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Electronics Labs - Digital Electronics
Zhenyu Ye
14-Nov-16 1
The Art of Electronics by Horowitz and Hill – Chapter 8
Boolean AlgebraBoolean algebra is the branch of algebra in which the values of the variables are the truth values:true and false, usually denoted as 1 and 0. Instead of elementary algebra where the values of the variables are numbers, and the main operations are addition and multiplication, the basic operations of Boolean algebra are n conjunction and denoted as 𝐴 ∧ 𝐵, 𝐴 $ 𝐵n disjunction or denoted as 𝐴 ∨ 𝐵, 𝐴+ 𝐵n negation not denoted as ¬𝐀, �̅�
October 24, 2016 Digit Electronics, Zhenyu Ye 4
Boolean Algebra –Truth Table
October 24, 2016 Digit Electronics, Zhenyu Ye 5
x y 𝐱 $ 𝒚 𝒙 + 𝒚0 0 0 01 0 0 10 1 0 11 1 1 1
𝐱 𝒙-0 11 0
Boolean Algebra – Secondary Ops.
NAND n 𝑥 $ 𝑦 = �̅� + 𝑦1
NORn 𝑥 + 𝑦 = �̅� $ 𝑦1
October 24, 2016 Digit Electronics, Zhenyu Ye 6
x y 𝒙 $ 𝒚 𝒙- + 𝒚-0 0 1 11 0 1 10 1 1 11 1 0 0
x y 𝒙 + 𝒚 𝒙- $ 𝒚-0 0 1 11 0 0 00 1 0 01 1 0 0
Boolean Algebra – Secondary Ops.
n Exclusive OR (XOR)𝒙⊕ 𝒚 = (𝒙 + 𝒚) $ (𝒙 $ 𝒚)
October 24, 2016 Digit Electronics, Zhenyu Ye 7
x y 𝒙⊕ 𝒚 (𝒙 + 𝒚) (𝒙 $ 𝒚)0 0 0 0 11 0 1 1 10 1 1 1 11 1 0 1 0
Boolean Algebra – De Morgan’s Laws
n Associativity of OR x + 𝑦 + 𝑧 = 𝑥 + 𝑦 + 𝑧n Associativity of AND x $ 𝑦 $ 𝑧 = 𝑥 $ 𝑦 $ 𝑧n Commutativity of OR x + 𝑦 = 𝑦 + 𝑥n Commutativity of AND x $ 𝑦 = 𝑦 $ 𝑥n Distributivity of AND over OR
x $ 𝑦 + 𝑧 = 𝑥 $ 𝑦 + (𝑥 $ 𝑧)n Distributivity of OR over AND
x+ 𝑦 $ 𝑧 = 𝑥 + 𝑦 $ (𝑥 + 𝑧)
October 24, 2016 Digit Electronics, Zhenyu Ye 8
Brownian Motionn Brownian Motion is the random motion of particles
suspended in a fluid (a liquid or a gas) resulting fromtheir collision with the fast-moving atoms ormolecules in the gas or liquid.
