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last edited 12 years ago by page |
Edit detail for SandBoxHermitianIsomorphisms3 revision 3 of 11
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Editor: Bill Page
Time: 2011/06/29 15:43:58 GMT-7 |
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| Note: characteristicPolynomial | ||
added: \begin{axiom} )set output tex off )set output algebra on \end{axiom} added: \end{axiom} \begin{axiom} changed: -s2:=solve(eval(imag trace ρ, s1),𝔍q2) s2:=solve(eval(imag trace ρ,s1),𝔍p1) s3:=solve(eval(eval(imag trace(ρ*ρ),s1), s2),ℜp1) added: \begin{axiom} C:=eval(eval(characteristicPolynomial ρ,s1),s2) C0:=zerosOf(C) #C0 imag(C0.1) imag(C0.2) \end{axiom} changed: -s3:=map((x,y)+->x=y,[a,b,c,e],c*N.1+e*N.2) -map(x+->eval(eval(eval(x,s1),s2),s3),H) s4:=map((x,y)+->x=y,[a,b,c,e],c*N.1+e*N.2) map(x+->eval(eval(eval(x,s1),s2),s4),H) removed: - - -\begin{axiom} -)set output tex off -)set output algebra on -\end{axiom} removed: -s1:=solve(imag determinant ρ,ℜp1) -s2:=solve(imag trace(ρ),𝔍p1) -s3:=solve(eval(imag trace(ρ*ρ),s2),ℜq2) -s4:=radicalSolve(eval(eval(imag trace(ρ*ρ*ρ),s2),s3),ℜr3) removed: -Given an operator $ρ \in End V$, one must find the tensor $H=0$ -for unknown manifold of hermitian isomorphisms $h$. changed: -h:Matrix ℂ:=matrix [[ℜa, complex(ℜb,𝔍b), complex(ℜc,𝔍c)], _ - [complex(ℜb,-𝔍b),ℜe, complex(ℜd,𝔍d)], _ - [complex(ℜc,-𝔍c),complex(ℜd,-𝔍d),ℜf ]] -test(h = htranspose h) -H:=htranspose(ρ)*h-h*ρ s1:=solve(imag determinant ρ,ℜp3) s2:=solve(eval(imag trace(ρ),s1),𝔍p1) s3:=solve(eval(eval(imag trace(ρ*ρ),s1),s2),ℜp1) eval(eval(eval(imag trace(ρ*ρ*ρ),s1),s2),s3) --s4:=radicalSolve(eval(eval(eval(imag trace(ρ*ρ*ρ),s1),s2),s3),𝔍q3) --#s4 --s4.1+s4.2 removed: -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 $h$. changed: -K:=concat( map(x+->[real x, imag x], concat(H::List List ?)))::List Polynomial Integer -)set output tex off -)set output algebra on -expr(K::InputForm) ---K2:=groebner(K) ---J:=jacobian(K2, [a,ℜb,𝔍b,ℜc,𝔍c,ℜd,𝔍d,e,f]::List Symbol) C:=eval(eval(eval(characteristicPolynomial ρ,s1),s2),s3); C0:=zerosOf(C); #C0 imag(C0.1) imag(C0.2) imag(C0.3) changed: -The null space (kernel) of the Jacobian -\begin{axiom} ---N:=nullSpace(map(x+->eval(eval(x,s1),s2),J)) -\end{axiom}
A complex vector ℂ-space 
possesses many different hermitian isomorphisms
. In quantum mechanics a given operator
may be said to be 
-hermitian if
 |
axiom
)set output tex off
axiom
)set output algebra on
axiom
ℂ:=Complex Fraction Polynomial Integer
(1) Complex(Fraction(Polynomial(Integer)))
Type: Type
axiom
