Why this combination and not another

1. May 6, 2014

M. next

Why did the author of the book choose the combinations in (9.66) and (9.67) as follows? Why didn't he choose some other combination; say 1/sqrt(3) and the other 2/sqrt(3) instead of 1/sqrt(2) and 1/sqrt(2)?

Please see attachment. Note that this is problem on stark effect.

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2. May 6, 2014

Matterwave

Different combinations lead to different vectors...the author found the "eigenvectors" corresponding to the specific eigenvalues, these eigenvectors for non-degenerate eigenvalues are unique up to normalization.

3. May 6, 2014

ChrisVer

I guess a physical explanation is that because they both build up $\psi$ state with equal contributions...

4. May 7, 2014

M. next

Then it is ok to choose whatever normalization I want? I can go for the 1/sqrt(3) and the other 2/sqrt(3) here?

5. May 7, 2014

Staff: Mentor

No, all eigenstates need to have the same nomalization, and in most cases you want the norm to be 1.

6. May 7, 2014

Bill_K

You need to look back in your textbook or your notes and review what an eigenvector is.

7. May 7, 2014

M. next

I know what an eigenvector is and I know how to normalize it. Let's say we have |+> I know it is already normalized. When I have (1 2)^T I know that 1/sqrt*5 is the normalization factor. But how would I know how to normalize |2 0 0>? |2 1 0>?
It is probably the notation of the ket including n, l, m that I am not being capable to translate into something familiar like a column vector....

8. May 7, 2014

Staff: Mentor

For the kets $|a,b,c \rangle$ to form a basis, they must have the same norm. Therefore, to form a new state $\psi$ as $|a,b,c \rangle + |a',b',c' \rangle$ with the same norm requires
$$\psi = \frac{1}{\sqrt{2}} \left( |a,b,c \rangle + |a',b',c' \rangle \right)$$

9. May 7, 2014

Fredrik

Staff Emeritus
I haven't looked at this specific case, but it's conventional to use notations like |210> only for vectors with norm 1. So these guys are almost certainly already normalized. If you're given a linear combination like $|200\rangle +|210\rangle$, and you want to normalize it, you need to keep in mind what it means to normalize a vector. It means to find a vector in the same 1-dimensional subspace that has norm 1. The vector $a|200\rangle +b|210\rangle$ isn't in the same 1-dimensional subspace as $|200\rangle +|210\rangle$ unless a=b. So you need to ask yourself this: For what values of $a$ does $a\big(|200\rangle +|210\rangle\big)$ have norm 1? The first step is of course to calculate the norm of this vector. Do you know how to do that? (Assume that |200> and |210> are already normalized).

There are infinitely many complex numbers $a$ that get the job done, but only one of them has imaginary part zero and a positive real part. It's convenient to choose that one.

10. May 7, 2014

M. next

Oh great!!!! Thank you for making this so clear Dr Claude!