q|ni = qn |ni
R∞
ρprob (x, t) = |Ψ(x, t)|2
⇐⇒
|Ψ(x, t)|2 dx = 1
−∞
Rb
2
(but a |Ψ(x, t)| dx is time-dep. in general.)
prob.
prob.
cn = hΨn |Ψi =⇒ |cn |2 • c(z) = hΨz |Ψi =⇒ |c(z)|2 dz
P
P
prob.
2
2
|cn |2En
En←→|cn |
•
nR
n |cn(t)| = 1P • hHi =
P
=1
•
Pz dz=1
Proj. op.: Pn ≡ |nihn|/|hn|ni|2 •
n
n
Schwarz ineq.: ha|aihb|bi ≥ |ha|bi|2 ≥ ((ha|bi−ha|bi∗)/(2i))2
2 2
1
~
Uncertainty: σA
σB ≥(2i
h[A,B]i)2 • σx σpx ≥ ~
2 • σt σE ≥ 2
Correspondence principle → “classical” limit agrees.
Schrö. Pict.: hQ(t)i = hΨ(x, t)|Q|Ψ(x, t)i iHt
iHt
Heisen. Pict.:
hQ(t)i = hΨ(x, 0)|e ~ Qe −~ |Ψ(x, 0)i
(no-degen or ψ is a
Feyn-Hllmn: ∂λEn = hψn|∂λH|ψni proper lin. comb.)
Ehrenfest’s Theorem / Virial Theorem
1
∂t hQi= i~
h[Q,H]i+h∂t Qi • h...i’s obey classical laws.
1
~
~ • ∂thLi=h
~ ~
~
h~
pi=h~
v i=∂th~
r i • ∂th~
pi=h−∇Vi=h
Fi
r×(−∇V)i
m
If V (λ~
r1 , λ~
r2 , ...) = λν V (~
r1 , ~
r2 , ...) for N particles
then hn|T |ni = ν2hn|V |ni (& En = hn|T |ni+hn|V |ni).
~
∂th~
r ·~
pi = 2hT i−h~
r · ∇Vi
• ∂thψn|Q|ψni = 0 if ∂t Q = 0
1D Scatter (L−∞ → R+∞ ) • R+T = 1
group
v
ψL = AeikL x +Be−ikL x
|F |2
|B|2
R
ikR x
−ik
x • R ≡ |A|2 • T ≡ v group |A|2
R
ψR = F e
+
Ge
L
1D Infinite Square Well
n
o
√
d2 ψ
0, 0 ≤ x ≤ L
V (x) = ∞,
=⇒
= −k2 ψ k ≡ 2mE
.
~
else
dx2
q
2
~2 kn
~2 n2 π 2
2
nπx
En = 2m = 2mL2 • ψn = L sin L
• n ∈ N>0
hn|x|ni =
2
L
2
hn|p |ni =
• hn|x2 |ni =
π 2 ~2 n2
L2
• σx,n =
L2
3
q
2
− 2πL2 n2
π 2 n2 −6 L
12
πn
•
σp,n = ~ πn
L
1D Harmonic Oscillator / 3D
[N,a ] = a • [a,N] = a • N |ni = n|ni • a|ni = n|n−1i
q
q
√
~
x = 2mω
a†+a • p = i mω~
a†−a • a†|ni = n+1|n+1i
2
[H,a†]=~ωa†
1D Free Particle
2
2
2π
V (x) = 0 • ∂x
ψ = −k2 ψ • p = ~k • λ = |k|
• ωk = ~k
2m
n
o
2
√
~k
k > 0 moves →
k ≡ ± 2mE
• Ψk ∝ ei kx− 2m t
~
k < 0 moves ←
(stationary states)
2
∞
R
~k
packet.”
Ψ(x, t) = √12π φ(k)ei kx− 2m t dk ← “Wave
General sol’n,
−∞
R
with ∼weights: φ(k) = √12π −∞
Ψ(x, t = 0)e−ikx dx.