-- dagger htranspose(h)==map(x+->conjugate(x),transpose h)
Type: Void
axiom
)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
axiom
p1:ℂ:=complex(ℜp1,𝔍p1)
(3) ℜp1 + 𝔍p1 %i
Type: Complex(Fraction(Polynomial(Integer)))
axiom
q1:ℂ:=complex(ℜq1,𝔍q1)
(4) ℜq1 + 𝔍q1 %i
Type: Complex(Fraction(Polynomial(Integer)))
axiom
p2:ℂ:=complex(ℜp2,𝔍p2)
(5) ℜp2 + 𝔍p2 %i
Type: Complex(Fraction(Polynomial(Integer)))
axiom
q2:ℂ:=complex(ℜq2,𝔍q2)
(6) ℜq2 + 𝔍q2 %i
Type: Complex(Fraction(Polynomial(Integer)))
axiom
ρ:Matrix ℂ := matrix [[p1,q1], [p2, q2]]
+ℜp1 + 𝔍p1 %i ℜq1 + 𝔍q1 %i+ (7) | | +ℜp2 + 𝔍p2 %i ℜq2 + 𝔍q2 %i+
Type: Matrix(Complex(Fraction(Polynomial(Integer))))
axiom
s1:=solve(imag determinant ρ,ℜp2)
ℜp1 𝔍q2 - ℜq1 𝔍p2 + ℜq2 𝔍p1 (8) [ℜp2= ---------------------------] 𝔍q1
Type: List(Equation(Fraction(Polynomial(Integer))))
axiom
s2:=solve(eval(imag trace ρ,s1), 𝔍p1)
(9) [𝔍p1= - 𝔍q2]
Type: List(Equation(Fraction(Polynomial(Integer))))
axiom
s3:=solve(eval(eval(imag trace(ρ*ρ),s1), s2), ℜp1)
(10) [0= 0]
Type: List(Equation(Fraction(Polynomial(Integer))))
axiom
eval(eval(imag trace (ρ*ρ),s1), s2)
(11) 0
Type: Fraction(Polynomial(Integer))
axiom
C:=eval(eval(characteristicPolynomial ρ,s1), s2)
(12) 2 2 𝔍q1 𝔍q2 + (ℜq1 ℜq2 - ℜp1 ℜq1)𝔍q2 + 𝔍p2 𝔍q1 + 2 2 ((ℜp1 - %A)ℜq2 - %A ℜp1 + %A )𝔍q1 + ℜq1 𝔍p2 / 𝔍q1
Type: Fraction(Polynomial(Complex(Integer)))
axiom
C0:=zerosOf(C)
(13) [ ROOT 2 2 - 4𝔍q1 𝔍q2 + (- 4ℜq1 ℜq2 + 4ℜp1 ℜq1)𝔍q2 - 4𝔍p2 𝔍q1 + 2 2 2 (ℜq2 - 2ℜp1 ℜq2 + ℜp1 )𝔍q1 - 4ℜq1 𝔍p2 / 𝔍q1 + ℜq2 + ℜp1 / 2 ,
- ROOT 2 2 - 4𝔍q1 𝔍q2 + (- 4ℜq1 ℜq2 + 4ℜp1 ℜq1)𝔍q2 - 4𝔍p2 𝔍q1 + 2 2 2 (ℜq2 - 2ℜp1 ℜq2 + ℜp1 )𝔍q1 - 4ℜq1 𝔍p2 / 𝔍q1 + ℜq2 + ℜp1 / 2 ]
Type: List(Expression(Complex(Integer)))
axiom
#C0
(14) 2
Type: PositiveInteger?
axiom
imag(C0.1)
(15) 0
Type: Expression(Integer)
axiom
imag(C0.2)
(16) 0
Type: Expression(Integer)
Given an operator 
, one must find the tensor 
for unknown manifold of hermitian isomorphisms .
axiom
h:Matrix ℂ:=matrix [[a,complex(b, c)], [complex(b, -c), e]]
+ a b + c %i+ (17) | | +b - c %i e +
Type: Matrix(Complex(Fraction(Polynomial(Integer))))
axiom
test(h = htranspose h)
axiom
Compiling function htranspose with type Matrix(Complex(Fraction(
Polynomial(Integer)))) -> Matrix(Complex(Fraction(Polynomial(
Integer))))
(18) trueType: Boolean
axiom
H:=htranspose(ρ)*h-h*ρ
(19) [ [(- 2b 𝔍p2 - 2a 𝔍p1 - 2c ℜp2)%i,
c 𝔍q2 + c 𝔍p1 - b ℜq2 - a ℜq1 + e ℜp2 + b ℜp1 + (- b 𝔍q2 - a 𝔍q1 - e 𝔍p2 - b 𝔍p1 - c ℜq2 + c ℜp1)%i ] ,
[ - c 𝔍q2 - c 𝔍p1 + b ℜq2 + a ℜq1 - e ℜp2 - b ℜp1 + (- b 𝔍q2 - a 𝔍q1 - e 𝔍p2 - b 𝔍p1 - c ℜq2 + c ℜp1)%i ,(- 2e 𝔍q2 - 2b 𝔍q1 + 2c ℜq1)%i] ]
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 .
