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GG = {N3*Conjugate[N3]+ N4*Conjugate[N4] == 1, (2 N3 + (-I + Sqrt[2]) N4)/(2 Sqrt[3]) == 0};
Sols = Solve[GG, {N3, N4}]

I need to solve the above equation. But I didn't get an answer!!! Can you please help. This N3 and N4 will be sometimes complex number or real number or imaginary number.

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    $\begingroup$ FullSimplify[Sols] I get on Mathematica 12.1.1.0: {N3 -> ConditionalExpression[-(1/2) (-I + Sqrt[2]) N4, And[ Or[Im[N4] + (Rational[4, 7] - Re[N4]^2)^Rational[1, 2] == 0, Im[N4] == (Rational[4, 7] - Re[N4]^2)^Rational[1, 2]], Inequality[(-2) 7^Rational[-1, 2], LessEqual, Re[N4], LessEqual, 2 7^Rational[-1, 2]]]]} $\endgroup$
    – flinty
    Nov 19, 2020 at 0:49

3 Answers 3

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Split the complex variables N3,N4 into real and imaginary part and ComplexExpand it.

GG = {N3*Conjugate[N3]+ N4*Conjugate[N4] == 1, (2 N3 + (-I + Sqrt[2]) N4)/(2 Sqrt[3]) == 0};

GG2 = GG /. {N3 -> n31 + I n32, N4 -> n41 + I n42}

ceRe = ComplexExpand[Re[GG2 /. Equal -> Subtract], 
         TargetFunctions -> {Re, Im}] // Simplify

ceIm = ComplexExpand[Im[GG2 /. Equal -> Subtract], 
         TargetFunctions -> {Re, Im}
] // Simplify

You get only solutions for three of the four variable, one you can choose free within a given range.

Here choose n42 from -2/Sqrt[7] to 2/Sqrt[7] and look at the solutions for the other variables.

sol = Solve[Join[Thread[ceRe == 0], {ceIm[[2]] == 0}], {n31, n32, n42, n41},
    Reals] // Simplify

Plot[Evaluate[{n31, n32, n41} /. sol[[1 ;; 2]]], {n42, -(2/Sqrt[7]), 
   2/Sqrt[7]}, PlotStyle -> {Red, Green, Blue
}]

enter image description here

Or choose n31 free in the range +-Sqrt[3/7]

sol2 = Solve[Join[Thread[ceRe == 0], {ceIm[[2]] == 0}], {n32, n41, n31, n42},
   Reals] // Simplify

Plot[Evaluate[{n32, n41, n42} /. sol2[[1 ;; 2]]], {n31, -Sqrt[(3/7)], 
  Sqrt[3/7]}, PlotStyle -> {Green, Blue, Magenta
}]
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Clear["Global`*"]

GG = {N3*Conjugate[N3] + N4*Conjugate[N4] == 
    1, (2 N3 + (-I + Sqrt[2]) N4)/(2 Sqrt[3]) == 0};

There are an unlimited number of solutions

Sols = Solve[GG, {N3, N4}][[1]]

enter image description here

The condition is

cond = Sols[[1, -1, -1]]

(* (Im[N4] == -(Sqrt[4 - 7 Re[N4]^2]/Sqrt[7]) && -(2/Sqrt[7]) <= Re[N4] <= 2/
    Sqrt[7]) || (Im[N4] == Sqrt[4 - 7 Re[N4]^2]/Sqrt[7] && -(2/Sqrt[7]) <= 
    Re[N4] <= 2/Sqrt[7]) *)

Some examples of N4 values

inst = FindInstance[cond, N4, 5, RandomSeeding -> 1234] // FullSimplify

(* {{N4 -> -((I (-805 I + Sqrt[2438247]))/2324)}, {N4 -> -(2/Sqrt[7])}, {N4 -> (
   I (1085 I + Sqrt[1909047]))/2324}, {N4 -> 2/Sqrt[
   7]}, {N4 -> -((I (-525 I + Sqrt[2810647]))/2324)}} *)

The associated {N3, N4} pairs are

{N3, N4} /. Sols /. inst // Simplify

(* {{((1 + I Sqrt[2]) (-805 I + Sqrt[2438247]))/
  4648, -((I (-805 I + Sqrt[2438247]))/2324)}, {(-I + Sqrt[2])/Sqrt[
  7], -(2/Sqrt[7])}, {((-I + Sqrt[2]) (1085 - I Sqrt[1909047]))/4648, (
  I (1085 I + Sqrt[1909047]))/2324}, {-((-I + Sqrt[2])/Sqrt[7]), 2/Sqrt[
  7]}, {((1 + I Sqrt[2]) (-525 I + Sqrt[2810647]))/
  4648, -((I (-525 I + Sqrt[2810647]))/2324)}} *)
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Reduce work.

GG = {N3*Conjugate[N3] + N4*Conjugate[N4] == 
    1, (2 N3 + (-I + Sqrt[2]) N4)/(2 Sqrt[3]) == 0};
Sols = Reduce[GG, {N3, N4}]

-Sqrt[(3/7)] <= Re[N3] <= Sqrt[3/
  7] && (Im[N3] == -(Sqrt[3 - 7 Re[N3]^2]/Sqrt[7]) || 
   Im[N3] == Sqrt[3 - 7 Re[N3]^2]/Sqrt[7]) && 
 N4 == -((2 N3)/(-I + Sqrt[2]))
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