∞
q
q
(&classical)dω
~|k|
~|k|
ω
E
vphase = |k| = 2m = 2m •
vgroup = dk = m = 2E
m
1D Delta Well/Barrierbound scatter
2
ψ ≡ κ2 ψ ≡ −k2 ψ.
V(x) = −αδ(x)
•
∂x
ψ = − 2mE
~2
√
2
mα|x|
mα
• E = − mα
Bound → κ • ψ(x) = ~ exp −~2
2
i
h 2~
h 2 i−1
−1
mα2
E
• T = 1+ 2~
Scatter → k • R = 1+ 2~
2E
mα2
V(x) = αδ(x)
•
No bound • Same scatter as δ-well.
1D Double Delta√Well
√
V(x) = −α(δ(x+a)+δ(x−a)) • κ ≡ −2mE
• k ≡ 2mE
~
~
(
2
o
Even: e−2κa = ~mακ −1
Sol’n: always
Bound:
−2κa
~2 κ Sol’n: sometimes
Odd: e
= 1− mα
Scatter: T =
1
(with z=αm/~2k)
1+2z 2 [(1+z 2)+(1−z 2)cos(4ak)−2zsin(4ak)]
1D Finite Square Well / Barrier
√
n
√
2m(E+V0)
−V
• κ ≡ −2mE
V(x) = 0,0 , |x| ≤ a • ` ≡
~
~
|x| > a
p
&
Even: tan z = q
(z/z0)2−1 z ≡ `a,
√
Bound:
a
Odd: tan z = −1/ (z/z0)2−1 z0 ≡ ~ 2mV0
π 2 ~2
Sol’ns: •Even: always, finite# •Odd: iff V0 ≥ 8mα
2.
p
V02
−1
2 2a
Scatter:
T = 1+ 4E(E+V ) sin ~ 2m(E+V0)
0
The 1D Finite Square Well, but −V(x). No bound.
p
V02
−1
2 2a
2m(V0−E)
Scatter:
TE<
V0= 1+ 4E(V0−E) sinh
~
p
V02
−1
−1
2 2a
2mE 2
TE=
2m(E−V0)
V =1+ ~2 a • TE>V =1+ 4E(E−V ) sin
~
0
~
2
•
• [H,x]=im p • [H,p]=i~mω x
2
p
1 mωx
1
mω 4
e −2~ Hn mω
En = ~ω(n+ 12)
•
ψn = √ n
~ x
2 n! π~
2
(hermite poly’s) 2
n −x
ψn = √1 (a†)nψ0 • n ∈ N≥0 • Hn(x) = (−1)nex ∂x
e
n!
p
√
√
hn|x|qi = ~/2mω( nδq,n−1 + qδn,q−1 )
•
hn|V |ni = 1 En
2
0
0
1D Sudden Step Up / Drop Down
n
o
≤0
V(x) = V0,, x
•
Reminder: ∆vgroup |x=0 6= 0.
0 x>0
q
2
√
E−V0 4E √
TE<V0= 0
•
TE>V0=
E − E −V0
2
E
V0
p
p
= 4( 1+V0/E)/(1+ 1+V0/E)2
Like Step Up, but −V(x) • T
3D Spherical Symmetry & ∂t V = 0
ψ ≡ R(r)Y(θ, φ) • u(r) ≡ rR(r) • Yl −m
= (−1)m(Ylm)∗
R∞ 2 2
R 2πR π m∗ m0
|R| r dr = 1 •
(Y ) Yl0 sinθdθdφ = δl,l0 δm,m0
0
0 0 l
q
n
(2l+1) (l−|m|)! imφ m
<0,
m
Yl (θ, φ) = 4π (l+|m|)! e
Pl (cosθ) = 1(−1if)mmelse.