axiom
J:=jacobian(concat( map(x+->[real x,imag x], concat(H::List List ?)) ), [a, b, c, e]::List Symbol)
+ 0 0 0 0 + | | |- 2𝔍p1 - 2𝔍p2 - 2ℜp2 0 | | | |- ℜq1 - ℜq2 + ℜp1 𝔍q2 + 𝔍p1 ℜp2 | | | |- 𝔍q1 - 𝔍q2 - 𝔍p1 - ℜq2 + ℜp1 - 𝔍p2 | (20) | | | ℜq1 ℜq2 - ℜp1 - 𝔍q2 - 𝔍p1 - ℜp2 | | | |- 𝔍q1 - 𝔍q2 - 𝔍p1 - ℜq2 + ℜp1 - 𝔍p2 | | | | 0 0 0 0 | | | + 0 - 2𝔍q1 2ℜq1 - 2𝔍q2+
Type: Matrix(Fraction(Polynomial(Integer)))
The null space (kernel) of the Jacobian
axiom
N:=nullSpace(map(x+->eval(eval(x,s1), s2), J))
- ℜq2 + ℜp1 ℜq1 𝔍p2 𝔍q2 (21) [[-----------,---, 1, 0], [- ---, - ---, 0, 1]] 𝔍q1 𝔍q1 𝔍q1 𝔍q1
Type: List(Vector(Fraction(Polynomial(Integer))))
gives the general solution to the problem.
axiom
s4:=map((x,y)+->x=y, [a, b, c, e], c*N.1+e*N.2)
- e 𝔍p2 - c ℜq2 + c ℜp1 - e 𝔍q2 + c ℜq1 (22) [a= -----------------------,b= ---------------, c= c, e= e] 𝔍q1 𝔍q1
Type: List(Equation(Fraction(Polynomial(Integer))))
axiom
map(x+->eval(eval(eval(x,s1), s2), s4), H)
+0 0+ (23) | | +0 0+
Type: Matrix(Fraction(Polynomial(Complex(Integer))))
Three-Dimensions
axiom
p1:ℂ:=complex(ℜp1,𝔍p1)
(24) ℜp1 + 𝔍p1 %i
Type: Complex(Fraction(Polynomial(Integer)))
axiom
q1:ℂ:=complex(ℜq1,𝔍q1)
(25) ℜq1 + 𝔍q1 %i
Type: Complex(Fraction(Polynomial(Integer)))
axiom
r1:ℂ:=complex(ℜr1,𝔍r1)
(26) ℜr1 + 𝔍r1 %i
Type: Complex(Fraction(Polynomial(Integer)))
axiom
p2:ℂ:=complex(ℜp2,𝔍p2)
(27) ℜp2 + 𝔍p2 %i
Type: Complex(Fraction(Polynomial(Integer)))
axiom
q2:ℂ:=complex(ℜq2,𝔍q2)
(28) ℜq2 + 𝔍q2 %i
Type: Complex(Fraction(Polynomial(Integer)))
axiom
r2:ℂ:=complex(ℜr2,𝔍r2)
(29) ℜr2 + 𝔍r2 %i
Type: Complex(Fraction(Polynomial(Integer)))
axiom
p3:ℂ:=complex(ℜp3,𝔍p3)
(30) ℜp3 + 𝔍p3 %i
Type: Complex(Fraction(Polynomial(Integer)))
axiom
q3:ℂ:=complex(ℜq3,𝔍q3)
(31) ℜq3 + 𝔍q3 %i
Type: Complex(Fraction(Polynomial(Integer)))
axiom
r3:ℂ:=complex(ℜr3,𝔍r3)
(32) ℜr3 + 𝔍r3 %i
Type: Complex(Fraction(Polynomial(Integer)))
axiom
ρ:Matrix ℂ := matrix [[p1,q1, r1], [p2, q2, r2], [p3, q3, r3]]
