MathAction SandBoxHermitianIsomorphisms

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A complex vector ℂ-space possesses many different hermitian isomorphisms $h^\dagger=h \in iso(V,V^\dagger)$ . In quantum mechanics a given operator $\rho \in End(V)$ may be said to be -hermitian if

$\rho^\dagger \circ h = h \circ \rho$

fricas

(1) -> ℂ:=Complex Fraction Polynomial Integer

$\label{eq1}\hbox{\axiomType{Complex}\ } \left({\hbox{\axiomType{Fraction}\ } \left({\hbox{\axiomType{Polynomial}\ } \left({\hbox{\axiomType{Integer}\ }}\right)}\right)}\right)$

(1)

Type: Type

fricas

-- dagger
htranspose(h)==map(x+->conjugate(x),transpose h)

Type: Void

fricas

)expose MCALCFN

   MultiVariableCalculusFunctions is now explicitly exposed in frame 
      initial

Theorem

The necessary conditions for an operator to possess hermitean isomorphism is that $trace ρ \in ℝ$ and $det ρ \in ℝ$ .

Two-Dimensions

fricas

p:ℂ:=complex(ℜp,𝔍p)

$\label{eq2}\begin{array}{@{}l} \displaystyle � � p +{ \begin{array}{@{}l} \displaystyle �� p \ \cdot \ \ \displaystyle i$

(2)

Type: Complex(Fraction(Polynomial(Integer)))

fricas

q:ℂ:=complex(ℜq,𝔍q)

$\label{eq3}\begin{array}{@{}l} \displaystyle � � q +{ \begin{array}{@{}l} \displaystyle �� q \ \cdot \ \ \displaystyle i$

(3)

Type: Complex(Fraction(Polynomial(Integer)))

fricas

r:ℂ:=complex(ℜr,𝔍r)

$\label{eq4}\begin{array}{@{}l} \displaystyle � � r +{ \begin{array}{@{}l} \displaystyle �� r \ \cdot \ \ \displaystyle i$

(4)

Type: Complex(Fraction(Polynomial(Integer)))

fricas

t:ℂ:=complex(ℜt,0)

$\label{eq5}� � t$

(5)

Type: Complex(Fraction(Polynomial(Integer)))

fricas

ρ:Matrix ℂ := matrix [[t/2+p,q],[r,t/2-p]]

$\label{eq6}\left[ \begin{array}{cc} {{\frac{� � t +{2 \ � � p}}{2}}+{�� p \ i}}&{� � q +{�� q \ i}} \ {� � r +{�� r \ i}}&{{\frac{� � t -{2 \ � � p}}{2}}-{�� p \ i}}$

(6)

Type: Matrix(Complex(Fraction(Polynomial(Integer))))

fricas

trace ρ

$\label{eq7}� � t$

(7)

Type: Complex(Fraction(Polynomial(Integer)))

fricas

d:=determinant ρ

$\label{eq8}\begin{array}{@{}l} \displaystyle {\frac{{4 \ �� q \ �� r}+{4 \ {{�� p}^{2}}}+{{� � t}^{2}}-{4 \ � � q \ � � r}-{4 \ {{� � p}^{2}}}}{4}}+ \ \ \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle -{ \begin{array}{@{}l} \displaystyle � � q \ \cdot \ \ \displaystyle �� r$

(8)

Type: Complex(Fraction(Polynomial(Integer)))

fricas

test(p^2+r*q=(1/4)*t^2-d)

$\label{eq9} \mbox{\rm true}$

(9)

Type: Boolean

fricas

s0:=solve(imag d,ℜr)

$\label{eq10}\left[ � � r ={\frac{-{� � q \ �� r}-{2 \ � � p \ �� p}}{�� q}}\right]$

(10)

Type: List(Equation(Fraction(Polynomial(Integer))))

fricas

eval(trace(ρ*ρ),s0)

$\label{eq11}\frac{{{\left(-{4 \ {{�� q}^{2}}}-{4 \ {{� � q}^{2}}}\right)}\ �� r}+{{\left(-{4 \ {{�� p}^{2}}}+{{� � t}^{2}}+{4 \ {{� � p}^{2}}}\right)}\ �� q}-{8 \ � � p \ � � q \ �� p}}{2 \ �� q}$

