Useful Equations · Useful Equations 2 zx H x 1 N= J B E=hν ν= c λ ν!= 1 λ hv=Φ+ 1 2 m e v...
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Useful Equations E = h ν ν = c λ ν ! = 1 λ hv = Φ + 1 2 m e v 2 λ = h p ΔxΔp ≥ ! 2 σ x σ p ≥ ! 2 H ! ψ = E ψ ψ n ( x ) = 2 a sin n π x a ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ p x ! = −i" ∂ ∂ x x ! = x E n = n 2 π 2 ! 2 2 ma 2 ψ ∗ ψ dτ τ ∫ = 1 a = ψ ∗ A ! ψ dτ τ ∫ σ a = a 2 − a 2 H ! = p ! 2 2 m = − ! 2 2 m ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 + ∂ 2 ∂ z 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = − ! 2 2 m ∇ 2 E n x ,n y ,n z = π 2 ! 2 2 m n x 2 a 2 + n y 2 b 2 + n z 2 c 2 ⎧ ⎨ ⎩ ⎪ ⎫ ⎬ ⎭ ⎪ H ! = − " 2 2 μ ∂ 2 ∂ x 2 + 1 2 kx 2 μ = m 1 m 2 m 1 + m 2 E n = n + 1 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ !ω ω = k μ = 2 πν ψ 0 = α π ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 1/4 e −α x 2 /2 ψ 1 = 4α 3 π ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 1/4 xe −α x 2 /2 ψ 2 = α 4π ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 1/4 (2α x 2 − 1)e −α x 2 /2 T = 1 2 nV ψ 3 = α 3 9π ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 1/4 (2α x 3 − 3x )e −α x 2 /2 α = kμ ! H ! = L " 2 2I Y 0 0 = 1 4π Y 1 0 = 3 4π ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 1/2 cosθ I = μr 2 Y 1 ±1 = 3 8π ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 1/2 sinθ e ± i φ Y 2 0 = 5 16π ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 1/2 (3cos 2 θ − 1) Y 2 ±1 = 15 8π ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 1/2 sinθ cosθ e ± i φ Y 2 ±2 = 15 32π ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 1/2 sin 2 θ e ±2 i φ ! B = h 8π 2 Ic E J = ! 2 2I JJ + 1 ( ) ⎡ ⎣ ⎤ ⎦ = hc " B JJ + 1 ( ) ⎡ ⎣ ⎤ ⎦ L ! z = −i" ∂ ∂ φ L ! zY l m = m!Y l m L ! 2 Y l m = l(l + 1) ! 2 Y l m P a→b = N a ψ b * H ! 1 ψ a dτ τ ∫ 2 δ ( E b − E a − h ν ) N j = g j e − E j / k B T Q N J N 0 = (2 J + 1)e − E J / k B T J max = 1 2 2 k B T hc ! B − 1 ⎧ ⎨ ⎩ ⎫ ⎬ ⎭
Transcript of Useful Equations · Useful Equations 2 zx H x 1 N= J B E=hν ν= c λ ν!= 1 λ hv=Φ+ 1 2 m e v...
Useful Equations
E = hν ν = cλ
ν! = 1λ
hv = Φ + 12mev
2 λ = hp
ΔxΔp ≥ !
2 σ xσ p ≥
!2 H
!ψ = Eψ
ψ n (x) =2asin nπ x
a⎛⎝⎜
⎞⎠⎟
px! = −i" ∂∂x x! = x
En =
n2π 2!2
2ma2ψ ∗ψ dτ
τ∫ = 1
a = ψ ∗A!ψ dτ
τ∫
σ a = a2 − a 2 H! = p!2
2m = −!2
2m∂2
∂x2+ ∂2
∂ y2+ ∂2
∂z2⎛
⎝⎜⎞
⎠⎟= −!2
2m ∇2
Enx ,ny ,nz
= π 2!2
2mnx2
a2+ny2
b2+nz2
c2⎧⎨⎩⎪
⎫⎬⎭⎪
H! = −"2
2µ∂2
∂x2+ 12kx2 µ = m1m2
m1 +m2
En = n + 12
⎛⎝⎜
⎞⎠⎟ !ω
ω = kµ=2πν ψ 0 =
απ
⎛⎝⎜
⎞⎠⎟1/4
e−αx2 /2
ψ 1 =4α 3
π⎛⎝⎜
⎞⎠⎟
1/4
xe−αx2 /2 ψ 2 =
α4π
⎛⎝⎜
⎞⎠⎟1/4
(2α x2 −1)e−αx2 /2 T = 1
2n V
ψ 3 =α 3
9π⎛⎝⎜
⎞⎠⎟
1/4
(2α x3 − 3x)e−αx2 /2
α =
kµ!