+ℜp1 + 𝔍p1 %i ℜq1 + 𝔍q1 %i ℜr1 + 𝔍r1 %i+ | | (33) |ℜp2 + 𝔍p2 %i ℜq2 + 𝔍q2 %i ℜr2 + 𝔍r2 %i| | | +ℜp3 + 𝔍p3 %i ℜq3 + 𝔍q3 %i ℜr3 + 𝔍r3 %i+
Type: Matrix(Complex(Fraction(Polynomial(Integer))))
axiom
s1:=solve(imag determinant ρ,ℜp3)
(34) [ ℜp3 = (𝔍p1 𝔍q2 - 𝔍p2 𝔍q1 - ℜp1 ℜq2 + ℜp2 ℜq1)𝔍r3 + (- 𝔍p1 𝔍q3 + 𝔍p3 𝔍q1 + ℜp1 ℜq3)𝔍r2 + (𝔍p2 𝔍q3 - 𝔍p3 𝔍q2 - ℜp2 ℜq3)𝔍r1 + (ℜp1 ℜr2 - ℜp2 ℜr1)𝔍q3 - ℜp1 ℜr3 𝔍q2 + ℜp2 ℜr3 𝔍q1 + (- ℜq1 ℜr2 + ℜq2 ℜr1)𝔍p3 + (ℜq1 ℜr3 - ℜq3 ℜr1)𝔍p2 + (- ℜq2 ℜr3 + ℜq3 ℜr2)𝔍p1 / ℜq1 𝔍r2 - ℜq2 𝔍r1 - ℜr1 𝔍q2 + ℜr2 𝔍q1 ]
Type: List(Equation(Fraction(Polynomial(Integer))))
axiom
s2:=solve(eval(imag trace(ρ),s1), 𝔍p1)
(35) [𝔍p1= - 𝔍r3 - 𝔍q2]
Type: List(Equation(Fraction(Polynomial(Integer))))
axiom
s3:=solve(eval(eval(imag trace(ρ*ρ),s1), s2), ℜp1)
(36) [ ℜp1 = 2 - 𝔍q2 𝔍r1 𝔍r3 + (𝔍q3 𝔍r1 + ℜq1 ℜr3)𝔍r2 + 2 (- 𝔍q2 - 𝔍p2 𝔍q1 - ℜq3 ℜr2 + ℜp2 ℜq1)𝔍r1 - ℜr1 ℜr3 𝔍q2 + ℜr2 ℜr3 𝔍q1 * 𝔍r3 + 2 ℜq1 ℜq3 𝔍r2 + (𝔍q2 𝔍q3 + 𝔍p3 𝔍q1 - ℜq2 ℜq3)𝔍r1 + ℜq1 ℜr2 𝔍q3 + (- ℜq3 ℜr1 + ℜq1 ℜq2)𝔍q2 + (ℜq3 ℜr2 + ℜp2 ℜq1)𝔍q1 + ℜq1 ℜr1 𝔍p3 + 2 ℜq1 𝔍p2 * 𝔍r2 + 2 (𝔍p2 𝔍q3 - 𝔍p3 𝔍q2 - ℜp2 ℜq3)𝔍r1 + 2 (- ℜq2 ℜr2 - ℜp2 ℜr1)𝔍q3 + (ℜq2 ℜr3 - ℜq3 ℜr2 - ℜq2 )𝔍q2 + (ℜp2 ℜr3 - ℜp2 ℜq2)𝔍q1 - ℜq1 ℜr2 𝔍p3 + (ℜq1 ℜr3 - ℜq3 ℜr1 - ℜq1 ℜq2)𝔍p2 * 𝔍r1 + 2 2 (- ℜr1 ℜr2 𝔍q2 + ℜr2 𝔍q1)𝔍q3 - ℜq2 ℜr1 𝔍q2 + 2 2 ((ℜq2 ℜr2 - ℜp2 ℜr1)𝔍q1 - ℜr1 𝔍p3 - ℜq1 ℜr1 𝔍p2)𝔍q2 + ℜp2 ℜr2 𝔍q1 + (ℜr1 ℜr2 𝔍p3 + ℜq1 ℜr2 𝔍p2)𝔍q1 / (ℜq1 𝔍r2 - ℜr1 𝔍q2 + ℜr2 𝔍q1)𝔍r3 + (- ℜq3 𝔍r1 + ℜq1 𝔍q2)𝔍r2 + 2 (- ℜr2 𝔍q3 + (ℜr3 - ℜq2)𝔍q2)𝔍r1 - ℜr1 𝔍q2 + ℜr2 𝔍q1 𝔍q2 ]
Type: List(Equation(Fraction(Polynomial(Integer))))
axiom
eval(eval(eval(imag trace(ρ*ρ*ρ),s1), s2), s3)
(37) 0
Type: Fraction(Polynomial(Integer))
axiom
C:=eval(eval(eval(characteristicPolynomial ρ,s1), s2), s3);
Type: Fraction(Polynomial(Complex(Integer)))
axiom
C0:=zerosOf(C);
Type: List(Expression(Complex(Integer)))
axiom
#C0
(40) 3
Type: PositiveInteger?
axiom
imag(C0.1)
(41) 0
Type: Expression(Integer)
axiom
imag(C0.2)
(42) 0
Type: Expression(Integer)
axiom
imag(C0.3)
(43) 0
Type: Expression(Integer)