Meaning of "Up to a Scale" for Eigenvectors in Quantum Mechanics

[Gary Roach]

Homework Statement

What is the meaning of the phrase "up to a scale" as applied to eigenvectors in quantum mechanics.

Homework Equations

None

The Attempt at a Solution

N/A did an extensive search of the web and my texts. No joy.

 

[Mark44]

I'll guess that this means that two eigenvectors point in the same or opposite direction. IOW, each one is a scalar multiple of the other.

 

[Gary Roach]

I guess I wasn't specific enough in my question. Sorry. The definition I would like is as the phrase applies to the following statement:

ie $\Lambda|\omega_i>$ is an eigenvector of $\Omega$ with eigenvalue $\omega_i$. Since the vector is unique $\underline{up\ to\ a\ scale}$,

$\Lambda|\omega_i > = \lambda_i | \omega_i >$

 

[HallsofIvy]

Yes, that is exactly what Mark44 was referring to. If v is an eigenvector of linear transformation, A, with eigenvector $\lambda$, then $Av= \lambda v$. If u is any "scalar multiple" of v, u= sv for some scalar, s, then, since A is linear, $Au= A= (sv)= s(Av)= s(\lambda v)= \lambda(sv)= \lambda u$ so that u is also an eigenvector with eigenvalue $\lambda$. That is, the eigenvector is unique "up to a scalar multiple" which is, I presume, what this physics text means by "up to scale". (You might want to recheck the exact wording. "Up to a scale" doesn't seem grammatically correct.)

 

[Gary Roach]

This is from the proof of theorem 13 page 44 of Principles of Quantum Mechanics- 2nd edition by R Shanker. The up to scale is a direct quote.

Thanks all for the help.

Gary R.

 

[HallsofIvy]

This is from the proof of theorem 13 page 44 of Principles of Quantum Mechanics- 2nd edition by R Shanker. The up to scale is a direct quote.

But what you said before was "up to a scale" which is not a direct quote.

Thanks all for the help.

Gary R.

 

[Gary Roach]

I rechecked the text. The actual statement is "up to a scale". I goofed in the second message. Sorry

Gary R.