# Confused about the spectrum of an observable

• I
In summary, the conversation discusses the concept of eigenvalues in quantum mechanics and their relationship to probability amplitudes and measurements. The spectrum of an observable is determined by the potential of the system, while the measurement of an eigenvalue is influenced by the actual state of the system. The state vector can predict the probability of a measured value based on the inner product with the corresponding eigenstate. The conversation concludes with a humorous reference to a popular Gary Larson cartoon.
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TL;DR Summary
Eigenvalues of an observable are probability amplitudes but also stated to be measurement values. What am I missing?
This is a very elementary question, from the beginnings of quantum mechanics.

For simplicity, I refer to a finite case with pure states.

If I understand correctly, the spectrum of an observable is the collection of eigenvalues formed by the inner product of states and hence equal to probability amplitudes; they are then associated to the possible values of a measurement.

Hence I am confused by statements such as the following

“Eigenvalues of observables are real and in fact are possible outcomes of measurements of a given observable.” (https://www.quantiki.org/wiki/observables-and-measurements, but not the only example.)

How can the square roots of probabilities, which are less than one, be values of measurements (which can be greater than one)?

Eigenvalues are not the same as probability amplitudes. The wave function that gives the prob. amplitudes is one representation of the eigenstate, and it can be either a position or momentum representation.

Summary:: Eigenvalues of an observable are probability amplitudes but also stated to be measurement values. What am I missing?

If I understand correctly, the spectrum of an observable is the collection of eigenvalues formed by the inner product of states and hence equal to probability amplitudes
Only the green part of the sentence is required. The possible eigenvalues of the system are determined from the potential independent of the actual state of the system. This is the spectrum.
The result of any measurement will be determined by the actual state of the system. The measurement will yield an eigenvalue. The state vector of the system will predict the probability of that value being measured (i.e if you repeated the measurement on similarly prepared states blah blah) which is given by the inner product of the state vector of the system with the corresponding eigenstate.

.

Thank you very much, hutchphd and hilbert2. An explanation similar to the last two sentences of hutchphd's explanation apparently was picked up by me somewhere and morphed into the version I posted. This clears it up.

hutchphd
This clears it up.
Good.
May I congratulate you on your Gary Larson cartoon. Perhaps my all-time favorite, and that's a difficult choice.

## 1. What is the spectrum of an observable?

The spectrum of an observable refers to the range of possible values that can be observed for a particular physical quantity. This can include things like energy levels, wavelengths, or frequencies.

## 2. How is the spectrum of an observable determined?

The spectrum of an observable is determined through experiments and observations. Scientists use specialized instruments and techniques to measure and record the different values that can be observed for a particular quantity.

## 3. What factors can affect the spectrum of an observable?

Several factors can affect the spectrum of an observable, including the properties of the material being observed, external forces or fields, and the measurement method used. Additionally, the spectrum can also be affected by the precision and accuracy of the measuring instruments.

## 4. Can the spectrum of an observable change over time?

Yes, the spectrum of an observable can change over time. This can occur due to various factors such as changes in the material being observed, external influences, or advancements in measurement techniques. It is important for scientists to regularly review and update the spectrum of an observable as new information and data become available.

## 5. How is the spectrum of an observable used in scientific research?

The spectrum of an observable is a crucial tool in scientific research. It allows scientists to accurately measure and analyze physical quantities, make predictions, and test theories. It also provides a framework for understanding the behavior of matter and energy in the natural world.

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