Quantum mechanics is our current model of the microscopic world.  Like all models, it is created by people for people. 

Quantum mechanics divides the world into two parts, commonly called the system and the observer.  The system is the part of the world that is being modeled.  The rest of the world is the observer.  An interaction between the observer and the system is called a measurement.  Properties of the system that can be measured are called observables.  Examples are position, momentum, angular momentum, energy, etc.  The initial information the observer has about the system comes from a set of measurements.  (This is the same as in classical physics.)  The state of the system represents this information, which can be cast into different mathematical forms.  It is often represented in terms of a wave function.  Quantum mechanics predicts how the state of the system evolves and therefore how the information the observer has about the system evolves with time.  Some information is retained, and some is lost.  The evolution of the state is deterministic.  The Schroedinger equation describes this evolution.  Measurements at a later time provide new information, and therefore the state of the system, in general, changes after the measurements.  The wave function of the system, in general, changes after a measurement.

The wave function has no direct physical meaning.  It is just one way of storing information.  It stores all the information available to the observer about the system.  To make predictions about the outcome of all measurements, at any time, one has to "do" something to the wave function to extract information.  One has to perform some mathematical operation on it, such a squaring it, multiplying it by a constant, differentiating it, etc.  One has to operate on the wave function with some operator.  (The operator is a specific instruction or set of instructions.)  Every observable is associated with its own operator.  Operations can change the information the observer has about the system and therefore can change the wave function, or they can preserve it.  If the initial information about a particle came from a particular measurement, then the resulting wave function is said to be an eigenfunction of the operator associated with this measurement.  If the operator associated with a different observable does not change this eigenfunction, then the two measurements are said to be compatible.  We can know the value of both observable simultaneously.  But if the operator associated with a different observable changes the eigenfunction of the first observable, then the two observables are incompatible.  We cannot know the value of both observable simultaneously with arbitrary precision.  The uncertainties in the values of both observables will be related by a generalized uncertainty principle.

In quantum mechanics, a measurement of an observable yields a value, called an eigenvalue of the observable.  Many observables have quantized eigenvalues, i.e. a measurement can only yield one of a discrete set of values.  Right after the measurement, the state of the system is an eigenstate of the observable, which means that the value of the observable is exactly known.  A state can be a simultaneous eigenstate of several observable, which means that the observer can exactly know the value of several properties of the system at the same time and make exact predictions about the outcome of measurements of those properties.  But there are also incompatible observables whose exact values cannot be known to the observer at the same time.  A state cannot be a simultaneous eigenstate of incompatible observables.  If it is in an eigenstate of one of the incompatible observables and the value of this observable is known, then quantum mechanics gives only the probabilities for measuring each of the different eigenvalues of the other incompatible observables.  The outcome of a measurement of any of the other incompatible observables is uncertain.  A measurement of one of the other incompatible observables changes the state of the system to one of its eigenstates and destroys the information about the value of the first observable.  To completely specify the initial state of a system with n degrees of freedom, we have to make up to n compatible measurements.  A single electron, for example, has four degrees of freedom, the three degrees of freedom associated with moving in a three-dimensional world, and one internal degree of freedom associated with its spin.  To specify the state of the electron, we have to make up to four compatible measurements.  The possible outcome of the measurements are usually not specified directly, but through labels called quantum numbers. 

• The quantum state of a particle is characterized by a wave function Ψ(r,t), which contains all the information about the system an observer can possibly obtain.
• The wave function Ψ(r,t) is interpreted as a probability amplitude of the particles presence.  |Ψ(r,t)|² is the probability density.  The probability that a particle is at time t in a volume element d³r situated at r is  dP(r,t) = C|Ψ(r,t)|²d³r.  For a single particle the total probability of finding it anywhere in space at time t is equal to 1.  (In non-relativistic Quantum Mechanics, material particles, unlike photons, are neither created nor destroyed.)
  ∫_{all space}P(r,t)d³r = 1,   C∫_{all space}|Ψ(r,t)|²d³r = 1,   ∫_{all space}|Ψ(r,t)|²d³r = finite.
  A proper wave function must be square-integrable.  The wave function must also be single valued and continuous for the probability amplitude interpretation to make sense.
• The principle of spectral decomposition applies to the measurement of an arbitrary physical quantity A.
  • The result of a measurement belongs to a set of eigenvalues {a}.
  • Each eigenvalue is associated with an eigenfunction Ψ_a(r).
    If Ψ(r,t₀) = Ψ_a(r) then a measurement of A at t = t₀ will yield the eigenvalue a.
  • Any Ψ(r,t₀) can be expanded in terms of eigenfunctions, Ψ(r,t₀) = ∑_a c_a Ψ_a(r).
    The probability that a measurement at t = t₀ will yield the eigenvalue a’ is P_a' = |c_a'|²/(∑_a|c_a|²).
  • If a measurement of A yields a, then the wave function immediately after the measurement is Ψ_a(r).
• The Schroedinger equation describes the evolution of Ψ(r,t).
  iħ∂Ψ(r,t)/∂t = -(ħ²/(2m))∇²Ψ(r,t) + U(r,t)Ψ(r,t)
  is the Schroedinger equation for a particle of mass m whose potential energy is given by U(r,t).