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In QM, is it possible to ask what the probability of finding a particle in a superposition will be?

Once a particle is in a superposition, it is possible to find out the probability that it will be in a particular state, and using the density matrix, it is also possible to find the probability of a mixed state.

As far as I am aware however, there is no way to ask $\mathrm{Pr}(|{\Psi}\rangle=c_1|{1}\rangle+c_2|{2}\rangle)$. Is this true?

$%I understand it is possible to find the probability that a particular system is in a mixed state (it's just the diagonal terms of the density matrix), but as far as I am aware $

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  • $\begingroup$ The spin state along the $x$ direction can be written $\sqrt(2)|x,+\rangle = |z,+\rangle + |z,-\rangle$. $\endgroup$ – jacob1729 May 23 at 9:41
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    $\begingroup$ As a pedantic point of terminology, it's not really correct to say eg $Pr(|1\rangle + |2\rangle) = |1\rangle) = 1/2 $ since those states are just not equal and the probability that they are equal is $0$. This may be pedantry, or it might be where the confusion is. $\endgroup$ – jacob1729 May 23 at 9:46
  • $\begingroup$ @jacob1729 in the question, the |1> and |2> are basis vectors and |Psi> is the wavefunction. For others, the question was asking for the probability of a specific superposition not the probability of any. $\endgroup$ – user400188 May 27 at 7:31
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Asking "what is $\mathrm{Pr}(|{\Psi}\rangle=c_1|{1}\rangle+c_2|{2}\rangle$?" is not a well-defined question, because it does not correspond to a specific physical procedure.

However, you can ask something like

if I perform a projective measurement on the basis $|\phi_1\rangle = c_1|{1}\rangle+c_2|{2}\rangle$, $|\phi_2\rangle = c_2^*|{1}\rangle-c_1^*|{2}\rangle$, what is the probability that the measurement outcome will be $\phi_1$?

where the crucial differences are

  • the question starts with "If I perform a projective measurement...", and
  • the question asks for the probability of an outcome of that projective measurement.

Once you do that, if the system starts in the pure state $|\Psi\rangle$, then the probability that the measurement outcome will be $\phi_1$ is given by $$ \mathrm{Pr}(\phi_1) = \big|\langle \phi_1|\Psi\rangle\big|^2 = \left|\left(c_1^*\langle 1| + c_2^* \langle 2|\right)|\Psi\rangle\right|^2. $$

Note, however, that this question is not equivalent to asking "Is the system in a superposition state?", as implied in your question title, particularly if you interpret the connector "a" to include all possible superposition states. (Where, moreover, the term "superposition states" needs to be specified as relative to a given basis. The term has no meaning of its own without that.) That question does not correspond to any specific projective measurement, so you cannot assign it to the Born-rule probability of any experiment. If you have a large ensemble of systems coming from the same preparation procedure, then you can perform a quantum-state-tomography procedure that will give you the exact density matrix of the system, including its purity, but even with that the question "what's the probability that the system is in a superposition?" still has no assignable meaning.

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