A complex vector ℂ-space possesses many different hermitian isomorphisms . In quantum mechanics a given operator may be said to be -hermitian if fricas ℂ:=Complex Fraction Polynomial Integer
Type: Type
fricas -- dagger htranspose(h)==map(x+->conjugate(x), Type: Void
fricas )expose MCALCFN TheoremThe necessary conditions for an operator to possess hermitean isomorphism is that and . Two-Dimensions fricas p:ℂ:=complex(ℜp,
Type: Complex(Fraction(Polynomial(Integer)))
fricas q:ℂ:=complex(ℜq,
Type: Complex(Fraction(Polynomial(Integer)))
fricas r:ℂ:=complex(ℜr,
Type: Complex(Fraction(Polynomial(Integer)))
fricas t:ℂ:=complex(ℜt,
Type: Complex(Fraction(Polynomial(Integer)))
fricas ρ:Matrix ℂ := matrix [[t/2+p,
Type: Matrix(Complex(Fraction(Polynomial(Integer))))
fricas trace ρ
Type: Complex(Fraction(Polynomial(Integer)))
fricas d:=determinant ρ
Type: Complex(Fraction(Polynomial(Integer)))
fricas test(p^2+r*q=(1/4)*t^2-d)
Type: Boolean
fricas s0:=solve(imag d,
Type: List(Equation(Fraction(Polynomial(Integer))))
fricas eval(trace(ρ*ρ),
Type: Fraction(Polynomial(Complex(Integer)))
Given an operator , one must find the tensor for unknown manifold of hermitian isomorphisms . fricas h:Matrix ℂ:=matrix [[a,
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))))
Type: Boolean
fricas H:=htranspose(ρ)*h-h*ρ
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,
Type: Matrix(Fraction(Polynomial(Integer)))
The null space (kernel) of the Jacobian fricas N:=nullSpace(map(x+->eval(x,
Type: List(Vector(Fraction(Polynomial(Integer))))
gives the general solution to the problem. fricas s1:=map((x,
Type: List(Equation(Fraction(Polynomial(Integer))))
fricas map(x+->eval(x,
Type: Matrix(Fraction(Polynomial(Complex(Integer))))
... --Bill Page, Mon, 27 Jun 2011 18:30:02 -0700 reply SandBoxHermitianIsomorphisms3
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