Finding the wave function of a quantum harmonic oscillator How can I find the wave function of a quantum harmonic oscillator? If I measure its energy several times, my measurements will change the state of a system. All I know are the possible states, given my Schrodinger equation.
 A: If I understand the question. Assume you have a quantum oscillator, then if the system is an eigenvector of the hamiltonian, then a series of measurements will give a series of results reflecting eigenvalues of the system. The appearance of these eigenvalues must obey at a number of many measurements the probabilities of each eigenvalue to appear. It's eigenvalue will appear as the probability factor of the Schroedinger equation predicts.
There is not a wave function in a sense that the system is this state but how I can find it? We take a measurement and verify whether or not the energy spectrum is finite- has a number of eigenvalues that satisfy the probabilities given by the equation and thus determining the state of the system. 
The total wavefunction is the sum of the eigenvectors  
with the constant in front that give the probability of measuring this state and n=0,1,2... and $E=(n+1/2) \hbar \omega $
Now, if you want to write down the wavefunction just by experiment and wish to know how to perform such a task, I have to say I'm not capable of posting an analytical answer. But my thinking would suggest that by measuring the energy you get the possibilities of the wavefunction and the eigenvalues of the hamiltonian operator, that is the energy spectrum. But for more information on the "right" mathematical form of the wavefunction I'd think you should make measurement of other operators to and from there start solving the eigenvalue equation of each operator. See the question commented at your post by ACuriousMind and also this for maybe more clarification: Is the wave function objective or subjective?
In the end you must have a function that will agree with the experiment. I don't know if there is a possibility of two person contacting this same experiment and getting in the end with two different wavefunctions that predict the same results for all the operators we would like to measure.
Hope this helps.
