What mathematics should I know to study quantum mechanics, particle physics? Tensor analysis? Is it as difficult as Finite Element Method?

Depends how far you want to go. Most aspects of the Bohr atomic model can be understood with high school algebra, but if you want to deal with topics like the Schrodinger equation and beyond, you'll definitely want to have a good background in multivariable and vector calculus, partial differential equations and linear algebra. The maths can be just as difficult as solving PDE's with the Finite Element method, and even more so, but again it depends on how deep you want to go in your understanding and application. Tensor analysis isn't really necessary but it can be useful in understanding certain results such as the Wigner-Eckart theorem or quantum electrodynamics form factors, but these are really fine details that aren't important if your goal is just to gain a general philosophical understanding of the theory.

You can learn how to write quantum software at IBM's cloud computing site. This is superficially just object-oriented programming, but with qubits. There are Python libraries and a developer's kit (Qiskit). On the other hand you could learn about the underlying hardware, how a superconducting Josephson junction is a 2-state qubit given the right design. The statistics in quantum programming is down to how many times a program is run (maybe a few hundred times). Running it once would mean you have a high confidence in the output, say, which would be a somewhat unusual situation.

Just a basic understanding of probability and how to work with continuous probability distributions should be sufficient for your needs. I forgot to mention some basic knowledge of complex variables is necessary too, and a more detailed background in complex analysis will help down the road when dealing with particle propagators and other more advanced concepts.

Mathematical Requirements for Basic Quantum Physics https://www.physicsforums.com/threads/mathematical-requirements-for-basic-quantum-physics.512109/

In my opinion, which I tend to keep to myself, studying quantum mechanics spans a lot of other disciplines, but mainly these are logic (and what quantum logic is or isn't), solid-state physics, optics, and information science. Modern digital computers use 2-valued logic and Boolean algebra, one correspondence with quantum information science is that it's more useful to have a 2-state quantum logic (roughly, spin-up or spin-down) than try to engineer access to higher eigenstates of individual qubits. One quite striking non-correspondence is that it's generally more utilitarian to use semiconductor technology which is relatively constrained (in the solid-state context, a handful of elements), but a qubit can be just about anything (which is small enough), it doesn't even have to be a single quantum particle like say, an electron. An electron is just one of the very many things that a qubit can be. Look also, at what a universal computer is, or a universal set of gates/operations is, compared to non-universality. Quantum information appears to be something we can only exploit if we don't look at it in any way, including letting the environment do this. This is explained in terms of inputs and outputs, and two important quantum principles: unitarity and locality. But that's just the start of your study, and, what you want to know might depend on what you want to do with that. These days you can teach yourself quantum logic and programming online, if you can demonstrate your knowledge of this to someone who cares (maybe they work for IBM), maybe they'll offer you a job. You don't really need to know a lot of mathematics to do this. Why did IBM make its very expensive machine open to the public? It wants to attract innovators, and it wants to make cloud quantum computing into a commodity (why else?).

Basic Algebra. Basic Calculus. Linear algebra will be very useful. Vector calculus. Some group theory could be useful. Differential equations, particularly partial differential equations.

Not only what is Quantum Mechanics, but also how do Quantum Mechanics work. How is a quantum event modelled? We know there is a threshold event involved which triggers the quantum change. At this threshold event the system is in a state of "quantum suspension" or "in superposition" before collapsing into a single state. If we are ever going to find an absolute universal time constant, it should be the duration of a "superposition" before the quantum collapse itself. AFAIK quantum events have a value, but I don't know of any equation for the duration of "superposition" before collapse of the function.

I don't think it's been completely resolved to everyone's satisfaction, but there is a theoretical mechanism governing the process of wave collapse in the quantum decoherence model, and the transition is continuous (albeit extremely rapid) as opposed to an instantaneous *poof* wave collapse just because a measurement was performed. I think the main sticking point is that it either leads into the many worlds parallel universe interpretation, or else some as yet unknown force is needed to make sure our universe is the only one to manifest.

There is basically tons of mathematics you need to master to become really good at doing mathematical and theoretical physics. http://math.ucr.edu/home/baez/books.html http://www.staff.science.uu.nl/~hooft101/theorist.html That's why I see that for someone who is not good at math studying theoretical physics can be a waste of time and money. For starters physics and math textbooks are highly expensive and so obviously not everyone can buy them. A better alternative to studying physics for someone who isn't good at math is to study economics and business administration and try to find a way to become rich quickly. Physics is a waste of time. Since money is everything these days I think that rather than studying physics I believe it is more productive to just find a way to become very rich.