n https://upload.wikimedia.org/wikipedia/commons/6/6d/Translational_motion.gif
n https://upload.wikimedia.org/wikipedia/commons/5/51/Brownianmotion5particles150frame.gif
14-Nov-16 12
Brownian Motion
14-Nov-16 13
Reproduced from the book of Jean Baptiste Perrin, Les Atomes, three tracings of the motion of colloidal particles of radius 0.53 µm, as seen under the microscope, are displayed. Successive positions every 30 seconds are joined by straight line segments (the mesh size is 3.2 µm)
Brownian Motion
14-Nov-16 15
𝑚𝑑9𝑥𝑑𝑡9 = −𝛼
𝑑𝑥𝑑𝑡 + 𝐹(𝑡) 𝛼 = 6𝜋𝜂𝑎
𝑚2𝑑9𝑥9
𝑑𝑡9 −𝑚𝑑𝑥𝑑𝑡
9= −
𝛼2𝑑𝑥9
𝑑𝑡 + 𝑥𝐹(𝑡)
Brownian Motion
14-Nov-16 16
𝑚𝑑9𝑥𝑑𝑡9 = −𝛼
𝑑𝑥𝑑𝑡 + 𝐹(𝑡) 𝛼 = 6𝜋𝜂𝑎
𝑚2𝑑9𝑥9
𝑑𝑡9 −𝑚𝑑𝑥𝑑𝑡
9= −
𝛼2𝑑𝑥9
𝑑𝑡 + 𝑥𝐹(𝑡)
Define 𝛽 = DEF
DG
𝑚2𝑑𝛽𝑑𝑡 − 𝑚
𝑑𝑥𝑑𝑡
9= −
𝛼2 𝛽 + 𝑥𝐹(𝑡)
Brownian Motion
14-Nov-16 17
𝑚𝑑9𝑥𝑑𝑡9 = −𝛼
𝑑𝑥𝑑𝑡 + 𝐹(𝑡) 𝛼 = 6𝜋𝜂𝑎
𝑚2𝑑9𝑥9
𝑑𝑡9 −𝑚𝑑𝑥𝑑𝑡
9= −
𝛼2𝑑𝑥9
𝑑𝑡 + 𝑥𝐹(𝑡)
Define 𝛽 = DEF
DG
𝑚2𝑑𝛽𝑑𝑡 − 𝑚
𝑑𝑥𝑑𝑡
9= −
𝛼2 𝛽 + 𝑥𝐹(𝑡)
𝑚2𝑑𝛽𝑑𝑡 − 𝑘I𝑇 = −
𝛼2 𝛽
Brownian Motion
14-Nov-16 18
𝑚𝑑9𝑥𝑑𝑡9 = −𝛼
𝑑𝑥𝑑𝑡 + 𝐹(𝑡) 𝛼 = 6𝜋𝜂𝑎
𝑚2𝑑9𝑥9
𝑑𝑡9 −𝑚𝑑𝑥𝑑𝑡
9= −
𝛼2𝑑𝑥9
𝑑𝑡 + 𝑥𝐹(𝑡)
Define 𝛽 = DEF
DG
𝑚2𝑑𝛽𝑑𝑡 − 𝑚
𝑑𝑥𝑑𝑡
9= −
𝛼2 𝛽 + 𝑥𝐹(𝑡)
𝑚2𝑑𝛽𝑑𝑡 − 𝑘I𝑇 = −
𝛼2 𝛽 𝛽 =
2𝑘I𝑇𝛼 + 𝐴𝑒L
MGN⇒
Brownian Motion
14-Nov-16 19
𝑚𝑑9𝑥𝑑𝑡9 = −𝛼
𝑑𝑥𝑑𝑡 + 𝐹(𝑡) 𝛼 = 6𝜋𝜂𝑎
𝑚2𝑑9𝑥9
𝑑𝑡9 −𝑚𝑑𝑥𝑑𝑡
9= −
𝛼2𝑑𝑥9
𝑑𝑡 + 𝑥𝐹(𝑡)
Define 𝛽 = DEF
DG
𝑚2𝑑𝛽𝑑𝑡 − 𝑚
𝑑𝑥𝑑𝑡
9= −
𝛼2 𝛽 + 𝑥𝐹(𝑡)
𝑚2𝑑𝛽𝑑𝑡 − 𝑘I𝑇 = −
𝛼2 𝛽 𝛽 =
2𝑘I𝑇𝛼 + 𝐴𝑒L
MGN
𝑥9 =2𝑘I𝑇𝛼 𝑡 =
2𝑘I𝑇6𝜋𝜂𝛼 𝑡
⇒
Brownian Motion
14-Nov-16 20
𝑚𝑑9𝑥𝑑𝑡9 = −𝛼
𝑑𝑥𝑑𝑡 + 𝐹(𝑡) 𝛼 = 6𝜋𝜂𝑎
𝑚2𝑑9𝑥9
𝑑𝑡9 −𝑚𝑑𝑥𝑑𝑡
9= −
𝛼2𝑑𝑥9
𝑑𝑡 + 𝑥𝐹(𝑡)
Define 𝛽 = DEF
DG
𝑚2𝑑𝛽𝑑𝑡 − 𝑚
𝑑𝑥𝑑𝑡
9= −
𝛼2 𝛽 + 𝑥𝐹(𝑡)
𝑚2𝑑𝛽𝑑𝑡 − 𝑘I𝑇 = −
𝛼2 𝛽 𝛽 =
