# 查看完整版本 : Mathematical Physics

Queenboy 2016-11-15 09:15 AM

## Mathematical Physics

Queenboy 2016-11-15 09:24 AM

map S x S  -> S = S^3  (algebraic)
or 2^S (geometric )

Queenboy 2016-11-15 09:55 AM

∮c f(z)dz = 0
f(z) (simple connected ) , Domain D 有解析
c :simple closed curve  ∈ D
curve 形式與積分無關.

Queenboy 2016-11-15 04:42 PM

E0  ?
H = Hb + He + Heb

He =1/2 e^2 ∑ [i=1 , N] ∑ [j=1, N] {(e^ -u|ri-rj|) /|ri -rj|

from  ∮c f(z)dz = 0 ........

biochemphy 2016-11-16 09:10 AM

∮c f(z)dz = 0  路徑積分
:smile_44:

biochemphy 2016-11-16 09:11 AM

Queenboy 2016-11-16 10:12 AM

E=( 1/2pi )∮c  g dA =(4pi/2pi) =2
Chern N =( 1/2pi )∮c  berry 曲率  dA  =2 spin
Geometric phase
Ψ(R) = ▽σ      (Quantum state |Ψ(R)>  )
∆γ = i<Ψ(R)|∇RΨ(R)> · dR
∆γ = i< ▽σ|∇R▽σ > · dR

biochemphy 2016-11-16 04:06 PM

complex variables and applications
:P

biochemphy 2016-11-16 04:07 PM

and
Adiabatic Approximation
:loveliness:

abctea 2016-11-18 08:40 AM

biochemphy 2016-11-18 12:02 PM

Condensed matter physics
:smile_44:

abctea 2016-11-19 08:25 AM

I: differential  invariant
DI = DxI dx

biochemphy 2016-11-19 02:24 PM

subspace  s1, s2 ∈  S , ∃f -> f(s1)=s2 and ∃ f ∈ S^1 -> R^3
s1 and s2 :  homeomorphism

Q 是一個 knot,

;P

[[i] 本帖最後由 biochemphy 於 2016-11-19 02:26 PM 編輯 [/i]]

Queenboy 2016-11-21 09:53 AM

N= (1/2pi)∫(bz) d^2 K dot i < ∇(k) u(K)| X |  ∇(k) u(K)> = 整數

ZPE 的 eigenstate 與 其他的 E 不能分類出來.

eigenstate 可用Z2 拓撲不變量分類:
Z2 = (1/2pi){∮(∂b+) dl dot  A(K) -  ∫(b+) dK^2 dot ∇X A(K)} /mod2

spin-orbit coupling:

from Shrodinger, 能量 E 的 eigenvalue 是e (假設為e)

X(up) ,Y (up) , Z(up) , X(down) ,Y(down) , Z(down)

eg : Ux(up)x(up)= e(系數a) + <x(up)|Hso|x(up)>

UC=EC ,
C=[ Cx(up),Cy(up),Cz(up),Cx(down),Cy(down),Cz(down)],

u1 =(1/root2)(X+iY)up
u2 =(1/root2)(X+iY)down
u3 =(1/root6)[(X+iY)down-2Zup]
u4 =(1/root6)[(X-iY)up+2Zdown]

u5 =(1/root3)[(X+iY)down+Zup]
u6 =(1/root3)[(X-iY)up+Zdown]

eg:  u1 transpose matrix = u1'
u1u1'= S
T=time reversal operator
TST^-1=-S

E degeneracy : 必為偶數 ,u1,u2,u3,u4

bjj123 2016-11-22 08:35 AM

biochemphy 2016-11-22 01:09 PM

;P

bjj123 2016-11-23 12:53 PM

biochemphy 2016-11-23 02:26 PM

use quantum mechanics for discussion.

bjj123 2016-11-23 03:06 PM

biochemphy 2016-11-23 09:23 PM

Quantum Entanglement :smile_27:

biochemphy 2016-11-24 11:41 AM

|ψ > = a1|HH> + a2 | HV > + a3|VH> +a4|VV>
and ,  |a1|^2 + |a2|^2+ |a3|^3+ |a4|^2 =1
two-state systems of photon polarized 的例子:
p1|ψ1 > = ( α1 |H >p1 + α2 | V > p1)
p2|ψ2 > = ( β1 |V>p2 + β2 | H >p2 )

p1|ψ1 > p2|ψ2 > = | ψ > = p1|ψ1 >  ⊗  p2|ψ2 >

=  α1β1 |H >p1 |V>p2 + α1β2|H >p1 | H >p2 +
α2β1| V > p1|V>p2  + α2 β2| V> p1 |H>p2
and , | α1β2 | ^2 + | α1β1|^2 + | α2 β2|^2 + | α2β1|^2 =1
:smile_35:

biochemphy 2016-11-24 01:59 PM

magnetic moments ordering .

testsample 2017-1-13 09:07 AM

QEPS 實驗講解?
SPDC 產生two-photon polarization-entangled states

biochemphy 2017-1-13 02:47 PM

yes

biochemphy 2017-1-13 04:02 PM

( 用 Mathematical  ) ;P

biochemphy 2017-1-23 03:43 PM

P(an) = | < un|Ψ> |^2
|Ψ >  = ∑ Cn |un>
P(an)  = |C n | ^2
:smile_30:

Queenboy 2017-1-23 07:00 PM

topological insulator 的研究 內容:
Quantum transport in  the  electronic  band  structure , Energy gap/topological  surface  states
spin-polarized currents in quantum spin Hall
bi2se3 crystal structure
Majorana particles  topological phase transition

bjj123 2017-1-25 12:44 PM

biochemphy 2017-1-25 05:20 PM