# Why should the integral of wavefunction squared be normalized for a Quantum Object?

According to the statistical interpretation of Quantum Physics, a particle does not have a precise position regardless of any measurements. But then, the interpretation imposes another condition on the wavefunction of a quantum object - The integral of the wavefunction squared should be equal to one i.e. the total probability of finding that particle anywhere in space should be one because the particle must exist somewhere. Isn't this contradictory? On one hand stating the particle does not have a position firsthand but then also stating that it got to exist somewhere. I am a complete novice in this subject so pardon me if I don't make sense.

• You know that it exists. Just not where. Thus the overall probability that it is somewhere has to be one Commented Aug 6, 2020 at 15:37
• @planetmaker I doubt it. Most physicists believe that the particle does not exist at all. Measurements performed on it forces it to manifest itself into existence. Prior to that its just a cloud of probability. Commented Aug 6, 2020 at 15:52
• @AjinkyaNaik Various conservation laws (energy, momentum, charge etc.) would be violated if fundamental particles appeared and disappeared at random. The normalisation criteria ensures that conservation laws hold. Commented Aug 6, 2020 at 16:01
• @AjinkyaNaik what are you talking about??? Are you debating the existence of protons and electrons? Commented Aug 6, 2020 at 16:22
• @Ajinkya Naik either you have a fundamental mis-understanding or you a trolling. No physicist will deny the existence of a particle, if it has a wave function. Don't confuse it with the wave-particle duality being true at the same time Commented Aug 6, 2020 at 22:20

How is it contradictory? The contradiction is with the assumption that a particle must have a position in order to exist. This is clearly not true. As Dirac put it

“In the general case we cannot speak of an observable having a value for a particular state, but we can … speak of the probability of its having a specified value for the state, meaning the probability of this specified value being obtained when one makes a measurement of the observable.” Dirac P.A.M., 1958, Quantum Mechanics, Clarendon Press, p.47.

Even in the macroscopic world position exists only as a relationship with other matter. You cannot say where you are unless you say where you are relative to something else. I am in this room. This room is in this building. The building is in this town. and so on. You cannot say where you are in space. In the macroscopic world, position always exists because objects are in continuous interaction with their environment. This is not true in the quantum world. A particle may have too few interactions with its environment to generate a precise property of position. Of course, this does not mean that the particle does not exist.

It has no precise position, but ist is somewhere, you can also calculate the probability of finding it in some specific Volume. Not knowing how your throw of a dice comes out, that you get any of the number 1 to 6 is sure so the probability for a number between and including 1 and 6 is 1.

• It is somewhere, but we don't know where. I presume this is the viewpoint of the realist interpretation of QM. The one which Einstein believed. Commented Aug 7, 2020 at 9:20

Just restating what trula said; when you roll a die, the probability of the outcome being $$1,2,3,4,5$$ or $$6$$ is $$\frac{1}{6}$$ each. $$\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}=1$$ Each $$\frac{1}{6}$$ above represents the probability of a possible outcome. They add up to one because the die must roll a $$1,2,3,4,5$$ or $$6$$. Although you don't know which outcome the die will show beforehand, you do know that it will be one among these 6 possibilities.

The integral you are referring to is doing a similar addition of probabilities, but all through space.

Even if we're not able to calculate the exact position of that particle, quantum physics helps to find out the probability. Measurements are not at all creating it's existence, rather it make the so called 'particle' or we can say wavefunction to choose a position. This is because in quantum physics we are dealing with microscopic particles. Even our small measurements can make a change in their path.