In quantum mechanics what was the theory or evidence supporting many world theory?

The Many Worlds Interpretation (MWI) is a response to the double-slit experiment's implications. We see particles break apart and become wave-like instead of particle-like, and then become particles again. In previous interpretations of quantum mechanics, such as the Copenhagen Interpretation, there was no real explanation of why a particle splits apart, and why we end up seeing it in one final location rather than any of the other possible final locations. So MWI states that perhaps the particle actually does appear at all other possible final locations, just in a different universe that is a branch off from our universe.

The reason this is considered a better explanation is then we don't have to explain a particle ended up in one place rather than any other place, because MWI says it ended up in all possible places, just in different sub-branches of the universe.

However, this interpretation has led to all kinds of woo-woo pseudo-scientific speculation about there being a multiverse, and there are multiple different versions of us in each of these universes, each of us leading slightly different lives, etc. This is BS, all MWI does is explain why quantum particles end up in the same positions even after breaking apart briefly. In the end all particles are constrained by classical physics to come together and form the classical physics world that we know. There aren't multiple universes, there is just one universe that is a bit blurry at the lowest level.

In quantum mechanics what was the theory or evidence supporting many world theory?

That an electron can appear TWO places at once in the two slit experiment---SUGGEST the THEORY of the many Worlds THEORY---Like other Advanced theories, they are just SUGGESTIONS of what COULD EXPLAIN the real observations.---Going into different universes explains it---as do other theories.   We just do not know what theories are correct that explain what is observed in the standard model. -----I prefer the "Holographic Universe Theory" of physics myself---Where all REAL particles exist on an "Upper Reality"  and space and position is an illusion in our "lower reality" based on how we perceive wave attributes of particles and arrange them in a 3D space.---- and it is quickly gaining favor..   HUT explains many aspects of quantum behavior and is based on David Bohm's  explanation of quantum entanglement.----Where there is no space between entangled particles in the upper reality but could be light years apart in our 3D lower reality--and thus respond to each other FTL.

You are right in the middle of a select group of people who also do not understand anything about Quantum Physics. They just wish they did, But, They are trying, very trying.

You should check out some of Sean Carroll's YouTube stuff or one of his books. He is a proper professor of physics who supports the many world's interpretation of QM and who also writes/talks for a general audience.

Everyone agrees the Schrodinger equation which describes the quantum mechanical wavefuntion is correct. And the wavefunction is a probabilistic distribution of outcomes for the future and how they evolve over time. If you were pick a point in the wavefunction you could consider that as being one "world", ie a particular set of outcomes.

The issue is not the Schrodinger equation itself but how we interpret what it physically means.

The traditional university teaching is the Copenhagen Interpretation, which says that when we measure the state of our world, it forces a collapse (partial collapse really) of the wavefunction into a particular state, where the collapsed state is probabilistic and determined by the Schrodinger equation.

If you think about this for a while it raises all sorts of philosophical problems about multiple observers and other things. Schrodinger himself thought it made no sense as an interpretation and famously invented "Schrodinger's Cat" to mock it. But it is  mathematically correct.

The many world's interpretation basically says; Well, we all agree on the Schrodinger equation, so lets just leave it at that and not add in all this wave collapsing nonsense.

People in one region of the wavefunction measure their own local state and people in another region measure a different state. And that's it. You can ask, "Why am I in this particular state?" and the answer is that there are also all the other states, you just happen to be in this one. It's kind of like asking why you are Alexis? and the answer is that everyone exists and you are Alexis. So all the other worlds exist and this is our world.

Everett developed the Many World's Interpretation because of the measurement problem in quantum mechanics. Nobody knows why the wavefunction collapses when a measurement is made. And how does nature define a measurement? Can an atom make a measurement? Or a photograph film? Or is a conscious observer required? Is a cat ok or must it be a human? Can it be a baby? Does the human need a PhD? There are no answers.

Hugh Everett thought that the wavefunction never collapses. When you make a measurement or observation, you become part of the wavefunction. You go into a superposition of states. But these states can not communicate with each other. So you are not aware of them. Each version of you is real, but each thinks he is the only you. 

This is an Interpretation, not a new theory. It makes exactly the same predictions for the outcome of experiments. There is no way to prove or disprove this idea. It is a matter of your opinion. 

The reason that many-worlds interpretation is popular is that it avoids a huge gap in our understanding of quantum mechanics - the notion of a state collapse (e.g. wave function collapse).

An unmolested quantum system can exist in a superposition of possibilities, each with its own probability of occurring. When the quantum system interacts with an external system (e.g. a measurement), one of those possibilities becomes ‘real’ - the superposed state collapses to a particular possibility. We have no mathematical description of that process. Even worse, it appears to be a non local process happening instantly over extended regions.

In the many worlds interpretation, there is no state collapse. All possibilities actually occur. It’s like a series of branching paths are created. Each of those in turn, leads to other branches so you can think of an evolving path structure continuing to branch over time. Somebody on one path can only see things in the past on paths that they have traversed, they cannot see things in the past of other branches nor interact with anything on other paths. Each sees a single path of consistent reality with the other branches hidden from them.

My take on this: "many worlds" is not a theory in the usual science sense. It's not a science explanation of quantum mechanics. It's what we call  an "interpretation", a way of taking the mathematics and putting it into common words.

The MATH of quantum mechanics is well understood, there's no reasonable disagreement about it. We can take that math, and make predictions about the results of experiments. It is when we try to talk about the MEANING of quantum mechanics, how it relates to reality, that we get different interpretations.

Some of these interpretations have names such as Copenhagen, quantum information, relational, quantum Bayesianism, and many worlds. They ALL have this in common: they have to match the same math and experiments. NONE of them make any experimental predictions that differ from the others (at least so far ...)

ADDED: What would many worlds look like? It would look exactly like Copenhagen, quantum information, relational, quantum Bayesianism, or any of the other interpretations. That's my point here: there's no difference in what we would observe.

What level of maths do you understand?

The minimum math background to start to understand quantum mechanics is linear algebra for discrete energy levels and calculus (differentiation and integration) for continuous energy levels. Complex numbers, partial and ordinary differential equations, integral calculus I-III, linear algebra, Fourier analysis. And then Field equations and a good grasp of statistics are usually required.

Plus an understanding of the "Standard Model" with all elementary particles, including quarks, spin, etc.

Once you're up to speed, we can talk n-dimensions, string theory, etc., to minimally discuss the many-worlds hypothesis.

That's too many questions, & too complex to be answered here.