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We represent the wave function in complex form. But complex number gives us plane i.e. two dimension. Than how its possibel to represent a three dimensional quantity in a plane? Please some one explain it.

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  • $\begingroup$ Don't confuse the range of a function with its domain. $\endgroup$ – Bert Barrois Apr 2 '18 at 16:47
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What three dimensional quantity? The function $\psi$ maps from $\mathbb{R}^4$ (if you include time dependence) to $\mathbb{C}$. However, the probability current $\mathbf{j}$ satisfying $\boldsymbol{\nabla}\cdot\mathbf{j}=-\partial_t (\psi^\ast\psi)$ is 3D.

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Each background space, whether 1D, 2D, 3D or some other curved spacetimes has its own version of the wave equation $$\frac{\partial^2 u}{\partial t^2}=c^2 \nabla^2 u$$ where the $\nabla^2$ Laplacian depends on the kind of background space. But the thing you are asking about is $u$, which is a scalar function of location and time.

Scalars in physics are physical quantities that can be described as a single member of a number field - a single "number" - as opposed as vector or tensor quantities that have several components. But while using real numbers is the most common case, there is in principle no reason not to use complex numbers depending on problem. The math of the wave equation works out the same. Note that you cannot as easily plug in (say) a 2D or 3D vector as $u$ in the equation since the Laplacian is not defined for vectors, only for scalars.

So in a way using complex numbers allows us to "cheat" by getting something that has vector-like properties into the wave equation. This is of course very convenient. Whether we interpret this merely as a clever mathematical trick (and in the end ensure that all results we get out of the equation are real numbers) or we actually view the complex $u$ as an appropriate description of the system (e.g. the wave function in quantum mechanics, where the complex values give it some of its magic).

To sum up, you can have wave equations for any dimension of the base space where the waves move about. The waves can be real, complex or one of the other fields (quaternions, octonions; I think nearly any differential algebra works) - the thing that determines which kind of scalar you use is the problem and how you try to model it.

(There is a slight irony here: representing a three-dimensional quantity using a scalar number field turned out to be much harder than expected and eventually led to the 4D quaternions. If you use the imaginary quaternions you can do 3D "vectors", which is sometimes a good trick in computer graphics, physics or neural networks.)

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