|m|
m|
l l!2l • ∂ hLi
~ =0
Plm(x) =(1−x2)2 ∂|x
Pl(x) • Pl(x) = ∂xl(x2−1)/
t
i
h
[H,L2]=[H,Lz]=0
−~2 2
~2 l(l+1)
•
2m ∂r u+ V + 2m r 2 u = Eu
m∈−l,...0,1,...l
• hn|p|ni = 0
1
1
V(x) = 12kx2 = 12mω 2 x2 • H = ~ω(a†a+
2) = ~ω(N+ 2)
pmω
ip
ip
† pmω
†
a = 2~ x+ mω
• a = 2~ x− mω
• [a, a ] = 1
√
†
†
[H,a]=−~ωa
p
√
√
hn|p|qi = i mω~/2( nδq,n−1 − qδn,q−1 )
•
hn|T |ni = 1 En
2 p
p
hn|x2 |qi = (~/2mω) (2q+1)δn,q + n(q+1)δn,q+2 + q(n+1)δn,q−2
p
p
hn|p2 |qi = (mω~/2) (2q+1)δn,q − n(q+1)δn,q+2 − q(n+1)δn,q−2
3
ψ = ψa(x)ψb(y)ψc(z) • En=~ω(n+ 2) • dn=(n+1)(n+2)/2
N
3D Rigid ing Circle with N Masses
~ = aM v
about ~
r◦ at r=a • H = T+
V = 12M v 2 • |L|
L2
2a2M
~2l(l+1)
2a2M
⇒ El =
• ψlm = Ylm • m∈−l,..0,1,..l
l∈0,1,2,...
3D Inf. Spherical Well
n
i
o
h
√
0, r ≤ a =⇒ ∂ 2 u = l(l+1) −k2 u k ≡ 2mE .
V (x) = ∞,
r
~
else
r2
1
u(r) = Arjl(kr)+Brnl(kr)
jl(x) = (−x)l x
∂x l sinx
x
•
x
x
1
j0 = sin
• n0 = − cos
nl(x) = −(−x)l x
∂x l cosx
x
x
x
H=
2l l!xl
1
jl(x) ≈ x
cos(x− π
(2l+1)!
2 (l+1)) x1
•
1
π
−(2l)! x1
n
(x)
≈
l
nl(x) ≈ l l+1
x sin(x− 2 (l+1))
2 l!x
∂x jl(x) = xl jl(x)−jl+1(x)
•
∂x nl(x) = xl nl(x)−nl+1(x)
2
~2βnl
jl(βnl)=0 (n roots/l)
Enl = 2ma2 E →(2l+1)-degen.
• ψnlm = jl(βnl r/a)Ylm
nl
jl(x) ≈
Hydrogen Atom – Bound States
√
−2mE
−e2 1
−m e2 2 n∈N>0 ,
• En = 2~
2n2 4π◦
4π◦ r • κ ≡
~
n2 -deg.
2
m∈−l,..0,1,..l
4π◦~
~
1
a◦ = αm
≈
0.529Å
•
=
=
2
c
nκ
l∈0,1,..,(
n−1)
e
n
mee
r
h i3
il
−rh
(
2l
+
1
)
(n−l−1)!
