State space and its subspace : finding a basis

[max_jammer]

Hello.

I really need help with this one:

Homework Statement

I have a 3 dimensional state space H and its subspace H₁ which is spanned with

|Psi> = a x₁ + b x₂ + c x₃
and
|Psi'> = d x₁ + e x₂ + f x₃

Those two "rays" are linearly independent and x₁, x₂, and x₃ is an (orthonormal) basis for H.

Now I need to find coefficients g, h and i so that
|psi_othogonal> = g x₁ + h x₂ + i x₃
is not a trivial element of the subspace H₁_orthogonal.

Homework Equations

not sure

The Attempt at a Solution

I think that I need to find another element in H which is orthogonal both to |Psi> and |Psi'>

In R³ I would normally use cross product to find the third base vector but how does this transforms to the complex valued coefficients?

I think it may be:

g = (bf~ - ce~)
h = (cd~ - af~)
i = (ae~ - bd~)

where f~ means complex conjugate of f.

but is this right?

Also how would I go about prooving this? What pops in my mind is to build scalar products and see if they give 0...

Thanks a lot

/Nathan

 

[vela]

I think it may be:

g = (bf~ - ce~)
h = (cd~ - af~)
i = (ae~ - bd~)

where f~ means complex conjugate of f.

but is this right?

Also how would I go about prooving this? What pops in my mind is to build scalar products and see if they give 0...

Try calculating the scalar products and see what you get. If they're 0, you have your answer.

 

[max_jammer]

Thanks for the replay.

I tried to calculate the scalar product but it doesn't work; it all boils down to where I put conjugate. if I put conjugate on

g = (~bf - ~ce)
h = (~cd - ~af)
i = (~ae - ~bd)

then the scalar product of this vector and |psi> is zero BUT
it is not zero with |psi'>...

Or am I missing something?

~z w is generally not equal to z ~w ?

any help is appreciated...

 

[vela]

Try going the other way. Set the inner products to 0 and to get two equations involving g, h, and i, which you can solve up to a multiplicative constant.

 

[max_jammer]

I did that before and I got expressions for g and h with i as a parameter. (1 equations and 3 unknowns - no surprise there) But I have no idea what i might be. If I were dealing with real numbers I would just set i = 1 and normalize the (g h i) vector, but these are complex numbers and I don't think I can do the same thing.

anyway I got:

g = i (b~ f~ - c~ e~) / (a~ e~ - b~ d~)
and
h = (-d~ g - f~ i) / e~

what am I missing? either my complex algebra is too rusty or there is some fundamental physical relations I didn't think of.

Please help me; this has been bugging me for a week now.

 

[vela]

I did that before and I got expressions for g and h with i as a parameter. (1 equations and 3 unknowns - no surprise there) But I have no idea what i might be. If I were dealing with real numbers I would just set i = 1 and normalize the (g h i) vector, but these are complex numbers and I don't think I can do the same thing.

Why not? You don't actually have to set i to 1. You could just normalize the vector, which will allow you to solve for |i|. You still have the freedom to set an arbitrary complex phase, which you can take to be 0.