H! = L"2
2I
Y00 = 1
4πY10 = 3
4π⎛⎝⎜
⎞⎠⎟1/2
cosθ I = µr2
Y1±1 = 3
8π⎛⎝⎜
⎞⎠⎟1/2
sinθe± iφ Y20 = 5
16π⎛⎝⎜
⎞⎠⎟1/2
(3cos2θ −1)
Y2±1 = 15
8π⎛⎝⎜
⎞⎠⎟1/2
sinθ cosθe± iφ Y2±2 = 15
32π⎛⎝⎜
⎞⎠⎟1/2
sin2θe±2iφ !B = h
8π 2Ic
E J =!2
2I J J +1( )⎡⎣ ⎤⎦ = hc"B J J +1( )⎡⎣ ⎤⎦
L! z = −i"
∂∂φ
L! zYlm = m!Yl
m
L!2Ylm = l(l +1)!2Ylm
Pa→b = Na ψ b
*H!1ψ a dττ∫
2
δ (Eb − Ea − hν )
N j =gje
−Ej /kBT
QNJ
N0
= (2J +1)e−EJ /kBT
Jmax =
12
2kBThc !B
−1⎧⎨⎩
⎫⎬⎭
Integrals:
Other formulas:
H! = − !2
2me
∇2 − e2
4πε0rψ nlm = Rnl (r)Yl
m (θ ,φ) En = − R∞
n2
ψ 100 =1π
1ao
⎛⎝⎜
⎞⎠⎟
3/2
e−r/ao ψ 200 =132π
1ao
⎛⎝⎜
⎞⎠⎟
3/2
2 − rao
⎛⎝⎜
⎞⎠⎟e−r/2ao
ψ 210 =132π
1ao
⎛⎝⎜
⎞⎠⎟
5/2
re−r/2ao cosθ ψ 21±1 =164π
1ao
⎛⎝⎜
⎞⎠⎟
5/2
re−r/2ao sinθe± iφ
ψ 300 =1
81 3π1ao
⎛⎝⎜
⎞⎠⎟
3/2
27 −18 rao
+ 2 r2
ao2
⎛⎝⎜
⎞⎠⎟e−r/3ao
Eφ =φ*H!φ dτ
τ∫φ*φ dτ
τ∫
2S+1LJ , J = L + S,L + S −1,..., L − S
S!2α = S(S +1)!2α Sz!α = !
2α α *α dσ
σ∫ = β *β dσ
σ∫ = 1
S!2β = S(S +1)!2β S! zβ = −!
