The Schrödinger equations have the term $\Psi$, which is the wave function.


I do not know what type of equation the wave function in the Schrödinger Equations is but I noticed that the symbol $\Psi$ is also used in Exact Equation First Order Differential Equations.


Does the symbol $\Psi$ mean the same thing in the Schrödinger Equations as it does in exact Equation First Order Differential Equations? If it doesn't mean the same thing what type of equation is the wave function in the Schrödinger Equations?


The Schrödinger equation is a partial differential equation. Its type depends on the Hamilton operator and the fact whether we have time-independent or time-dependent equation. In fact, it is not even strictly speaking a wave-equation in the mathematical sense, because it is at most first order in the time derivative.

Arguably the most important Schrödinger equation is the harmonic oscillator, which is a second order partial differential equation. Most reasonable Hamiltonians will be at least second order, since they contain a term for the kinetic energy, which is of second order.

Hence, no, it is hardly ever (never?) an exact equation first order differential equation.

Also note that the number of symbols is limited so you are bound to find the same symbols in similar but different locations. But that's okay, they are only symbols and their meaning should be clear from the context.

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  • $\begingroup$ Of course the Schrödinger equation is a wave equation. $\endgroup$ – my2cts Jul 12 '18 at 8:20
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    $\begingroup$ The symbol $\PSI$ can be used for anything you please, just like $X$. $\endgroup$ – my2cts Jul 12 '18 at 8:21
  • $\begingroup$ @my2cts the term “wave equation” is usually reserved for $\partial_t^2 u = c^2\nabla^2 u$. It is generally not possible to write the Schrödinger equation in this form. $\endgroup$ – leftaroundabout Jul 12 '18 at 9:01
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    $\begingroup$ @JasonBorn time-dependent Schrödinger is an equation that has wave solutions – but so does a simple advection equation. Neither are wave equations. Mathematically, the Schrödinger equation is rather a complex-valued diffusion equation. $\endgroup$ – leftaroundabout Jul 12 '18 at 9:11
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    $\begingroup$ @JasonBorn that equation has minus before the second time derivative, unlike the one in your comment. But anyway, that paper doesn't assert that this is the time dependent Schrödinger equation: it names it plasma wave equation. Moreover, its solutions will differ from those of Schrödinger's one at least by phase velocities of waves of different frequencies. $\endgroup$ – Ruslan Jul 12 '18 at 11:38

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