Mean Wave Function

What does an average wave function look like? I mean a quantum wave function, and I'm applying my own personal brand of averaging where spaces with symmetries align before being combined. 

The simplest wave other than trivial planar waves are the quantum harmonic oscillators. These are just like harmonic motion of a point on a spring but apply to wave functions instead, which are complex valued. The raw simplest case is based on the Hamiltonian 

which has the interesting property that the physics is invariant to rotations in phase space, between position x and momentum p, known as Born reciptocity. This is interesting as it gives us another symmetry from which some structure can emerge. 

A single standing wave of this oscillator is 

and if we sample cn from a complex Gaussian distribution then the equivalent of white noise in this quantum phase space is called a Gaussian Entire Function:

and its corresponding white-noise wave function is:

This covers the entire x axis and its shape in phase space looks like this sample:

A complex-valued function with the physical distance x in the horizontal and momentum p in the vertical. Brightness is its complex magnitude and hue is its complex phase. Those dark spots are zeros of the function and they are roughly evenly spaced throughout the entire phase space. 

The wave itself is the x axis and it evolves over time in a very simple way, by rotating the image. This function has many titles, here I've listed some as the keywords can help in looking it up:

• Phase-Space White Noise Wave function
• Bargmann-Fock Gaussian Random Field 
• Fock-Space Gaussian Analytic Function
• random Coherent State Superposition
• random LLL wave function  (Lowest Landau Level)
• random vector in the Bargmann-Fock Hilbert space
• vacuum white noise field

So what does an average white-noise wave function look like?

If we average all possible samples it would be a zero wave function. Not very interesting.

But global phase is an unobservable in quantum theory, the physics is invariant to shifting the phase everywhere by a constant. So if we should align the phases optimally before calculating the average.

This reveals a problem, we can't align the phases of functions of infinite extent. Two plausible fixes are:

1. we perform a weighted best-fit alignment. Here I have chosen to weight each standing wave coefficient cn by its respective multiplier:

This weights evenly at unit radius from the origin, which is the average radius territory of each zero.
Simulating the first 40 terms (larger make no obvious difference) and averaging over 23 layers, the phase symmetry generates a simple average wave function (left):

On the right we simulate rotation symmetry (Born reciprocity) instead, yielding a torus shape.
Here with both phase and Born symmetry. The offst zero gives the function a crescent shape:

I'm not entirely sure but I think the physics might be invariant to conjugating the coefficients cn. So with this added symmetry we lose bilateral symmetry and see two of the zeros in the average wave function:

2. The second option is to reduce the temperature of the system. The white noise is equivalent to infinite temperature (q=1) where cn is drawn from CN(0,q). But for lower temperatures (q<1) the wave function decays with distance from the origin. 

This second option doesn't give us a single mean shape as it depends on temperature. The higher the temperature the more zeros the mean shape will have. But it does allow us to avoid the probably adhoc weighting on the coefficients. 

It does however need to weight the coefficients exactly according to how much they affect the magnitude and shape of the resulting function. This isn't trivial to do.

However I wouldn't be surprised if the resulting mean wave functions looked like the top but with the zeros located in similar configurations to the previous mean clusters post.