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

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Homework Statement


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


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The Attempt at a Solution


N/A did an extensive search of the web and my texts. No joy.
 
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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.
 
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 >
 
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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.)
 
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.
 
Gary Roach said:
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.
 
I rechecked the text. The actual statement is "up to a scale". I goofed in the second message. Sorry

Gary R.
 
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