m
2r
2
2r
ψnlm= na◦ 2n[(n+l)!]3 ena◦ na◦ L(n+l)−(2l+1) na◦ Yl (θ, φ)
V(r) =
p
q −x q
e−r/na◦
Lqp−p(x) = (−1)p∂x
Lq(x) • Lq(x) = ex∂x
e x • ψn00= √
π(na◦)3
2
hVi=2E
|nlmi ⇒ hTi=−En • hr1i = a 1n2 • hr12 i= a2n3(1l+1/2) • hp2i= a◦~n
n
◦
◦
s+1hrsi−(2s+1)a hrs−1i+ s[(2l+1)2−s2 ]a2hrs−2i=0
◦
◦
2
4
• n
q
q
(1+(−1)q)(q+2)!a◦
hz 2i=a2
(q+2)!a◦
q
Most
q
◦
hz i=
hr i= q+1
hzi=0
(q+1)2q+2
2
|100i ⇒
•
• prob
r=a◦ •
a
hri= ◦ [3n2−l(l+1)]
2
Angular Momentum, Intrinsic Spin
~ r ×~
L=~
p
~ ~~
S=
σ
2
~,r2]=0
[L
~,p2]=0
[L
~ L
~
L2=L·
•S
2=S·
~ S
~
•
[L ,L ]=i~L
• [Sx,Sy]=i~S z
x y
z
~,S]=0
~
[L
~
[L2,S]=0
•
~
[L2,L]=0
~
[S2,S]=0
•
[L ,L ]=i~L
• [Sy,Sz]=i~S x
y z
x
L± ≡Lx±iLy
S± ≡Sx±iSy
•
[L ,L ]=i~L
• [Sz,Sx]=i~S y
z x
y
[Lz,L±]=±~L±
[Sz,S±]=±~S±
~·S,
~ L]=i~
~
~×S
~ [L
~·S,
~ J]=0
~
[Lz,x]=i~y
[Lz,y]=−i~x
[Lz,z]=0 [L
L
•
•
•~~ ~
•~~ 2
~ L
~ [L
[Lz,px]=i~py [Lz,py]=−i~px [Lz,pz]=0 [L
·S,S]=i~S×
·S,L ]=0
√
m±1
L± Ylm=~ (l∓m)(l±m+1)Y
~·S,S
~ 2 ]=0
L2 Y m=~2 l(l+1)Ylm
[L
l
√
• 2l
•
~·S,J
~ 2 ]=0
S |smi=~2s(s+1)|smi
[L
S±|smi=~ (s∓m)(s±m+1)|sm±1i
Lz=−i~∂φ
Lz Ylm=~mYlm
s=0,1/2,1,...
• (Ym)∗ xyY
• ms =−s,..,0,1,..,s
m =0
Sz|smi=~m|smi
l
l
h i P (123)
1
1
~
σ = 01 10 , 0i −0i , 10 −01 → ±1
, ±i
, 10 / 01 • σjσk= δjk+i njknσn
P j1j2j
|j1m1i|j2m2i= Cm
|jmi
1m2m
j
Jz →m≡m1 +m2
If s1= 1/2 then:
(with s = s2 ±1/2)
•
P j1j2j
|jmi= Cm
m m |j1m1i|j2m2i
m1,m2 1 2
2
J →j∈|j1−j2|,..(++1)..,(j1+j2)
|smi=A|1 1i|s2 (m−1)i+B|1 −1i|s2 (m+1)i
22
2
2
r
r2 2
s2 ±m+1/2
s2 ∓m+1/2
A=
•B=±
2s2 +1
2s2 +1
•
Corrections to hydrogen
En∼α2
4
5
4
struc.∼α
∼α
Hyperfine∼α me/mp
• Fine
• Lamb∆
~
~ •µ
~ ⇒µ
~p
γ(v)6=1 & S↔
quant’zd E
~p →B
~e ↔B
0 Fine struc:
E0
4
2 n
0 +H 0 = −p + ~α S·
~L
~
1+ α
Enj = 2
−3
H 0=Hrel
n j+1/2 4
s−o 8m3c2
n
2m2 r 3
•
~ L=(
~ J 2−L2−S 2)/2•p2 ψ=2m(E−V)ψ Good #’s: n,l,s,j,mj
S·
e ~ ~ ~
~
Zeeman: H 0=−(µ
~ l+µ
~ s)·B
ext=2m(L+2S)·Bext •conflicts w/ fs
If: Bext Bint ⇒fine struc domn8’s⇒E 1 =hnljmj|H 0|nljmji
2
2
2
1
~
~~ ~ 2 ~ ~
use: S
t-avgd =J(S·J)/J •S·J=(J −L +S )/2•Egs ∼±µB Bext .
