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A complex vector ℂ-space V 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 h-hermitian if


\rho^\dagger \circ h = h \circ \rho
 
fricas
ℂ:=Complex Fraction Polynomial Integer

\label{eq1}\hbox{\axiomType{Complex}\ } (\hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{Polynomial}\ } (\hbox{\axiomType{Integer}\ })))(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 ρ to possess hermitean isomorphism h is that trace ρ \in ℝ and det ρ \in ℝ.

Two-Dimensions

fricas
p:ℂ:=complex(ℜp,𝔍p)

\label{eq2}� � p +{�� � p \  i}(2)
Type: Complex(Fraction(Polynomial(Integer)))
fricas
q:ℂ:=complex(ℜq,𝔍q)

\label{eq3}� � q +{�� � q \  i}(3)
Type: Complex(Fraction(Polynomial(Integer)))
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r:ℂ:=complex(ℜr,𝔍r)

\label{eq4}� � r +{�� � r \  i}(4)
Type: Complex(Fraction(Polynomial(Integer)))
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t:ℂ:=complex(ℜt,0)

\label{eq5}� � t(5)
Type: Complex(Fraction(Polynomial(Integer)))
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ρ:Matrix ℂ := matrix [[t/2+p,q],[r,t/2-p]]

\label{eq6}\left[ 
\begin{array}{cc}
{{{� � t +{2 \  � � p}}\over 2}+{�� � p \  i}}&{� � q +{�� � q \  i}}
\
{� � r +{�� � r \  i}}&{{{� � t -{2 \  � � p}}\over 2}-{�� � p \  i}}
(6)
Type: Matrix(Complex(Fraction(Polynomial(Integer))))
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trace ρ

\label{eq7}� � t(7)
Type: Complex(Fraction(Polynomial(Integer)))
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d:=determinant ρ

\label{eq8}{{{4 \  �� � q \  �� � r}+{4 \ {{�� � p}^{2}}}+{{� � t}^{2}}-{4 \  � � q \  � � r}-{4 \ {{� � p}^{2}}}}\over 4}+{{\left(-{� � q \  �� � r}-{� � r \  �� � q}-{2 \  � � p \  �� � p}\right)}\  i}(8)
Type: Complex(Fraction(Polynomial(Integer)))
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test(p^2+r*q=(1/4)*t^2-d)

\label{eq9} \mbox{\rm true} (9)
Type: Boolean
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s0:=solve(imag d,ℜr)

\label{eq10}\left[{� � r ={{-{� � q \  �� � r}-{2 \  � � p \  �� � p}}\over �� � q}}\right](10)
Type: List(Equation(Fraction(Polynomial(Integer))))
fricas
eval(trace(ρ*ρ),s0)

\label{eq11}{{{\left(-{4 \ {{�� � q}^{2}}}-{4 \ {{� � q}^{2}}}\right)}\  �� � r}+{{\left(-{4 \ {{�� � p}^{2}}}+{{� � t}^{2}}+{4 \ {{� � p}^{2}}}\right)}\  �� � q}-{8 \  � � p \  � � q \  �� � p}}\over{2 \  �� � q}(11)
Type: Fraction(Polynomial(Complex(Integer)))

Given an operator ρ \in End V, one must find the tensor H=0 for unknown manifold of hermitian isomorphisms h.

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

\label{eq13} \mbox{\rm true} (13)
Type: Boolean
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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 h in terms of the components of ρ. To do this we will determine how the components of H depend on the components of h.

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

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

\label{eq16}\left[{\left[{{2 \  � � p}\over �� � q}, \:{� � q \over �� � q}, \: 1, \: 0 \right]}, \:{\left[ -{�� � r \over �� � q}, \:{�� � p \over �� � q}, \: 0, \: 1 \right]}\right](16)
Type: List(Vector(Fraction(Polynomial(Integer))))

gives the general solution to the problem.

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

\label{eq17}\left[{a ={{-{e \  �� � r}+{2 \  c \  � � p}}\over �� � q}}, \:{b ={{{e \  �� � p}+{c \  � � q}}\over �� � q}}, \:{c = c}, \:{e = e}\right](17)
Type: List(Equation(Fraction(Polynomial(Integer))))
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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))))

SandBoxHermitianIsomorphisms3




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