(11)

Type: Fraction(Polynomial(Complex(Integer)))

Given an operator $ρ \in End V$ , one must find the tensor for unknown manifold of hermitian isomorphisms .

fricas

h:Matrix ℂ:=matrix [[a,complex(b,c)],[complex(b,-c),e]]

$\label{eq12}\left[ \begin{array}{cc} a &{b +{c \ i}} \ {b -{c \ i}}& e$

(12)

Type: Matrix(Complex(Fraction(Polynomial(Integer))))

fricas

test(h = htranspose h)

fricas

Compiling function htranspose with type Matrix(Complex(Fraction(
      Polynomial(Integer)))) -> Matrix(Complex(Fraction(Polynomial(
      Integer))))

$\label{eq13} \mbox{\rm true}$

(13)

Type: Boolean

fricas

H:=htranspose(ρ)*h-h*ρ

$\label{eq14}\left[ \begin{array}{cc} {{\left(-{2 \ b \ �� r}-{2 \ a \ �� p}-{2 \ c \ � � r}\right)}\ i}&{{e \ � � r}-{a \ � � q}+{2 \ b \ � � p}+{{\left(-{e \ �� r}-{a \ �� q}+{2 \ c \ � � p}\right)}\ i}} \ {-{e \ � � r}+{a \ � � q}-{2 \ b \ � � p}+{{\left(-{e \ �� r}-{a \ �� q}+{2 \ c \ � � p}\right)}\ i}}&{{\left(-{2 \ b \ �� q}+{2 \ e \ �� p}+{2 \ c \ � � q}\right)}\ i}$

(14)

Type: Matrix(Complex(Fraction(Polynomial(Integer))))

We wish to find expressions for in terms of the components of . To do this we will determine how the components of depend on the components of .

fricas

J:=jacobian(concat( map(x+->[real x, imag x], concat(H::List List ?)) ),
     [a,b,c,e]::List Symbol)

$\label{eq15}\left[ \begin{array}{cccc} 0 & 0 & 0 & 0 \ -{2 \ �� p}& -{2 \ �� r}& -{2 \ � � r}& 0 \ - � � q &{2 \ � � p}& 0 & � � r \ - �� q & 0 &{2 \ � � p}& - �� r \ � � q & -{2 \ � � p}& 0 & - � � r \ - �� q & 0 &{2 \ � � p}& - �� r \ 0 & 0 & 0 & 0 \ 0 & -{2 \ �� q}&{2 \ � � q}&{2 \ �� p}$

(15)

Type: Matrix(Fraction(Polynomial(Integer)))

The null space (kernel) of the Jacobian

fricas

N:=nullSpace(map(x+->eval(x,s0),J))

$\label{eq16}\begin{array}{@{}l} \displaystyle \left[{ \begin{array}{@{}l} \displaystyle \left[{\frac{2 \ � � p}{�� q}}, \right. \ \ \displaystyle \left.\: \right. \ \ \displaystyle \left.{\frac{� � q}{�� q}}, \right. \ \ \displaystyle \left.\: 1, \: 0 \right]$

(16)

Type: List(Vector(Fraction(Polynomial(Integer))))

gives the general solution to the problem.

fricas

s1:=map((x,y)+->x=y,[a,b,c,e],c*N.1+e*N.2)

$\label{eq17}\begin{array}{@{}l} \displaystyle \left[{ \begin{array}{@{}l} \displaystyle a = \ \ \displaystyle {\frac{-{e \ �� r}+{2 \ c \ � � p}}{�� q}}$

(17)

Type: List(Equation(Fraction(Polynomial(Integer))))

fricas

map(x+->eval(x,concat(s0,s1)),H)

$\label{eq18}\left[ \begin{array}{cc} 0 & 0 \ 0 & 0$

(18)

Type: Matrix(Fraction(Polynomial(Complex(Integer))))

... --Bill Page, Mon, 27 Jun 2011 18:30:02 -0700 reply

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