# Orthogonal eigenvectors and measurement

• I
• entropy1
In summary: No. The infinite dimensions in the Hilbert space are already there before the measurement, and the wave function is a function on the same Hilbert space after the measurement as before. In the MWI, "measurement" is just another unitary evolution process, and unitary evolution doesn't "create" anything; it's an information-preserving reversible process.
entropy1
An outcome of a measurement in a (infinite) Hilbert space is orthogonal to all possible outcomes except itself! This sounds related to the measurement problem to me, for we inherently only obtain a single outcome. So, to take a shortcut I posted this question so I quickly get to hear where I'm wrong, if I am.

entropy1 said:
An outcome of a measurement in a (infinite) Hilbert space is orthogonal to all possible outcomes except itself! This sounds related to the measurement problem to me, for we inherently only obtain a single outcome. So, to take a shortcut I posted this question so I quickly get to hear where I'm wrong, if I am.
You're not even wrong!

Your post mixes up measurement outcomes with states. States are vectors and may be orthogonal to each other. Measurement outcomes are real numbers.

vanhees71
PeroK said:
Your post mixes up measurement outcomes with states. States are vectors and may be orthogonal to each other. Measurement outcomes are real numbers.
Yes, the outcomes are eigenvalues and those are paired with eigenvectors, so you can map the measurement outcome (eigenvalue) onto the eigenvector on which the wavefunction collapsed, right? I mean to indicate that the math is already implicating a single outcome, except if you assume all outcomes simultaneously, like in MWI.

entropy1 said:
Yes, the outcomes are eigenvalues and those are paired with eigenvectors, so you can map the measurement outcome (eigenvalue) onto the eigenvector on which the wavefunction collapsed, right? I mean to indicate that the math is already imposing a single outcome, except if you assume all outcomes simultaneously, like in MWI.
The maths doesn't impose a single outcome, in the sense that we chose that mathematics because we only see a single outcome.

In general the wave function collapses onto an eigenspace, which may be spanned by several orthogonal eigenstates. So, although we get one result, we don't necessarily have a 1D eigenstate as the result of a single measurement.

Only by observing all relevant compatible observables do you get a definite eigenvector.

PeroK said:
The maths doesn't impose a single outcome, in the sense that we chose that mathematics because we only see a single outcome.

In general the wave function collapses onto an eigenspace, which may be spanned by several orthogonal eigenstates. So, although we get one result, we don't necessarily have a 1D eigenstate as the result of a single measurement.

Only by observing all relevant compatible observables do you get a definite eigenvector.
So I think you are saying that we don't necessarily have collapse in the math, which would suit for instance MWI, am I right?

Is it just the way you look at it?

entropy1 said:
An outcome of a measurement in a (infinite) Hilbert space is orthogonal to all possible outcomes except itself! This sounds related to the measurement problem to me, for we inherently only obtain a single outcome. So, to take a shortcut I posted this question so I quickly get to hear where I'm wrong, if I am.
I assume you mean infinite-dimensional? I don't think there are cardinality-wise finite Hilbert spaces.

entropy1 said:
So I think you are saying that we don't necessarily have collapse in the math, which would suit for instance MWI, am I right?
Yes, collapse is Copenhagen, for example. In many ways it's the simplest interpretation.

Collapse is part of the interpretation of the maths.

If the operator of the measurement is defined in infinite-dimensional Hilbert space, does that mean that if we apply MWI, that an infinite number of world-branches is created by the act of measurement?

entropy1 said:
I think you are saying that we don't necessarily have collapse in the math

We have a thing in the math that is called "collapse", but the math itself, the machinery that calculates predictions for experimental results, doesn't tell you whether this "collapse" is a real physical process or just a mathematical technique that let's you make correct predictions.

The mathematical technique that is sometimes called "collapse" is step 7 in the following Insights article:

https://www.physicsforums.com/insights/the-7-basic-rules-of-quantum-mechanics/
PeroK said:
Collapse is part of the interpretation of the maths.

In the sense that "collapse" refers to something more than just the mathematical technique that let's you make correct predictions, yes. See above.

Moderator's note: Moved thread to QM interpretations and foundations forum.

entropy1 said:
If the operator of the measurement is defined in infinite-dimensional Hilbert space, does that mean that if we apply MWI, that an infinite number of world-branches is created by the act of measurement?

No. The infinite dimensions in the Hilbert space are already there before the measurement, and the wave function is a function on the same Hilbert space after the measurement as before. In the MWI, "measurement" is just another unitary evolution process, and unitary evolution doesn't "create" anything; it's an information-preserving reversible process.

## 1. What is the significance of orthogonal eigenvectors in measurement?

Orthogonal eigenvectors are important in measurement because they represent the directions along which a system's properties are independent of each other. This means that measurements along these directions will not affect each other, allowing for more accurate and reliable data.

## 2. How are orthogonal eigenvectors used in quantum mechanics?

In quantum mechanics, orthogonal eigenvectors are used to represent the possible states of a quantum system. These states are then used to calculate the probabilities of different outcomes when the system is measured.

## 3. Can orthogonal eigenvectors be non-orthogonal in some cases?

No, by definition, orthogonal eigenvectors must be perpendicular to each other. If they were not, then they would not be considered orthogonal.

## 4. How are orthogonal eigenvectors determined experimentally?

Orthogonal eigenvectors can be determined experimentally by performing measurements in different directions and observing the results. If the measurements are independent of each other, then the corresponding eigenvectors are orthogonal.

## 5. What is the relationship between orthogonal eigenvectors and eigenvalues?

Orthogonal eigenvectors and eigenvalues are closely related. The eigenvalues represent the magnitude of the change in the system's properties when measured along the corresponding eigenvector. Orthogonal eigenvectors ensure that these changes are independent of each other.

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