# How do we know (BΨ) is not equal to zero?

• PFfan01
In summary, in the given scenario, it is stated that operators A and B have complete sets of eigenfunctions and that Ψ is an eigenfunction of A with the eigenvalue a. It is then mentioned that A(BΨ) = BAΨ = a(BΨ), leading to the conclusion that (BΨ) is also an eigenfunction of A. However, there is a question regarding whether (BΨ) could potentially be equal to zero. The response states that this is not an issue as long as Ψ is the domain of B. Furthermore, it is noted that Ψ and (BΨ) are both eigenfunctions of A with the same non-degenerate eigenvalue a, and that they
PFfan01
Suppose that operators A and B have complete sets of eigenfunctions, [A, B] = 0, and Ψ is an eigenfunction of A with the eigenvalue a, namely AΨ=aΨ. Then we have A(BΨ) = BAΨ = a (BΨ). They say (BΨ) is also the eigenfunction of A. Why? How do we know (BΨ) is not equal to zero?

Well Psi is the domain of B, so that the only way that B psi =0 is that psi =0, which is trivial, the null vector is excluded in any analysis.

dextercioby said:
Well Psi is the domain of B, so that the only way that B psi =0 is that psi =0, which is trivial, the null vector is excluded in any analysis.

Thanks, but I am thinking in a different way.
Ψ and (BΨ) are both eigenfunctions of A with the same non-degenerate eigenvalue a, and they must be linearly dependent, namely BΨ=b*Ψ. Ψ is a common eigenfunction of A and B. But the eigenvalue b could be zero (b=0), so that BΨ=0. Right?

If Bpsi =0, then it's no problem, the 0 vector is a trivial eigenvector for any linear operator.

## 1. How do we measure the value of (BΨ)?

The value of (BΨ) can be measured through experiments and observations in the field of physics. Various instruments and techniques, such as magnetic field sensors and spectroscopy, can be used to measure the magnetic field strength and energy levels of particles.

## 2. What is the significance of (BΨ) not being equal to zero?

(BΨ) represents the magnetic field strength and energy levels of particles. If it were equal to zero, it would mean that there is no magnetic field or energy present, which would have significant implications on our understanding of the universe and how it functions.

## 3. How do we know that (BΨ) is not just a result of chance or random fluctuations?

Scientists use statistical analysis and repeat experiments to ensure that their findings are not just due to chance. Additionally, the consistency of (BΨ) measurements across different experiments and observations further supports the idea that it is not a result of random fluctuations.

## 4. Can (BΨ) ever be equal to zero?

In certain theoretical models, (BΨ) can be equal to zero. However, in the real world, there is always some level of magnetic field and energy present, even if it is very small. This is supported by the fact that even in the vacuum of space, there is still a measurable magnetic field.

## 5. How has our understanding of (BΨ) evolved over time?

Our understanding of (BΨ) has evolved over time as our technology and scientific methods have improved. Early scientists, such as William Gilbert, first discovered the concept of magnetism in the 16th century. Since then, our understanding of magnetic fields and energy has advanced significantly through the use of new instruments and experiments.

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