What is Quantum Mechanics?

 Quantum mechanics is a framework in physics that describes systems of particles as probabilistic states. These states are aptly referred to as quantum states and unlike classical mechanics, are a level of abstraction above the literal spatial position of particles which are classically the variables of primary interest. The key point is that whereas we speak about position as a function of time in classical physics, we instead think in terms of the quantum state as a function of time in quantum mechanics as it becomes the critical construct or dynamic variable of interest in the formalism. The quantum state is loosely the square root of the probability distribution for finding the system in a particular (eigen)state.

Quantum states are represented mathematically in the language of linear algebra. In other words, they can be represented by vectors acted on by matrices. These vectors can be represented in different bases and the components of the vectors in a particular (eigen)basis can have their norm-squared to yield the probability of the quantum state being measured in a particular eigenstate (of a corresponding physical operator such as energy, position, momentum etc.) Since these state vectors must have a norm of 1 (as they describe probabilities which must sum to 1, that is, the physical system must be in some state), these state vectors are evolved through the action of unitary matrix transformations which have the mathematical property that they preserve the length of the vectors on which they act. These unitary transformations are in turn generated by Hermitian operators which have the property that they are equivalent to their conjugate transpose implying that their eigenvalues are real, these Hermitian operators and their eigenvalues have a natural interpretation then as real observable quantities like energy levels e.g., of the hydrogen atom. 

In particular, time evolution of the quantum state is generated by the Hamiltonian operator which is a fancy way of saying the energy operator. The celebrated Schrodinger equation is simply the statement that a Hermitian operator H (the Hamiltonian) generates the unitary time evolution of the quantum state. Indeed, equivalent counterpart equations exist for Hermitian operators p (momentum) and J (angular momentum) generating the unitary spatial and angular evolution of the quantum state. Though most introductory physics textbooks begin with the Schrodinger equation and its solutions for spatial wavefunctions, I find it more holistic and conceptually sound to begin with the core assumptions of the quantum formalism: unitarity, linearity and probabilistic constructs or states. From those core ingredients, much of the remaining content emerges or can be meaningfully motivated.