Theory is developed based on experiments. I like the way Professor Shankar talked about how quantum mechanic physicists nailed the coffin of Newtonian physics by showing the famous double slit experiments.

When they observed the interference phenomena, it’s a natural and immediate attempt to try to explain by applying typical wave function using sin or cos, however this is a failed attempt because it leads to conflict of Heisenberg’s uncertainty principle.

Complex or imaginary number in math comes to rescue. Hence, there is this so far the best math description/function:

I was quite puzzled by the seems to be a convention in this area that sai square is the probability of particle landed. The peculiarity in quantum mechanics “multiple of sai is equivalent to sai”, such that “double sai” stands for same physical situation. the only job of sai is to give the odds, since sai amplitude doesn’t vary the odds, it’s like a vector. The vector multiplication is same indication no matter how you stretch or squish it along that direction.

I searched quora, to no avail of any compelling, reasonable explanation, until Professor Shankar explains using two concrete examples. Basically it’s just a normalization/disciplining introduced by physicists to make computation or theory development later on more convenient.

Bearing in mind the peculiarity of quantum particles, they behave like waves, but not describable by Newtonian waves, what the complex formula (rather than cos or sin formula) denotes is the probability, it’s like a vector, what you care is only the direction not the amplitude, hence, we can pick one of those (normalization) to make the square of sai integrated by dx = 1.

Another example, if the wave function given is Gaussian distribution:

From above two example, one can see squarely that depending on whatever the sai distribution is, using math technique to normalize the CDF (cumulative density function) to be 1, job is done.