Representation of vectors in a new basis using Dirac notation?

[Vitani11]

Homework Statement

I have a vector V with components v₁, v₂in some basis and I want to switch to a new (orthonormal) basis a,b whose components in the old basis are given. I want to find the representation of vector V in the new orthonormal basis i.e. find the components v_a,v_b such that |v⟩ = v_a|a⟩+v_b|b⟩.

Homework Equations

Original vector |v⟩ = (1+i)₁ (√3+i)₂
Vector |a⟩ = [(1+i√3)/4]₁, [(-√3(1+i))/√8]₂
Vector |b⟩ = [(√3(1+i))/√8]₁[(i+√3)/4]₂
new vector |v⟩ = v_a|a⟩+v_b|b⟩.
Where the subscripts denote the row number.

The Attempt at a Solution

I took the inner product between the given vectors |a⟩ and |v⟩ for v_a and |b⟩ and |v⟩ for v_b: v_a = ⟨a|v⟩ and v_b=⟨b|v⟩ I don't think this is right - because for this to be true the new vector v using these components would have a norm which is invariant under the transformation and it does not.

 

[kuruman]

I don't think this is right - because for this to be true the new vector v using these components would have a norm which is invariant under the transformation and it does not.

Can you show what you did that made you think the norm is not invariant?

 

[Vitani11]

I just thought this was true - if I transform an orthonormal basis to a new one then I thought the norm shouldn't change.

 

[kuruman]

just thought this was true - if I transform an orthonormal basis to a new one then I thought the norm shouldn't change.

It shouldn't, I agree. Why did you imply in your first post that it is not invariant? Or did I miss what you asked?

 

[Vitani11]

I mentioned that the norm should be invariant because if I took the norm of the new vector generated by the process I stated above then it should be the same as the norm of the original vector v which it is not. So it is basically a way to check whether or not I performed the right calculation to find the new vector. The norm of of the new vector that I generated is not invariant with respect to the original vector v. So I know I did something wrong there. I just want to know how to find the representation of the given vector v in terms of a new orthonormal basis a and b.

 

[Vitani11]

Why would the inner product between the original vector v and the new vectors a or b not give the correct answers for the components v_a and v_b of the new basis (respectively)?

 

[kuruman]

Why would the inner product between the original vector v and the new vectors a or b not give the correct answers for the components va and vb of the new basis (respectively)?

How do you know the inner product between the original vector v and the new vectors does not give the correct answers? Where is your work that supports this claim? Maybe you made a mistake somewhere.

 

[Vitani11]

I took the inner product between the new vector and itself and the inner product of the original vector with itself and this gave answers that are really different. Is that not right?

 

[Vitani11]

I have added the vectors into the first post.

 

[kuruman]

Original vector |v⟩ = (1+i)₁ (√3+i)₂

It looks like you are unsure about DIrac notation.
This should be written as $|v> = (1+i)|1> + (\sqrt{3}+i)|2>$. Kets are not subscripts. Then
$<v|v> =[(1+i)^*<1| + (\sqrt{3}+i)^*<2| ][(1+i)|1> + (\sqrt{3}+i)|2>]$
Perhaps you did not realize that $(a|v>)^*= a^*<v|$.

 

[Vitani11]

I know that. When you find that norm it turns out to be 6. I'm working out another trial right now.