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I'm trying to understand what superposition of potentials means.
For example, let be $$V_0(x) = \begin{cases} 0 &x \in [0,2a]\\ +\infty & \text{otherwise}\end{cases}$$ and $$V_1(x)=-\lambda\delta(x-a), \qquad\lambda>0. $$ If I want to determine the superposition between $V_0$ and $V_1$, what is the result? I thought that it is a simple "union" of the two potentials, so
$$V(x) =V_0(x)+V_1(x)= \begin{cases} -\lambda\delta(x-a) &x \in [0,2a]\\ \\ +\infty & \text{otherwise}\end{cases} $$ Is this correct?

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    $\begingroup$ Are you talking about superposition of states? "superposition of potentials" just sounds like a strange way to say that they are additive. And that has little to do with it being quantum. Maybe you are using the right terms, and I simply have no idea what they mean. But maybe you are using the wrong terms. $\endgroup$ Jul 9 at 16:24
  • $\begingroup$ I’m sure of what I wrote because I reported an exam text. $\endgroup$
    – Gyro
    Jul 9 at 16:39
  • $\begingroup$ Right. $V_1$ with BCs imposed by $V_0$. People abuse language. $\endgroup$ Jul 9 at 16:44
  • $\begingroup$ Yes. In the context of it being from an exam test, they most likely meant two additive potentials, but the phrasing of the question "superposition of potentials in quantum mechanics" was misleading. $\endgroup$ Jul 9 at 16:48
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If the potentials are in superposition, in the quantum mechanical sense, then the potentials themselves must be quantum mechanical. This means that you are dealing with a quantum field theory. In quantum field theory, fields can be in a state of superposition just like ordinary quantum mechanics: the overall state is just the sum of the two states of the potentials (note: it’s the sum of two states, not one state given by adding the two potentials).

After all, potentials can be caused by quantum objects or even directly related to quantum mechanics. This means that the final solution can be a superposition of states with different potentials.

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    $\begingroup$ Potentials are not in superposition: they are simply additive in the same way that the potential of mass-spring system in a gravitational field is the sum of the potential for the spring and the gravitational potential. This has nothing to do with anything quantum. $\endgroup$ Jul 10 at 14:49
  • $\begingroup$ @ZeroTheHero The question itself is titled "Superposition of potentials", so why are you saying that "Potentials are not in superposition"? After all, potentials can be caused by quantum objects or even directly related to quantum mechanics (e.g. an electric potential is created by an electron and the electric potential is itself a quantum field, called a photon). $\endgroup$ Jul 10 at 15:00
  • $\begingroup$ @ZeroTheHero I think you need to think about the big, complete picture to understand my answer at all. $\endgroup$ Jul 10 at 15:02
  • $\begingroup$ With due respect: there is confusion in the mind of the OP and possibly in your on mind. You can superpose states or solutions, but nobody speaks of superpositions of potentials. There is no quantum field here: just a single potential with is defined piecewise. You can solve for this potential using ordinary quantum mechanics, without invoking fields. And the solution is NOT the superposition of solutions for each potential alone. $\endgroup$ Jul 10 at 15:04
  • $\begingroup$ @ZeroTheHero I see... I thought the OP is asking about superposition of potentials in the solutions/states. I'll edit my question to clarify it. $\endgroup$ Jul 10 at 15:08

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