What are the requirements in physics and mathematics that somebody must have in order to start learning Quantum Mechanics by himself?
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A firm understanding of classical physics is essential. This means understanding the qualitative and quantitative aspects of:
1) Newton's three laws of motion (Kinematic and Dynamic perspective)
2) Rotational motion dynamics
3) Electromagnetism
4) Newtonian mechanics (Newtonian gravity)
5) Principle of superposition and waves
6) Classical thermal physics
7) Particle physics
8) Harmonic oscillations
9) Scenarios where classical physics breaks down (differences between macroscopic and microscopic levels)
In mathematics, you'll most certainly benefit from knowing:
1) Calculus (preferably including differential geometry)
2) Calculations in 3D Euclidean space
3) A working knowledge of trigonometric functions
4) Vector calculus
5) Probability distributions and uncertainties
6) Linear algebra
Admittedly, this is a rather trivial summary of what you might need to start tackling quantum physics. In general, if you've completed high school physics and mathematics courses, you should first go through this link: http://en.wikipedia.org/wiki/Introduction_to_quantum_mechanics I personally took the A level educational qualification (which I believe helps in establishing a good foundation to begin with quantum physics), and pardon me if I direct you to the CIE links for Mathematics (I recommend going through all 3: Pure, Mechanical and Statistical) and Physics:
(Physics books) http://www.cie.org.uk/programmes-and-qualifications/whats-new/revised-qualifications/cambridge-international-as-and-a-level-redevelopment-programme/support/as-a-level-physics/ (Mathematics syllabus) http://www.cie.org.uk/images/128623-2015-syllabus.pdf (Physics syllabus) http://www.cie.org.uk/images/164526-2016-2018-syllabus.pdf
A brief knowledge about relativity is also sometimes beneficial.
