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Powerpoint TemplatesPage 1
Powerpoint Templates
Quantum Chemistry
Revisited
Powerpoint TemplatesPage 2
Wave Equation
Non Relativistic Limit
∇2𝜓 = ൬𝑘𝜔൰2 𝜕2𝜓𝜕𝑡2
𝜓ሺ𝑥,𝑦,𝑧,𝑡ሻ= 𝜓ሺ𝑥,𝑦,𝑧ሻ.𝜓ሺ𝑡ሻ Possible solution: Plane waves𝜓ሺ𝑡ሻ= 𝜓𝑜𝑒−𝑖𝜔𝑡
Powerpoint TemplatesPage 3
∇2𝜓 = −൬2𝜋𝜆൰
2 𝜓
λ= ℎ𝑝
∇2𝜓 = −൬2𝜋𝑝ℎ ൰
2 𝜓
𝐸= 𝑇+ 𝑉= 𝑝22𝑚+ 𝑉
Powerpoint TemplatesPage 4
𝑝= ඥ2𝑚(𝐸− 𝑉)
∇2𝜓 = −൬2𝜋ℎ൰
2 2𝑚(𝐸− 𝑉)𝜓
(− ℎ28𝜋2𝑚∇2 + 𝑉)𝜓 = 𝐸𝜓
Time Independent Schrödinger Equation
Powerpoint TemplatesPage 5
𝐸= ℎν= ℎ 𝜔2𝜋
𝜓ሺ𝑡ሻ= 𝜓𝑜𝑒−𝑖𝜔𝑡
𝑑𝜓ሺ𝑡ሻ𝑑𝑡 = −𝑖2𝜋𝐸ℎ 𝜓𝑜𝑒−𝑖𝜔𝑡
𝑖 ℎ2𝜋𝑑𝑑𝑡𝜓ሺ𝑡ሻ= 𝐸𝜓ሺ𝑡ሻ Time dependent Schrödinger Equation
Powerpoint TemplatesPage 6
ቆ− ℎ28𝜋2𝑚∇2 + 𝑉ቇ𝜓(𝑥,𝑦,𝑧)𝜓(𝑡) = 𝑖 ℎ2𝜋𝑑𝑑𝑡𝜓(𝑥,𝑦,𝑧)𝜓(𝑡)
Lousy relativistic equation
2nd derivative in space
1st derivative in time
Many fathers equation
(Klein, Fock, Schrödinger, de Broglie, ...)
Klein-Gordon Equation (1926)
Powerpoint TemplatesPage 7
(E – V)2 = p2c2 + m2c4
ℎ2𝑐24𝜋2 𝛿2𝜑(𝑞)𝛿𝑞2 +ሾሺ𝐸− 𝑉ሻ2 − 𝑚𝑜2𝑐4ሿ𝜑ሺ𝑞ሻ= 0
ℎ2𝑐24𝜋2 𝛿2𝜓ሺ𝑞,𝑡ሻ𝛿𝑞2 − ℎ24𝜋2 𝛿2𝜓ሺ𝑞,𝑡ሻ𝛿𝑡2 – 𝑖ℎ𝑉𝜋 𝛿𝜓(𝑞,𝑡)𝛿𝑡 + ሺ𝑉2 − 𝑚𝑜2𝑐4ሻ𝜓(𝑞,𝑡) = 0
Free Electron (V = 0)
ℎ2𝑐24𝜋2 𝛿2𝜓ሺ𝑞,𝑡ሻ𝛿𝑞2 − ℎ24𝜋2 𝛿2𝜓ሺ𝑞,𝑡ሻ𝛿𝑡2 − 𝑚𝑜2𝑐4 𝜓(𝑞,𝑡) = 0
Powerpoint TemplatesPage 8
Klein-Gordon Equations
Eigen values for E2
± E solutions
Matter
Antimatter
Carl Anderson discovers the positron in 1932
KG works well for bosons (integer spin particles)
Powerpoint TemplatesPage 9
(𝑎𝛻+ 𝑏𝑖𝑐 𝑑𝑑𝑡)2 = 𝑎2𝛻2 − 𝑏2 1𝑐2 𝑑𝑑𝑡2
a = b = 1
ab + ba = 0
𝑎 = ቂ1 00 −1ቃ 𝑏 = ቂ0 11 0ቃ 𝑜𝑟 𝑐 = ቂ0 −𝑖𝑖 0ቃ
Powerpoint TemplatesPage 10
𝛼1 = 0 00 0 0 11 00 11 0 0 00 0 𝛼2 = 0 00 0 0 −𝑖𝑖 00 −𝑖𝑖 0 0 00 0