If: Bext Bint ⇒zeeman domn8’s⇒non-pert with|nlml msi
1 =µ B
align ||z⇒E
b
B ext (ml+2ms).(note: can also pert the fs,
~L
~i=hSxihLxi+hSyihLyi+hSzihLzi=~2 m ms if including H 0 .)
use: hS·
l
fs
If: Bext ∼Bint ⇒use full-blown degen pert (→ W eigenprob)
~
~
~ b then H 0=ezE
Linear Stark: H 0=E
pe . If E
ext ·~
ext =|E|z
ext=
eEext r cos θ. Use degen pert. Most integrals→0.
0
1
3
~ p ⇒E =Ah[3(S
~p·b
~e·b
~p·S
~e ]/r i+
Hyperfine: H =−~
µe·B
r)(S
r)−S
~p·S
~ei • C=µ◦ gp e2/3mp me =8πA/3 • S
~p·S
~e=(S 2−S 2−S 2)/2
C|ψ(0)|2hS
e
p
2 4
If: l = 0 ⇒ A term→0 ⇒ E 1 =(S2−3~2/2)2gp ~2/3mp m2
e c a◦ .
Breaks spin degen of the gs: S2 = 0 for singlet, 2~2 triplet.
−
Stationary e
~
µ
~ =γ S
~
~
τ =~
µ ×B
•
~
H=T−µ
~ ·B
~
if B=B
b
◦z
•
in Magnetic Field
E± =∓γB◦~/2
hSzi= ~ cos α
2
•
hSxi∝cos ω
hSyi∝sin ω
•
ω=γB◦
Larmor
~
~ ∇(~
~ µ ·B)6
~ =0
e− Stern-Gerlach: B(x,y,z)→
F=
b
~ −αxb
B=
i+(B◦+αz)k
~ t∈[0,T]
µ
~ ·B
H(t)= 0
else
•
E±=∓γ(B◦+αz)~/2 • pz=αγ~/2 • q≡iγB◦/2

i
aχ+ +bχ−
t<0
χ(t)=
ae(qT+ipzz)(χ+) + be−(qT+ipzz)(χ−)t>T
T.I. Perturbation Theory: H ≡ H +λH
0
0 0
0 0
P hψ0
H 0|ψ0
m|H |ψni|ψ0 i
ni=En|ψni |ψ1
ni=
m
0−E 0
1 =hψ0 |H 0|ψ0 i
En
En
m
=
6
n
m
n
n •
0
0
0 0 2
2 = P |hψm|H |ψni|
En
0−E 0
En
m
=
6
n
m
•
We need to determine the α, β that will
0 0
If: H 0|ψ0
ai=E |ψai split the degen. ψ’s into distinct e-val’s
0
0
0
0
0
and H |ψ i=E |ψ i cleanly when H is applied. There are two
b
b
and maybe more... ways. [1] find A =A† s.t. [A,H 0] = [A,H 0] = 0
•
so H 0|ψ0i=E 0|ψ0i which are NOT degen. for the ψ’s in E0 s
©
©
0
sub-space (ψ0
w/ |ψ0i=α|ψ0
a,ψb, ...) and use non-degen.
ai+
©
0
[2] solve W eigenprob for α, β and E 1 (s).
+β|ψ i+...
b
You hopefully get N E’s and req’d α, β.
h i
h i W =hψ0|H 0|ψ0i
Waa Wab α
ij
i
j
Solve N×N:
1α
=
E
⇒ det(
β
0
Wba Wbb β
W−λ)=0
|ψ0i=α|ψ0
ai+β|ψbi+...
©
Variational Principle
Egs ≤ hψ|H|ψi for any normalized |ψi. Simply minimize!
