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Feynman says, "It doesn't matter how beautiful your theory is, it doesn't matter how smart you are. If it doesn't agree with experiment, it's wrong."

So, is superposition state of Simple Harmonic Oscillator(SHO) ever observed? If yes, what is that "particle" which was put in the potential energy well (e.g. 1-dim) $V(x) = {1 \over 2}kx^2$ ? Was it electron or some atom or some molecule or something else ?

EDIT: Consider, for example,

$$ \psi_1(x) + \psi_2(x) $$ where

$ \psi_1(x) = N_1 exp(-\alpha^2x^2/2)H_1(\alpha x)$

and

$ \psi_2(x) = N_2 exp(-\alpha^2x^2/2)H_2(\alpha x)$

where $H(\alpha x)$ are Hermite polynomials.

Somebody may say - "Superposition state can not be "measured"...when we measure some physical quantity of a "particle" which is in superposition state, wave-function collapses, and we get one of the eigenvalues of that physical quantity..." But, I then would like to say this: having a particle in a superposition state and measuring some physical quantity (e.g. momentum, energy) of that particle in that superposition state are two different things.

Heisenberg says that there is no particle trajectory since experimentally he can not have position of the particle as a continuous function of time. Heisenberg considered only those quantities (e.g. intensities of spectral lines) which can be measured experimentally while developing his theory.

If we can not achieve experimentally the superposition state of a particle, then we should neglect this concept of superposition, as per Heisenberg's logic.

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closed as unclear what you're asking by knzhou, AccidentalFourierTransform, Gert, user36790, John Rennie May 27 '16 at 5:07

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ What exactly do you mean by a 'superposition state'? This is in general not a well-defined term, because for you $|\psi⟩=|\phi⟩+|\chi⟩$ might look as a superposition state, but for me $|\psi⟩$ might be a legitimate state that's a building block for other superpositions. The ground state of the SHO, for example, is in a superposition of different position states; does that count? The question is equivalent to asking "Is this vector in $\mathbb R^3$ the sum of two other vectors?", to which the answer is trivially positive always. $\endgroup$ – Emilio Pisanty May 23 '16 at 11:31
  • $\begingroup$ That said, cat states have been produced and experimentally confirmed, as have explicit superpositions of number states. If those fit the bill then I'm happy to provide more details. $\endgroup$ – Emilio Pisanty May 23 '16 at 11:32
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    $\begingroup$ The question is a mess. We don't measure states, we only measure observables with the outcome probabilities determined by that state. For example of the state that is a superposition of the energy eigenstates google coherent state. Most of all I don't like the tone. $\endgroup$ – OON May 23 '16 at 11:53
  • $\begingroup$ @EmilioPisanty: I am concerned only with SHO and no q-bits and quantum optics stuff and other fancy stuff. I am concerned only with ELECTRONS or some other MATERIAL PARTICLES. Those electrons, if you remember, which flows through minute tracks in PCBs and which in turn run computers one of which you are using! $\endgroup$ – atom May 23 '16 at 11:54
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    $\begingroup$ @atom You misunderstood the query. What states do you count as superposition states? As a trivial answer, one could say "the ground state has been realized for system X, and the ground state is a superposition of being on the right plus being on the left". Presumably you want to rule that out, but to do that you need to be more precise with your question. $\endgroup$ – Emilio Pisanty May 23 '16 at 11:56
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OP explicitly asks whether a material object (i.e. not a state of light) has been placed in the superposition $|\psi⟩=|\psi_1⟩+|\psi_2⟩$, where $|\psi_n⟩$ is the $n$th Fock state of a harmonic oscillator.

Perhaps the clearest example of this is the achievement of precisely that superposition in the quantum state of a cantilever microwave resonator (described here and formally published here), which consists of a mechanical cantilever that's electromechanically coupled to a microwave resonator:

There are plenty of other realizations of this superposition in a wide variety of systems, both light-based and material-particle-based. However, this site is not really the place to ask for a complete enumeration of them.

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