2β α *β dσ = β *α dσ
σ∫ = 0
σ∫
sin2(bx)dx = x2− 14bsin(2bx)∫ xsin2(bx)dx = x2
4− x4bsin(2bx)− 1
8b2cos(2bx)∫
x2 sin2(bx)dx = x3
6− x4b2
cos(2bx)− x2
4b− 18b3
⎧⎨⎩
⎫⎬⎭sin(2bx)∫
cos2(bx)dx = x2+ 14bsin(2bx)∫ xcos2(bx)dx = x2
4+ x4bsin(2bx)+ 1
8b2cos(2bx)∫
cos(bx)dx = 1bsin(bx)∫ sin(bx)dx = − 1
bcos(bx)∫
xne−ax dx0
∞
∫ = n!an+1
e−ax2
dx = π4a0
∞
∫
x2ne−ax2
dx = 1⋅3⋅5 ⋅⋅⋅(2n −1)2n+1an
πa0
∞
∫ x2n+1e−ax2
dx = n!2an+10
∞
∫
udv = uva
b
∫ − vdua
b
∫ cosn xsin xdx∫ = −1n +1
cosn+1 x
sin2θ = 2sinθ cosθ cos2θ = cos2θ − sin2θ sin(θ / 2) = ± 1− cosθ2
Constants: h = 6.626 x 10-34 J•s kB = 1.381 x 10-23 J/K me = 9.11 x 10-31 kg R = 8.315 J/(K•mol) NA = 6.022 x 1023 mol-1
e = 1.602 x 10-19 C R∞ = 2.180 x 10-18 J c = 3.00 x 108 m/s 1hartree = 4.360 x 10-18 J
cos(θ / 2) = ± 1+ cosθ2
sin(x ± y) = sin xcos y ± cos xsin y tanθ = sinθ / cosθ
cos(x ± y) = cos xcos y ∓ sin xsin y e± iθ = cosθ ± isinθ
! = 1.054 ×10−34 J i s
4πεo = 1.113×10−10C 2 / (J im)
1H
1.008
3Li
6.941
11Na
22.99
19K
39.10
37Rb
85.47
55Cs
132.91
87Fr
(223)
4Be
9.012
12Mg
24.31
20Ca
40.08
38Sr
87.62
56Ba
137.33
88Ra
(226)
21Sc
44.96
39Y
88.91
57La
138.91
89Ac
(227)
22Ti
47.87
40Zr
91.22
72Hf
178.49
104Rf
(267)
23V
50.94
41Nb
92.91
73Ta
180.95
105Db
(268)
24Cr
52.00
42Mo
95.94
74W
183.84
106Sg
(271)
25Mn
54.94
43Tc(98)
75Re
186.21
107Bh
(270)
26Fe
55.85
44Ru
101.07
76Os
190.23
108Hs
(277)
27Co
58.93
45Rh
102.91
77Ir
192.22
109Mt
(278)
28Ni
58.69
46Pd
106.42
78Pt
195.08
110Ds
(281)
29Cu
63.55
47Ag
107.87
79Au
196.97
111Rg
(281)
30Zn
65.39
48Cd
112.41
80Hg
200.59
112Cn
(285)
5B
10.81
13Al
26.98
31Ga
69.72
49In
114.82
81Tl
204.38
113Nh
(286)
6C
12.01
14Si
28.09
32Ge
72.61
50Sn
118.71
82Pb
207.19
114Fl
(289)
7N
14.01
15P
30.97
33As
74.92
51Sb
121.76
83Bi
208.98
115Mc
(289)
8O
16.00
16S
32.07
34Se
78.96
52Te
127.60
84Po
(209)
116Lv
(293)
9F
19.00
17Cl
35.45
35Br
79.90
53I
126.90
85At
(210)
117Ts
(294)
2He
4.003
10Ne
20.18
18Ar
39.95
36Kr
83.80
54Xe
131.29
86Rn
(222)
118Og
(294)
71Lu
174.97
103Lr
(262)
70Yb
173.04
102No
(259)
69Tm
168.93
101Md(258)
68Er
167.26
100Fm(257)
67Ho
164.93
99Es
(252)
66Dy
162.50
98Cf
(251)
65Tb
158.93
97Bk
(247)
64Gd
157.25
96Cm(247)
63Eu
151.96
95Am(243)
62Sm
150.36
94Pu
(244)
61Pm(145)
93Np
(237)
60Nd
144.24
92U
238.03
59Pr
140.91
91Pa
231.04
58Ce
140.12
90Th
232.04
1
2
3
4
5
6
7
1A 8A
2A 3A 4A 5A 6A 7APERIODIC TABLE OF THE ELEMENTS
For elements Tc, Pm, and those beyond Bi, a number in parentheses is the mass number (not the atomic mass) of the most stable or the only known isotope.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
3B 4B 5B 6B 7B 8B 1B 2B
