What are postulates and what are derived?

[I_am_learning]

I have now understood special Relativity and now going to study quantum mechanics. But I am having hard time to understand the proofs of different phenomenas like barrier tunneling. I am also not understanding the wave function and the de-Broglie waves.
So please anyone here could list me what are the basic postulates one has to assume in order to read-on quantum mechanics.

 

[jtbell]

A Google search on "postulates of quantum mechanics" led me quickly to good old Hyperphysics:

http://hyperphysics.phy-astr.gsu.edu/Hbase/quantum/qm.html#c2

 

[muppet]

You'll probably struggle to understand either the wave function or de broglie waves if you just know the postulates of quantum mechanics. It sounds like you really need a proper textbook, or at the very least a decent set of lecture notes. There's billions of discussions of QM textbooks in the academic guidance/book discussion forum.

 

[humanino]

It looks to me that wikipedia is more rigorous here
Postulates of quantum mechanics

The following summary of the mathematical framework of quantum mechanics can be partly traced back to von Neumann's postulates.

• Each physical system is associated with a (topologically) separable complex Hilbert space H with inner product $\langle \phi | \psi \rangle$. Rays (one-dimensional subspaces) in H are associated with states of the system.
• The Hilbert space of a composite system is the Hilbert space tensor product of the state spaces associated with the component systems.
• Physical symmetries act on the Hilbert space of quantum states unitarily or antiunitarily due to Wigner's theorem.
• Physical observables are represented by densely-defined self-adjoint operators on H.

The expected value (in the sense of probability theory) of the observable A for the system in state represented by the unit vector $| \psi \rangle \in H$ is
$$\langle\psi|A|\psi\rangle$$
By spectral theory, we can associate a probability measure to the values of A in any state $\psi$. We can also show that the possible values of the observable A in any state must belong to the spectrum of A. In the special case A has only discrete spectrum, the possible outcomes of measuring A are its eigenvalues.

More generally, a state can be represented by a so-called density operator, which is a trace class, nonnegative self-adjoint operator $\rho$ normalized to be of trace 1. The expected value of A in the state $\rho$ is
$$\text{tr}\left(A\rho\right)$$
If $\rho_\psi$ is the orthogonal projector onto the one-dimensional subspace of H spanned by $\psi$, then
$$\text{tr}\left(A\rho_\psi\right)=\langle\psi|A|\psi\rangle$$
Density operators are those that are in the closure of the convex hull of the one-dimensional orthogonal projectors. Conversely, one-dimensional orthogonal projectors are extreme points of the set of density operators. Physicists also call one-dimensional orthogonal projectors pure states and other density operators mixed states.

To that we only need to add the dynamics. This can be defined via a Hamiltonian, a Lagrangian, (if the system has a familiar classical counterpart) the correspondance rule (replace the Poisson bracket by a(n anti)commutator), or even directly by a dispersion rule or wave equation such as the Shrodinger equation.