𝛼3 = 0 00 0 1 00 −11 00 −1 0 00 0 𝛽 = 1 00 1 0 00 00 00 0 −1 00 −1
3 dimensions and time
Powerpoint TemplatesPage 11
൭− 𝛼𝑖 𝑖ℎ𝑐2𝜋 𝛿𝛿𝑞3
𝑖=1 + 𝛽𝑚𝑜2𝑐4൱𝜓(𝑞,𝑡) = 𝑖ℎ2𝜋𝛿𝜓ሺ𝑞,𝑡ሻ𝛿𝑡
Dirac Equation
2 positive solutions
2 negative solutions
Matter / Antimatter
Spin ± ½
Powerpoint TemplatesPage 12
𝜎1 = ቂ1 00 −1ቃ 𝜎2 = ቂ0 11 0ቃ 𝜎3 = ቂ0 −𝑖𝑖 0ቃ Pauli Matrices
𝑝= −𝑖 ℎ2𝜋∇− 𝑒𝑐𝐴
𝜎.𝑝= 𝜎.(−𝑖 ℎ2𝜋∇− 𝑒𝑐𝐴)
Powerpoint TemplatesPage 13
𝐻= 12𝑚(𝜎.𝑝)2 + 𝑉
𝜎.𝑝 𝜎.𝑝= 𝑝2𝐼+ 𝑖𝜎.(𝑝𝑥𝑝)
ℎ2𝑚(−𝑖 ℎ2𝜋∇− 𝑒𝑐𝐴)2 + 𝑉− 𝑒ℎ4𝜋𝑚𝑐𝜎1.𝐵൨𝛹= 𝑖𝜎1 ℎ2𝜋𝑑𝑑𝑡𝛹
𝛹= ቀ10ቁ𝜓 𝛹= ቀ01ቁ𝜓
Powerpoint TemplatesPage 14
ቆ− ℎ28𝜋2𝑚 𝛿2𝛿𝑞2 + 𝑉ቇ𝜓ሺ𝑞,𝑡ሻ= 𝑖 ℎ2𝜋 𝛿𝜓(𝑞,𝑡)𝛿𝑡
ቆ− ℎ28𝜋2𝑚 𝛿2𝛿𝑞2 + 𝑉ቇ𝜓∗ሺ𝑞,𝑡ሻ= − 𝑖 ℎ2𝜋 𝛿𝜓∗(𝑞,𝑡)𝛿𝑡
𝜓∗ቆ− ℎ28𝜋2𝑚 𝛿2𝛿𝑞2 + 𝑉ቇ𝜓 = 𝜓∗𝑖 ℎ2𝜋 𝛿𝜓𝛿𝑡
𝜓ቆ− ℎ28𝜋2𝑚 𝛿2𝛿𝑞2 + 𝑉ቇ𝜓∗= − 𝜓𝑖 ℎ2𝜋 𝛿𝜓∗𝛿𝑡
Powerpoint TemplatesPage 15
𝑖 ℎ2𝜋 𝛿ሾ𝜓𝜓∗ሿ𝛿𝑡 = − ℎ28𝜋2𝑚𝜓∗ 𝛿2𝛿𝑞2 𝜓+ ℎ28𝜋2𝑚𝜓 𝛿2𝛿𝑞2 𝜓∗
𝛿𝜌𝛿𝑡 = − ℎ4𝜋𝑚𝑖 𝛿𝛿𝑞𝜓∗ 𝛿𝛿𝑞 𝜓− 𝜓 𝛿𝛿𝑞 𝜓∗൨
𝐽= ℎ4𝜋𝑚𝑖 ቂ𝜓∗ 𝛿𝛿𝑞 𝜓− 𝜓 𝛿𝛿𝑞 𝜓∗ቃ 𝛿𝜌𝛿𝑡 + ∇𝐽= 0
𝜌𝑉ሺ𝑡ሻ+ න 𝐽𝑆 .𝑛ሬԦ𝑑𝑆= 0
Powerpoint TemplatesPage 16
There cant be flow in pure real and pure imaginary wave
functions.
In stationary states the flow is either zero or constant.
div D = ρ implies that stationary states create static
electric fields.
rot H = J + D/t implies that stationary states with J≠0
create static magnetic fields.
Static magnetic fields induce currents J which create
induced magnetic fields.
Time dependent magnetic fields induce time dependent
electric fields (rot E = - B/t), which means time
dependent charge densities to which correspond non
stationary states.
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