Consider the Schrodinger equation $i\partial_t u=-\Delta u.$ Special solutions of the form $$u(t,x)=u_k(x)e^{ikt}$$ have many names which I've seen used, such as stationary states, solitons, standing waves, etc. I am wondering what one would call such a solution to the wave equation $\partial_t^2u=\Delta u.$ If it matters, assume that the spatial domain is unbounded. I suppose one could call them a mode solution since they feature a particular Fourier mode.
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2$\begingroup$ “soliton” certain does not apply (this is not a travelling wave that keeps its shape, solution of a nonlinear differential equation), and it’s not a standing wave either since (as written) the wave is travelling in only one direction. $\endgroup$– ZeroTheHeroCommented Jun 11, 2022 at 3:55
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$\begingroup$ I would call it a Helmholtz eigenfunction. $\endgroup$– Emilio PisantyCommented Jun 11, 2022 at 6:41
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$\begingroup$ @ZeroTheHero gotcha. Do you have any ideas on a standard name? I'm more interested in the technical mathematical phrasing, so I am not worried if it is somewhat inaccurate with the standard physical interpretation. $\endgroup$– user900940Commented Jun 11, 2022 at 16:21
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$\begingroup$ @EmilioPisanty that makes sense for how the question was phrased. However, that feels less robust, especially when considering operators who only equal the d'Alembertian at the principal level (you no longer get a simple Helmholtz equation when converting to a stationary problem). $\endgroup$– user900940Commented Jun 11, 2022 at 16:23
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1 Answer
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There is no universal standard for naming those solutions.
For mechanical or electromagnetic waves, a stationary wave is a wave that doesn't travel, typically of the form:
$$s(x,t)=A\cos(kx)\cos(\omega t)$$
However, in quantum mechanics, a stationary state is a wavefunction that has a probability amplitude that doesn't vary in time:
$$\psi(x,t)=\varphi(x)e^{-i\omega t}$$
You can have solutions like that for the d'Alembert equation, but their name would depend on the exact form of $u_k$.
- If $u_k$ is a real decreasing exponential, the wave is called evanescent.
- If $u_k$ is a real sine or cosine, the wave is called standing.
- If $u_k$ is a complex of the form $e^{ikx}$, the wave is called traveling.
If the spatial domain is bounded, then $k$ is quantized, and the solutions are called modes.
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$\begingroup$ So, it would be inaccurate to use the phrase "stationary state." For evanescent, does $u_k$ need to be exponentially decreasing, or can it simply be spatially localized (e.g. in a Sobolev sense)? What if $u_k$ is complex-valued but localized in the manner described? Is there no longer a name which people use? $\endgroup$ Commented Jun 11, 2022 at 16:26
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$\begingroup$ I used this expression only in a quantum context, where it has a very specific meaning. I also pointed out the discrepency that can exist between different parts of physics. "Stationary state" is an expression that I'm sure is in current use in quantum physics. $\endgroup$– MiyaseCommented Jun 11, 2022 at 16:40