2𝑘I𝑇𝛼 + 𝐴𝑒L
MGN
𝑥9 =2𝑘I𝑇𝛼 𝑡 =
2𝑘I𝑇6𝜋𝜂𝛼 𝑡 𝑟9 =
4𝑘I𝑇𝛼 𝑡 =
4𝑘I𝑇6𝜋𝜂𝛼 𝑡
⇒
⇒
Brownian Motion
14-Nov-16 25
𝑟9 =4𝑘I𝑇𝛼 𝑡 =
4𝑘I𝑇6𝜋𝜂𝛼 𝑡 ⇒ 𝑘I =
𝑀𝑆𝐷𝑡
6𝜋𝜂𝑎4𝑇 ~1.1×10L9a𝐽/𝐾
𝜂 = 8.90×10LX Pa $ 𝑠𝑎 = 1.1 𝜇𝑚
Advanced Labs - Zeeman Effects
Zhenyu Ye
14-Nov-16 26
Experiments in Modern Physics – A. Melissinos Chapter 6
Modeling of Hydrogen Atoms
14-Nov-16 27
n Schrodinger equation in 1926
i! ∂∂tΨ!r, t( ) = −!2
2m∇2 +V !r, t( )
⎡
⎣⎢
⎤
⎦⎥⋅Ψ
!r, t( )
Ψ!r( ) = 1
r⋅ χ l r( ) ⋅Ylm θ,φ( )
En = −e2
!c⎛
⎝⎜
⎞
⎠⎟
2mec
2
2n2
m = 0,±1,!,±l
L = l(l +1)! Lz =m!
l = 0,1,!,n−1n =1,2,!
See Adv.Lab.2
Electron Spin
14-Nov-16 28
S = s(s+1)! s = 12
1925: G.Uhlenbeck, S.Goudsmit
Sz =ms! ms = ±12
𝐽=𝐿+𝑆
𝑚g=𝑚h +𝑚i
Electron Spin
14-Nov-16 29
S = s(s+1)! s = 12
Ag Shell Structure: 2, 8, 18, 18, 1
1925: G.Uhlenbeck, S.Goudsmit
Sz =ms! ms = ±12
Stern-Gerlach Experiment 1922
𝐽=𝐿+𝑆
𝑚g=𝑚h +𝑚i
Electron Spin
14-Nov-16 30
S = s(s+1)! s = 12
Ag Shell Structure: 2, 8, 18, 18, 1
1925: G.Uhlenbeck, S.Goudsmit
Sz =ms! ms = ±12
Stern-Gerlach Experiment 1922
Bohr magneton 𝜇I =jℏ9N
𝑔m = 1
𝑔n = 2
𝜇h = 𝑔m𝑚m𝜇I
𝜇i = 𝑔n𝑚n𝜇I
𝐸p,Nr,Ns = −𝑒9
ℏ𝑐
9𝑚j𝑐9
2𝑛9 + 𝜇h𝐵 + 𝜇i𝐵
L, S and J
14-Nov-16 32
𝑔g =𝑔i $ 𝑆 + 𝑔h $ 𝐿
𝑆 + 𝐿
2S+1LJ𝐸 = 𝐸Ivw + 𝜇g𝐵
∆𝐸 = ∆(𝑔g𝑚g)𝜇I𝐵
541.6nm
𝐽=𝐿+𝑆
𝑚g=𝑚h +𝑚i
𝜇g = 𝑔g𝑚g𝜇I
L, S and J
14-Nov-16 33
ΔJ=±1, Δmj=0, ±1
2S+1LJ
541.6nm
𝐽=𝐿+𝑆
𝑚g=𝑚h +𝑚i
𝑔g =𝑔i $ 𝑆 + 𝑔h $ 𝐿
𝑆 + 𝐿
𝐸 = 𝐸Ivw + 𝜇g𝐵
∆𝐸 = ∆(𝑔g𝑚g)𝜇I𝐵
𝜇g = 𝑔g𝑚g𝜇I
Wave-length Shift Calculation
14-Nov-16 38
∆𝜆 =𝜆9
2𝑑𝐷~a9 − 𝐷~99
𝐷~L�9 − 𝐷~99=𝜆9
2𝑑𝐷~99 − 𝐷~�9
𝐷~L�9 − 𝐷~99
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