I The wave function is an exponential function, if I plot the

[LSMOG]

The wave function is an exponential function, if I plot the real part of it, I don't get a wave graph like sine or cosine function, Why the wave function is not represented by a trigonometric ratio instead.
Also, the wave function cannot be plotted since it is imaginary, why is it imaginary?

Thanks

 

[Matternot]

The wave function need not be a pure sinusoid, but instead a solution to Schrodinger equation for the system in question which may be exponentials e.g. a free particle with energy higher than a constant potential or sinusoids (a free particle with energy lower than a constant potential) or a Hermite polynomial multiplied by Gaussian exponentials (a harmonic potential)... You might be able to convince yourself from Euler's equation that exponentials are very closely related to waves and even provide valid solutions to the standard wave equation.

The wave function cannot be plotted, in general, on the real axis alone. But any measurement you make using the wave function must be real and can be plotted on the real axis. The wave function is imaginary for mathematical convenience/necessity. It stems from using complex exponentials to compactly describe both the amplitude and phase of a wave.

 

[Khashishi]

In the WKB approximation, you get sinusoidal wave patterns where the energy is positive, and exponential patterns where the energy is negative.

 

[Matterwave]

No real wave function can be a pure sinusoid. The function f(x)=sin(x) or f(x)=cos(x) is not normalizable. It makes sense that real wave functions decay spatially since a particle is at least partially localized to some region of space. A particle should not be spread out throughout all of space.