Incompatible observables

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In summary, incompatible observables like x and p_x, S_x and S_y, do not share a complete set of common eigenfunctions due to their operators not commuting. However, there are some rare cases, like the state of zero orbital angular momentum, where they may have a common eigenfunction but it may not span the entire Hilbert Space. An example of this is the spherical harmonic Y00 with a zero eigenvalue, which is a common eigenfunction for the incompatible operators Lx, Ly, and Lz.
  • #1
kakarukeys
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Incompatible observables do not share a complete set of common eigenfunctions, because their operators do not commute.

It seems that, incompatible observables like x and p_x, S_x and S_y do not have any common eigenfunction at all.

Can anyone give a concrete example of a pair of incompatble observables that have common eigenfunctions but incomplete (do not span the Hilbert Space)?
 
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  • #2
kakarukeys said:
Can anyone give a concrete example of a pair of incompatble observables that have common eigenfunctions but incomplete (do not span the Hilbert Space)?

The only example I can think of is the state of zero orbital angular momentum. The eigenfunction common to the incompatible operators Lx, Ly, and Lz is the spherical harmonic Y00, with a zero eigenvalue.
 
  • #3


One example of incompatible observables that have common eigenfunctions but incomplete set of eigenfunctions is the position and momentum operators in quantum mechanics. The position operator, denoted as x, and the momentum operator, denoted as p, do not commute with each other. This means that there is no single set of common eigenfunctions that can simultaneously diagonalize both operators. However, there are certain wavefunctions that are eigenfunctions of both x and p, such as the Gaussian wavefunction.

But even though these eigenfunctions are shared by both operators, they do not span the entire Hilbert Space. This is because there are other wavefunctions that are eigenfunctions of x or p, but not both. For example, a plane wave is an eigenfunction of the momentum operator but not the position operator. Therefore, the set of common eigenfunctions of x and p is incomplete and does not span the entire Hilbert Space.

This shows that incompatible observables, while having some common eigenfunctions, do not have a complete set of common eigenfunctions. This is due to the fact that their operators do not commute, leading to a lack of shared eigenfunctions and an incomplete set of eigenfunctions.
 

What are incompatible observables?

Incompatible observables are physical quantities that cannot be simultaneously measured with complete precision. This means that the measurements of these observables will be mutually exclusive.

Why are incompatible observables important in science?

Incompatible observables are important in science because they help us understand the fundamental principles of quantum mechanics. They also have practical applications, such as in the development of quantum technologies.

What is the uncertainty principle and how does it relate to incompatible observables?

The uncertainty principle states that it is impossible to know the exact value of certain pairs of observables, such as position and momentum, at the same time. This is because when one observable is measured with high precision, the other becomes more uncertain. This is directly related to incompatible observables, as they are observables that cannot be simultaneously measured with high precision.

Can incompatible observables ever be measured simultaneously?

No, incompatible observables can never be measured simultaneously with complete precision. This is a fundamental principle of quantum mechanics and has been proven through experimental evidence.

What are some examples of incompatible observables?

Some examples of incompatible observables include position and momentum, energy and time, and spin in different directions. These observables cannot be measured simultaneously with high precision, and the uncertainty principle applies to them.

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