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Marrrrrrr

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In summary, the identity matrix represents the observable of "do nothing at all" in quantum mechanics, with an eigenvalue of 1 for any state-vector. This means the result of any measurement performed with the identity matrix will always be 1, and it is conserved in any system. The expectation value and average value of this observable is also 1. It can be thought of as a black box that always gives back the same result, regardless of the input.

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Marrrrrrr

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PeterDonis

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Marrrrrrr said:I (a matrix whose elements are all 1)

The symbol ##I## is usually used to denote the identity matrix (which has 1's all along the diagonal and 0's elsewhere). Is that what you meant? Or did you actually mean a matrix with every single element (off diagonal as well as on) 1?

I suspect you mean the identity matrix, since you say this:

Marrrrrrr said:for any state-vector A, A would be an eigenvector of I with the eigenvalue of 1.

Which is true for the identity matrix, but false for a matrix with all elements 1.

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Gene Naden

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So if you have a black box that, whenever you apply it to any system, it gives you back a 1. That is the measurement. Not a very interesting black box, I might add, since no matter what you do with it it gives you back the same result.

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PeterDonis

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Gene Naden said:Not a very interesting black box, I might add, since no matter what you do with it it gives you back the same result.

Yes. The identity matrix is the mathematical description of the physical operation "do nothing at all". Which just gives you back whatever state you hand it, multiplied by the eigenvalue ##1##, i.e., the same state.

QM, or quantum mechanics, is a branch of physics that deals with the behavior of matter and energy at a very small scale, such as atoms and subatomic particles. It is important because it helps us understand the fundamental laws of nature that govern the behavior of particles, which in turn can be applied to various fields such as chemistry, biology, and technology.

"I" refers to the identity operator in QM, which represents the measurement of a physical quantity or property of a quantum system. In other words, "I" allows us to observe or measure a specific characteristic of a particle, such as its position or momentum, and obtain a numerical value.

In QM, eigenvectors are mathematical objects that represent the possible states of a quantum system, while eigenvalues are the corresponding numerical values associated with these states. They are important because they allow us to describe the behavior and properties of particles in a quantum system.

Eigenvectors and eigenvalues are closely related in QM. The eigenvalue represents the result of measuring the corresponding eigenvector, and the eigenvector represents the state of the particle that will give us that eigenvalue when measured. In other words, the eigenvector is the "input" and the eigenvalue is the "output" of a measurement.

Yes, eigenvectors and eigenvalues can change over time in QM. This is because quantum systems are constantly evolving and changing, and measurements of particles can also change their states. However, certain properties of a quantum system, such as the total energy, remain constant over time.

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