2
V∝x: ∂x
ψ = α3xψ → ψ = CAi(x)+DBi(x) (Airy’s)
√
3
/
2
2
− x √
Ai(x0)∼sin 2(−x)3/2+π /( π(−x)1/4)
Ai(x0)∼e 3 /(2 πx1/4)
4
3
•
√
2
2
π
3
/
2
+2x3/√
Bi(x0)∼cos (−x) + /( π(−x)1/4)
3
4
Bi(x0)∼e 3 /( πx1/4)
0
T.D. Perturbation Theory: H ≡ H +λH 0(t>0)
P
0=E 0 ψ 0
H 0 ψn
Ψ(t)= cn(t)ψn e−iEn t/~ • ∆mn ≡Em−En
n n
0 ≡hψ 0 |H 0 |ψ 0i • ∂ c n
1P
−i∆mnt/~ (exact)
0
Hm
n
m
n
t m = i~ cn Hmn e
n
Rt 0
Rt 0 0
0 −i∆mnt0/~dt0
0
1
cm6=n(t)≈ 1 Hm
(F.O.) cn(t)≈1+
Hnn(t )dt
n(t )e
i~
i~
•
0
0
0
∂t H 0 =0: sin2(∆ t/2~)
H =V cosωt: sin2((∆ ±ω~)t/2~)
nm
nm
Pn→m =
Pn→m =
2 /(4|H 0 |2)
∆n
m
mn
• iff ∆nm =±~ω (∆nm±ω~)2/(|Vmn|2)
3 |P|2
ω0
Spon. emiA=
ssion rate:
3π◦ ~c3
Life- τ =
1
r |ψai
A1+A2+...
• P=qhψb|~
• time:
WKB(J) – (Classical endpoints a, b; L → R)
R
√
b p(x0)dx0 / p(x)
ψ(x)≈C exp ±i a
Assume ψ=A(x)eiφ(x)
~
√
•
2
2
2
2
& ∂x [A]/A((∂xφ) & p /~ )
p(x)= 2m(E−V◦) “classical” p.
Rb√
2
2
−2γ
• γ~≡ a |p(x)|2 dx
Rb
Rb
no |walls: a
p(x)dx=(n−1)π~
2 |walls: a
p(x)dx=nπ~
2
• w/n=1,2,.. • Best when:
Rb
1
n1
1 |wall @a: a p(x)dx=(n− )π~
4
(semi-classical regime)
1
1
hT i= ∂n[En](n− ) when there are no |walls.
2
2
Tunneling a → b: T =|F| /|A| ≈e
Scattering: Partial wave, Born
ikr
D(θ)= dσ =|f(θ)|2 ψ(r,θ)≈A[eikz+f(θ)e
] u(r)=Arj (kr)+Brn (kr)
r
l
l
R dΩ
• Use u near
• (1) ix (1)
origin,
σ= D(θ)dΩ
h0 = e •h ≡jl(x)+inl(x) match with expansion.
ix
l
P
P∞
l
2
eikz = ∞
i=0 i (2l+1)jl(kr)Pl(cosθ) • σ=4π i=0(2l+1)|al|
P∞
(1)
l
ψ(r,θ)=A
i (2l+1)[jl(kr)+ikal h (kr)]Pl(cosθ) ←impose b.c.’s.
i=0
l
1. V(~
r0) is localized about ~
r0 2. incoming wave ∼ not altered.
R ik|~
r−~
r0|
ψ(~
r )=ψ0(~
r )− m2 e
V(~
r0)ψ(~
r0)d3~
r0 s.t. (∇2+k2)ψ0 =0
|~
r−~
r0|
2π~
R
i(~
k0−~
k)·~
r 0 V(~
0)d3~
0 s.t. ~
Born→f(θ,φ)≈ −m
e
r
r
k=kr
b,~
k0 =kz
b
2π~2
Spherical sym:
κ=2k sin(θ/2)
Low-E: −m R
R
V(~
r 0)d3~
r0
•
f(θ)≈ −2m
rV(r)sin(κr)dr
f(θ,φ)≈
2π~2
κ~2
1. V(~
r0) is localized about ~
r0 2. incoming wave ∼ not altered.
