Why Does the Quantum Harmonic Oscillator's Equation Yield a Gaussian Curve?

[exponent137]

Dimensionless equation for quantum harmonic oscilator in the lowest energy state is:

d²u/dx²=(x²-1)u

u means wave function and solution is:

u = exp(-x²/2)

As we can see, solution is the Gauss curve.

But, what is special in the above equation that it give the Gauss curve?
Maybe some special way of deriving solution for u can give answer, why there is the Gauss curve, which is curve with the largest entropy?

 

[exponent137]

I tried with a derivation
d²u/dx²=u'du'/du, (1)
where u'=du/dx
So the above equation becomes:
u'du'=(x²-1)udu (2)
if du=u'dx
then
du'=(x²-1)udx (3)
The above equation (3) can be solved, if we try with Wolfram integrator. OK, the last equation can follow directy from the input equation, but maybe the pre-last equation (2) can be useful, because left side is u'²/2.
Or we rewrite:
u'du'=(-2ln(u)-1)udu
and it gives:
u'²=-ln(u²)u²

Because I think that Gauss curve is something special for quantum mechanics (Fourier transform...) and there should exist some simplified derivation which gives it?
The above is not one way derivation, but I seems to me that should exist some.

 

[soothsayer]

Are you just trying to derive the expression for the wave function or are you wondering why the QHO wavefunction is Gaussian in nature?

 

[exponent137]

Are you just trying to derive the expression for the wave function or are you wondering why the QHO wavefunction is Gaussian in nature?

The second of that.
Fourier transformation of Gaussian curve is also Gaussian curve and this give principle od uncertainty.

But, how simply present that "Fourier transformation of Gaussian curve is also Gaussian curve" or anything else?

 

[soothsayer]

perhaps this will help?
http://mathworld.wolfram.com/FourierTransformGaussian.html