I Physical Meaning of the Imaginary Part of a Wave Function

AI Thread Summary
The discussion centers on the physical meaning of the imaginary part of the wave function, particularly in the context of wave equations and quantum mechanics. The wave function, expressed as a complex exponential, includes both real and imaginary components, with the real part corresponding to observable plane waves. The inquiry highlights that while the real parts of physical variables often have clear meanings, the significance of the imaginary part remains less understood. It is noted that complex solutions to linear differential equations yield two real solutions, emphasizing the role of both components in fulfilling initial and boundary conditions. Ultimately, the conversation seeks to clarify the relevance of the imaginary part beyond quantum mechanics, suggesting a broader inquiry into its implications in various physical contexts.
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As far as I've encountered, the imaginary part of functions describing physical phenomena have a physical meaning too. What is the physical meaning of the imaginary part of the wave function for the plane wave?
We know the wave function:
$$ \frac {\partial^2\psi}{\partial t^2}=\frac {\partial^2\psi}{\partial x^2}v^2,$$

where the function ##\psi(x,t)=A\ e^{i(kx-\omega t)}## satisfies the wave function and is used to describe plane waves, which can be written as:

$$ \psi(x,t)=A\ [\cos(kx-\omega t)+i\sin(kx-\omega t)]$$

Here, the real part of the equation alone, ##\Re(\psi)=A\cos(kx-\omega t)##, also describes a plane wave, however what is the physical meaning of the imaginary part? I know that in QM, since ##|\psi|## depends on the imaginary part too, it has some physical relevance, but my question is not necessarily limited to quantum mechanics. I have seen other similar questions, but I unfortunately haven't seen a satisfying answer
The motivation behind my question is that so far the complex parts of physical variables I have encountered also have a physical meaning: The complex part of the refraction index corresponds to the absorption, the complex part of the scattering amplitude indicates the existence of inelastic processes; that's why I am curious

If it doesn't have a meaning, why don't we say that ##A\sin(kx-\omega t),\ A\cos(kx-\omega t)## and ##A\exp[i(kx-\omega t)]## all satisfy the wave equation, where we don't know the physical meaning of the exponential one?
 
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It depends of course on the physics you consider. If your field, ##\psi##, is a real quantity you look of course only for real solutions. Since it's a linear differential equation with real coefficients for any complex solution you get two real solutions by taking ##\mathrm{Re} \psi## and ##\mathrm{Im} \psi##.

As an initial-value problem the solution is uniquely determined by giving initial values ##\psi_0(t=0,x)=f(x)## and ##\partial_t \psi_0(t=0,x)=g(x)##.

Sometimes you have in addition also boundary constraints (e.g., if ##\psi## displacement of a string of length ##L## you have ##\psi(t,0)=\psi(t,L)=0##).

Note that the general solution of the (1+1)d wave equation is given by
$$\psi(t,x)=\psi_1(x-vt) + \psi_2(x+vt)$$
with arbitrary functions ##\psi_1## and ##\psi_2##, i.e., you have enough "freedom" to fulfill the initial and boundary conditions.
 
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