Defining Quantum Mechanics

Does

• Schrödinger's Equation (Operator form)

• $[\hat{X},\hat{P}]=i$

• Born Rules

define Quantum Mechanics?

• Definitely not. There are quantum systems that don't have position and momentum. For example, consider the spin of an electron. – DanielSank Apr 15 '17 at 4:26
• I'm not sure what you mean by "defining" quantum mechanics. For comparison, what do you think "defines" classical mechanics? – ACuriousMind Apr 15 '17 at 11:38
• Quantum mechanics is based on solutions of quantum mechanical equations , plus a number of postulates. see this link hyperphysics.phy-astr.gsu.edu/hbase/quantum/qm.html – anna v Apr 17 '17 at 3:36

Quantum Mechanics is a framework to describe phenomena according to a set of postulates that tells you how to describe states of the physical system, how to describe physical quantities of the system and how to understand the time evolution of that system.

Important to understand is: Quantum Mechanics is not a theory of a particular phenomenon but a framework that can describe lots of phenomena.

The whole point is: there are systems often said "quantum" which cannot be described according to the rules of Classical Mechancs. Atoms, molecules and subatomic particles for example, all fall in this category. These kind of systems motivated the construction of this new framework.

Whether or not this is the right framework to describe a certain phenomena can only be judged by experiment.

Anyway, the frameowrk can be concisely stated as a set of rules. Quantum Mechanics can be described by six postulates plus some ideas carried over from prior developments in Classical Mechanics (that is the procedure to quantize systems that have classical descriptions). The postulates are:

1. The states of a system are described by unit vectors in a Hilbert space.
2. The physical quantities associated to the system are described by hermitian operators on said Hilbert space, which are called observables.
3. The possible values that an observable can take are its eigenvalues.
4. If an observable $A$ has discrete spectrum with basis $|\varphi_i^{(j)}\rangle$ with $A|\varphi_i^{(j)}\rangle = a_i |\varphi_i^{(j)}\rangle$ then the probability of measuring $a_i$ in the state $|\psi\rangle$ is $P(a_i)=\sum_{i}|\langle \varphi_i^{(j)}|\psi\rangle|^2$. If $A$ has continuous spectrum with basis $|a\rangle$, the probability density of the values of $A$ in the state $|\psi\rangle$ is $\rho(a)=|\langle a|\psi\rangle|^2$.
5. Immediately after the measurement of $A$ in the state $|\psi\rangle$, the system will be in the state obtained by projecting and normalizing $|\psi\rangle$ to the eigenspace corresponding to the measured value.
6. The time evolution of the state is such that the energy is the generator of such transformation. In other words, the equation $i\hbar \partial_t |\psi(t)\rangle = H|\psi(t)\rangle$ holds where $H$ is the energy observable, called Hamiltonian.

This defines Quantum Mechanics. The quantization procedure you outline - that is, turn $X,P$ to observables with $[X,P]=i$ - is just a requirement that: if you have a classical description of a system and want a quantum one, then momentum will also be the generator of translations in the canonicaly conjugate variable in the quantum sense as it was in the classical.

But quantization is just a particular situation where you get a quantum description of a system. Spin for example is not the result of any quantization.

In summary, Quantum Mechanics is a whole set of rules/ideas to describe specific phenomena within that context, and cannot be tied down utimately to one or a few equations only.