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

• Gary Roach
In summary, "up to a scale" in quantum mechanics refers to the fact that eigenvectors are unique up to a scalar multiple. This means that any scalar multiple of an eigenvector is also an eigenvector with the same eigenvalue. This concept is used in the proof of theorem 13 on page 44 of Principles of Quantum Mechanics by R Shanker.
Gary Roach

## Homework Statement

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

None

## The Attempt at a Solution

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

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 $\underline{up\ to\ a\ scale}$,

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

Last edited:
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.

## 1. What does it mean for an eigenvector to be "up to a scale" in quantum mechanics?

"Up to a scale" refers to the fact that eigenvectors in quantum mechanics can be multiplied by a scalar (a number) without changing their physical significance. This is because the magnitude of an eigenvector is not important, only its direction matters in determining the state of a quantum system.

## 2. How does the concept of "up to a scale" apply to the measurement of quantum systems?

In quantum mechanics, measurements are represented by operators, and the eigenvectors of these operators represent the possible outcomes of the measurement. The concept of "up to a scale" allows us to consider all possible scalar multiples of these eigenvectors as valid measurement outcomes.

## 3. Can the "up to a scale" property of eigenvectors be applied to all quantum systems?

Yes, the concept of "up to a scale" applies to all quantum systems, as it is a fundamental property of eigenvectors in linear algebra. However, it is particularly relevant in quantum mechanics where the physical interpretation of an eigenvector is determined by its direction rather than its magnitude.

## 4. How does the "up to a scale" property impact the calculations of quantum states?

The "up to a scale" property allows us to simplify calculations in quantum mechanics, as we can choose to represent a quantum state using the eigenvector with the most convenient magnitude, without affecting the physical interpretation of the state. This concept is particularly useful in complex calculations involving multiple quantum systems.

## 5. Are there any limitations to the use of the "up to a scale" property in quantum mechanics?

The "up to a scale" property is a useful tool in quantum mechanics, but it is important to remember that it only applies to eigenvectors, not all vectors in a quantum system. Additionally, this property may not hold in certain scenarios, such as when dealing with non-linear operators or when considering the normalization of states.

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