Why two orbitals having same phase is not a random phenomenon? I have been reading about Molecular Orbital Theory for Chemistry.

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*I tend to believe that when two Hydrogen atoms approach each other, whether the $1s$ orbitals are in-phase or out-phase is a random phenomenon. However I know that this is not so. Please provide some arguments to counter it.


Please try not to indulge in complex Mathematics as I am relatively new to the subject. My mathematical understanding is "negligible".
 A: The total wavefunction is a function of both electron positions.  There is no phase between them.  The wave function is also a function of the electron spins.  The total wave function (both spin and spacial part) must be anti-symmetric.  Although a clever experimenter could prepare a pair of hydrogen atoms which had specified spins, in most cases the total (spin) angular momentum will be random, and if the spins are the same then the spacial part of the wavefunction will be antisymmetric and one of the electrons will be forced into an anti-bonding orbital.  In a spin 0 case, the spin wavefunction is ant-symmetric, so to spacial wave function is symmetric, and both electrons can fit into a bonding orbital.  The attractive force between the atoms will depend on the spins.
You may want to look up the Woodward-Hoffmann rules of orbital symmetry, which predict the stereochemistry of reactions based on simple physical descriptions of the electron orbitals.
A: I have been going through several articles on the matter. Based on that, I have found that the Linear Combination of Atomic Orbitals is a mathematical model with little correlation to the actual process of formation of Molecular Orbitals. Hence the concept of phase/sign of atomic orbitals are simply for the process of denoting that whether our mathematical model will "add" these orbitals or "subtract" these orbitals.
In short, what I am trying to look for in my question (and I suspect a lot others who are new to the subject) is simply not there "physically" but only mathematically.
