What is the use of representing wavefunctions as an exponential?

[pardesi]

what is the use of representing wavefunctions say $$\psi(x,t)=A\cos(kx-\omega t)$$ by $$\psi(x,t)=Ae^{i(kx-\omega t)}$$ when we actually mean the real part

 

[jtbell]

It's purely a matter of mathematical convenience. In principle, we can solve any calculation involving a classical wave using only sines and cosines. However, some calculations are easier to do using complex exponentials.

 

[pardesi]

yes i saw some but how cwn we be so sure that whatever we do using the complex numbers is right withoput checking them with the cos and sin

 

[OOO]

yes i saw some but how cwn we be so sure that whatever we do using the complex numbers is right withoput checking them with the cos and sin

the reason is simple: generally you only use the complex plane wave representation in case you are dealing with a linear(wave) equation. But if it's linear you may may compose from or decompose into real and imaginary part (sine and cosine) without loss of generality.

If you have a nonlinear equation (even if it's one that "naturally" involves complex numbers) then using sin/cos won't give you an exact solution anyway.

 

[jostpuur]

Some authors first give the solution as a complex function, and then say "we choose the real part, because only it has physical meaning." That is something that, I would say, is closer to poetry than to science. Of course you cannot just split arbitrary solution f into sum of two functions, f=f_1+f_2, and hope that these functions f_1 and f_2 alone would still be solutions. Complex numbers and physical meanings are there only to add confusion.

If the PDE is linear, then that gives the correct result because the real trigonometric functions can be written as linear combinations of complex exponential functions.

Sometimes the easiness seems to stem from the fact that in can be frustrating to use the trigonometric formulas such as cos(A+B)=cos(A)cos(B)-sin(A)sin(B) and sin(A+B)=sin(A)cos(B)+sin(B)cos(A), when the formula exp(A+B)=exp(A)exp(B) is much easier. Although I don't think I have example of this right now.

 

[Count Iblis]

You don't necessarily need to take the real part. You can just as well represent the wave by a complex number by identifying the amplitude with the absolute value of the complex number and the phase by the phase of the complex number.

 

[OOO]

If the PDE is linear, then that gives the correct result because the real trigonometric functions can be written as linear combinations of complex exponential functions.

I shall add: if the complex conjugate is also a solution then adding the wave and its complex conjugate amounts to taking the real part.

I agree that physics authors are too reluctant to mention the seemingly obvious, which can be quite irritating for a novice.

 

[OOO]

You don't necessarily need to take the real part. You can just as well represent the wave by a complex number by identifying the amplitude with the absolute value of the complex number and the phase by the phase of the complex number.

If you are discribing for example a mechanical string, which is described by a real wave equation, you won't get what you really want (i.e. the time dependent elongation) by calculating the complex modulus (that's always a positive number !). What you would get instead is the RMS time average over one period, and this is indeed a very useful property if this is what you are interested in. This is used for example in the Poynting theorem in Fourier space (see Jackson, Electrodynamics).