In classical physics, the momentary state of a particle is given by it's location and it's velocity .

When we talk about the temporal behavior of dynamic systems, however, this notion of ``state'' is somewhat cumbersome to deal with, since by definition, the momentary state of the system changes constantly. This is especially true when it comes to periodic movement, so it is often more adequate to talk about the current orbit of a satellite (which remains constant until it is actively altered by outside intervention) than to give the actual coordinates (which permanently change).

So in a more abstract definition, the states of an isolated classical
system are the positions
of all included
particles as a function of time .^{1.2}

The above definition implies that the state of a system can only change when an interaction with another system occurs.

Typically, the duration of the interaction (e.g. the collision of 2 billiard-balls) is very small compared to the duration of the isolated states, so for practical purposes the interaction can often be assumed as instantaneous.

Isolated systems preserve their total energy and momentum^{1.3} ,
which are given as^{1.4}

(1.4) |

Legal physical states must obey a movement law which characterizes
the dynamics of a system. For classic one-particle systems, the
dynamic equation is known as Newton's Second Law

(1.5) |

Any momentary state of the system can be used as an initial value for the above equation to determine its temporal behavior.

In quantum physics, the state of a one-particle system is
characterized by a complex distribution function
with the normalization

(1.7) |

Two states differing by a constant phase factor are considered equivalent.

Particle Location

The classical notion of particle location is replaced by a
spatial probability distribution
,
which can be characterized by its expectation value
and its uncertainty , which are
defined as

(1.8) |

Time Dependency

When a classical system involves moving particles, the location of the particles is time dependent. This is not necessarily the case with quantum systems and the describing probability distribution : If the quantum state is of the form with , then is time independent.

Figure 1.1 shows a particle that is trapped between two reflecting
``mirrors''.^{1.5}A classical particle will move periodically from on end to another at a
constant speed, it's location can be described by a periodic
triangle-function of the time.
An undisturbed quantum particle in a similar trap, however, doesn't
have a defined location; the probability to ``meet'' (i.e. measure)
the particle at a certain location remains constant over
time^{1.6}but changes throughout space,
or in more physical terms, the particle forms a standing *wave* just
as a vibrating piano-string between 2 fixed ends.

A constant probability distribution is typical for bound states of defined energy, i.e. for particles trapped in a constant potential well, e.g. an electron in the electric field of a proton.

Expectation Values

It has been shown above how the classical concept of a well defined
particle location has been replaced by the quantum concept of a
statistical expectation value.
This correspondence, however, is not just restricted to
space. In fact, all classical physical quantities of a system can be
described as the expectation value of an appropriate *operator*
(see table 1.1 for some examples).

In analogy to equation 1.2.2.1, the expectation value
and the
uncertainty for an observable are defined as

(1.9) |

The quantum analogy to Newton's Third Law (see equation 1.6)
is the *Schrödinger Equation*

The Time-Independent Schrödinger Equation

If we take the simple case of a particle in a static
potential field , equation 1.10 can be written as

The time part is solved by with . is the energy of the state, since

(1.13) |

The remaining eigenvalue problem
is
also called the *time-independent Schrödinger Equation*.^{1.7}

Depending on the imposed boundary conditions, the Schrödinger
Equation is often only solvable for particular values of ,
i.e. it has a discrete *energy spectrum* and the possible
eigenvalues (also called energy terms) can be enumerated.
The solution for the lowest eigenvalue is called the ground-state
of the system.

Since for most physical applications, only the value of the energy terms is of importance, it is hardly ever necessary to actually compute the eigenstates.

It has been the discovery of discrete energy states, which gave quantum physics its name, as any state change from eigenstate to involves the exchange of an energy quantum .

Electron in a Capacitor

As an example, let's consider an electron in a capacitor. To keep
things simple, the capacitor should be modeled by an infinitely
deep, one-dimensional potential well (see also 1.2.2.2),
thus

(1.14) |

(1.15) |

(1.16) |

(1.17) |

(1.18) |

Figure 1.2 shows the first 3 eigenstates and and their corresponding spatial probability distributions .

3-dimensional Trap

The above example can easily be extended to 3 dimensions, using the
potential

(1.21) |

and the energies

(1.23) |

- ....
^{1.2} - Note, that since , this also includes the velocities.
- ... momentum
^{1.3} - There are several other conservation laws for angular momentum, electric charge, baryon count, etc. which are not mentioned here
- ... as
^{1.4} - The form of the potential energy actually defines the physical problem and can also depend on particle velocities, time, spins, etc.
- ...
``mirrors''.
^{1.5} - Mathematically, such ideal ``mirrors'' are described by infinitely deep potential-wells.
- ...
time
^{1.6} - This is only the case with eigenstates; in mixed states, the local probability can oscillate due to the different periods of the involved phase functions
- ... Equation.
^{1.7} - Note that this requires the Hamilton operator to be time-independent, which is not necessarily the case