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Some particles are described as a quantum superposition of quark states. For example, the pion meson: $$ |\pi^0⟩=\frac{|u\bar u⟩−|d \bar d⟩}{\sqrt{2}} $$ Is it possible to design an experiment giving us access to some information about the system and make the wave function collapse to a single eigenstate $|u\bar u⟩$ or $|d\bar d⟩$ ? What will be the implications concerning the particle and its mass ?

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The words "single eigenstate $|u\bar u\rangle$ or $|d\bar d\rangle$" in the question is invalid or meaningless. Every state in the Hilbert space is the eigenstate of some operators but without saying what the operator is, the statement that the state is an "eigenstate" is vacuous.

The state $|u\bar u \rangle$ is an eigenstate of some operators such as the number of $u$-quarks, with the eigenvalue $N_u=1$, but what's important is that this state (and the similar state with the $d$-quarks) is not an energy or mass eigenstate. The states $|u\bar u\rangle$ and $|d\bar d\rangle$ may look "simpler" but to construct a mass eigenstate, Nature chooses not them but the "seemingly more complicated" superposition you wrote.

Because $|u\bar u\rangle$ is not a mass/energy eigenstate, if we prepare a particle in this initial state, it won't stay in it. Instead, it will oscillate between $|u\bar u\rangle$ and $|d\bar d\rangle$. In fact, because the absolute values of the coefficients in front of both terms are equal, we deal with the maximum oscillation that may change from the pure $|u\bar u\rangle$ state to the pure $|d\bar d\rangle$ and back, back and forth, just like when the harmonic oscillator changes all the energy from the potential one to the kinetic one and back.

The frequency of these oscillations may be theoretically predicted and verified; the off-diagonal element of the energy between the states $|u\bar u\rangle$ and $|d\bar d\rangle$ is the cause of the oscillation (and of the need to have a superposition if we want an energy eigenstate), it is proportional to the frequency of the oscillations. It is possible to measure whether the particle is in the $|u\bar u\rangle$ state or the $|d\bar d\rangle$ state: it is possible to measure observables such as $N_u$. For example, bombard the pion with a huge number of $u$-quarks (and make sure that there are no $d$ quarks in the beam). Such $u$-quarks may annihilate with the $\bar u$ in the $u\bar u$ state of the pion and annihilate to photons, while they won't be able to do the same with the pion in the state $d\bar d$.

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Is it possible to design an experiment giving us access to some information about the system and make the wave function collapse to a single eigenstate |uu¯⟩ or |dd¯⟩ ? What will be the implications concerning the particle and its mass ?

The description of the pi0 as a linear combination of quark antiquark pairs is useful in the classification of the constituent quarks that make it up. Due to QCD interactions which are holding it together ( not for long, as the pi0 decays fast electromagneticaly to two gammas) those are not the only quarks in the game, they are the constituent quarks. Due to QCD there exists a sea of quarks, antiquarks and gluons tying up the whole system into the pi0mass. And all that mess has to balance the color quantum number that in addition to upness and downness characterizes the quarks.

The complexity is such that usually weakly and/or electromagnetically interacting particles are used as probes of the quark nature , which is the way that the existence of quarks has been verified: deep inelastic scattering. There are no free up or down quarks, due to the nature of QCD, for experimenters to use, and pi0 decays into two gammas very fast. Thus your proposed experiment cannot be done.

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