Electrodynamics
General
~ ·E=ρ/
~
Gauss’ ∇
◦
~ ×E=−∂
~
~
Faraday’s ∇
tB
•
~ ·B=0
~
No monopoles ∇
~
~
~
~
Ampere’s ∇×B=µ◦(J+◦∂tE)
In matter
~ ·D=ρ
~
∇
f
~ ×E=−∂
~
~
∇
tB
~ ·B=0
~
∇
~
~
~
~
∇×H=Jf +∂t D
Aux. & Pot.s
~
~ P
~
D=
◦ E+
1 B−
~
~ M
~
H=
µ◦
~
~ V−∂ A
~
E=−
∇
t
~ ∇×
~ A
~
B=
•
1 B.
~ =◦ χe E
~ so D=
~
~ & M
~ =χm H
~ so H=
~
~
Linear media: P
E
µ
R
1 (E×
~
~ B)
~
Enrg: U = 1 (◦E 2+ 1 B 2)dV Poynting: S=
C≡Q/V
2
µ◦
µ◦
R
•
•
µ
~ =◦ (E×
~ B)
~ dV
W=CV 2/2
Larmor: P = ◦ q 2 a2
Momntm: P
6πc
~ =q(E+~
~ v ×B)
~
~ nσ
F
W=Q[V(~
r )−V(O)]
F=
b 2/2◦
• P =E 2 ◦/2 •
•
R
−σ =n·
~V
W= 1 V dq
b∇
2
surf. press.
◦
@surf
0
Spherical image: |rq|=a→q =−qR/a & |r 0|=R2/a•V(~
r )=k((q/)+(q0/ 0))
q
2 X=C→B.c.’s!
−∇2V=ρ/◦ solve via sep.vars: V≡X(x)Y(y)Z(z)→ 1 ∂x
X
(Laplace) N
H
r
R~
Pqi b
~ a= Qenc
E·d~
~ ~
~ ~
dq={λdl,
E(
r )= 1
V(
~
r )=− E·d
l
2
4π◦
~
~V ◦
σda,ρdV}, E=−
i i
∇
O
•
•
R 1
R1
0
1
1
(
∞
~
~=~
r −~
r ,
b
bdq
E(~
r )=
V(~
r )=
dq
~ σn
E=
2
4π◦
4π◦
2◦ plane)
q at ~
r0 .
V
V
~~
Geometry – 4π◦V(~
r )•4π◦E(
r)
1D wire. ||b
x. Len: L
⊥ above cent.
2D ring. x−b
b y.
⊥ above cent.
2D disk. x−b
b y.
•
⊥ above cent.
3D sph. shell.
At origin.
•
q
λ ln L + 1+(L/z)2
z
•
q(λL)
z
b
z z2+L2/4
(λ2πR)
z(λ2πR)
•
z
b
(R2+z 2)1/2
(R2+z 2)3/2
q
2
(σπR )
−|z|
q
2
2
z +R −|z| • 2
1+ q
z
b
R2
R2
z 2+R2
•
q
q
(out),
(in)
r
R
•
•
(σ4πR2)
r>R,
r
b
0else
r2
Magnetic Dipoles (~
µ – proton, electron)
gp e ~
2µ
~e • B
~µ = µ0 [3(~
Sp • µ
~e = −eS
µ ·b
r)b
r −~
µ]+ 0 µ
~ δ 3(~
r)
µ
~p =
2mp
me
4πr3
3
~
Electric Dipoles (±|q| sep. by d~ ⇒ p
~e = −q d)
R 0 0 3 0 V
~
~·b
r E
1
4π 3
p
~e= ~
rρ(~
r )d ~
r • k = q− q ≈ p
•
=
[3(~
p
·b
r
)b
r
−~
p
]−
~δ (~
r)
k
3p
r2
r3
+ −