login  home  contents  what's new  discussion  bug reports help  links  subscribe  changes  refresh  edit

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 (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 and .

Two-Dimensions

fricas
p:ℂ:=complex(ℜp,𝔍p) (2)
Type: Complex(Fraction(Polynomial(Integer)))
fricas
q:ℂ:=complex(ℜq,𝔍q) (3)
Type: Complex(Fraction(Polynomial(Integer)))
fricas
r:ℂ:=complex(ℜr,𝔍r) (4)
Type: Complex(Fraction(Polynomial(Integer)))
fricas
t:ℂ:=complex(ℜt,0) (5)
Type: Complex(Fraction(Polynomial(Integer)))
fricas
ρ:Matrix ℂ := matrix [[t/2+p,q],[r,t/2-p]] (6)
Type: Matrix(Complex(Fraction(Polynomial(Integer))))
fricas
trace ρ (7)
Type: Complex(Fraction(Polynomial(Integer)))
fricas
d:=determinant ρ (8)
Type: Complex(Fraction(Polynomial(Integer)))
fricas
test(p^2+r*q=(1/4)*t^2-d) (9)
Type: Boolean
fricas
s0:=solve(imag d,ℜr) (10)
Type: List(Equation(Fraction(Polynomial(Integer))))
fricas
eval(trace(ρ*ρ),s0) (11)
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,complex(b,c)],[complex(b,-c),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)))) (13)
Type: Boolean
fricas
H:=htranspose(ρ)*h-h*ρ (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) (15)
Type: Matrix(Fraction(Polynomial(Integer)))

The null space (kernel) of the Jacobian

fricas
N:=nullSpace(map(x+->eval(x,s0),J)) (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) (17)
Type: List(Equation(Fraction(Polynomial(Integer))))
fricas
map(x+->eval(x,concat(s0,s1)),H) (18)
Type: Matrix(Fraction(Polynomial(Complex(Integer))))

SandBoxHermitianIsomorphisms3

 Subject:   Be Bold !! ( 15 subscribers )