Postulating the existence of photons and matter waves were the first steps towards developing our current model of the microscopic world. That model is quantum mechanics. It is a mathematical model, and exploring its details requires a fair amount of mathematical preparation. The following is a largely non-mathematical summary of the model's main features.
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 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. Quantum mechanics does not really describe the system, but the information that the rest of the world can possibly have about the system.
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.
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 the 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.
If you want to make predictions about the energy of a particle, you have to
operate on its wave function with the energy operator.
If you want to make predictions about the momentum of a particle, you have
to operate on its wave function with the momentum operator.
Operations (measurements) can change the information that the observer already has about the system and therefore
can change the wave function, or they can preserve it.
(When you do something to a wave function, you may change it in the process.)
The wave function immediately after a measurement is said to be an eigenfunction or eigenstate 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. But if the operator associated with a different observable changes the eigenfunction of the first observable, then the two observables are incompatible.
In quantum mechanics, a measurement of an observable yields a value,
called an eigenvalue of the observable. 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 values 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 second
measurement of an observable incompatible with the first one changes the
state of the system to an eigenstates of the second observable and destroys
the information about the value of the first observable. The observer cannot know
the value of both incompatible observables simultaneously with arbitrary precision.
The uncertainties in the values of both observables will be related by a
generalized uncertainty principle.
[In summary, a state can be a simultaneous eigenstate of several compatible observable, which means that the observer can exactly know the values of several properties of the system at the same time and make exact predictions about the outcome of measurements of those properties. But a state cannot be a simultaneous eigenstate of incompatible observables. If a system 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.]
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 a measurement. The wave function of the system, in general, changes after a measurement.
Many observables have quantized eigenvalues, i.e. a measurement can only yield one of a discrete set of values. 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 total energy of an electron in a hydrogen
atom is quantized.
The values we will measure will always be equal to -13.6 eV/n2,
where n is and integer, n = 1, 2, 3, ... .
We can measure E1 = -13.6 eV, E2 = -13.6/4 eV, E3
= -13.6/9 eV, etc.
Instead of writing out the numerical values, we label these energy level by
the quantum number n, n = 1, 2, 3, etc.
Four quantum numbers labeling the outcome of 4 different measurements
completely specify the state of an electron in the hydrogen atom.