Chapter 5

Schrödinger equation

Complément : A conference on Quantum mechanics, by Serge Haroche (nobel Prize 2012)

A conference on Quantum mechanics, by Serge Haroche (nobel Prize 2012)

Fondamental : Link with light interferences

The following picture describes the double slit experiment realized for the first time in 1801 by Young.

Double slit experiment

The intensity in a point M of coordinate on the screen is expressed by :

The light source has emitted photons during a time period of .

The intensity of the source is written .

Let be the number of photons that arrive in point M on the screen during this period of time .

This number is proportional to the intensity received by the point M.

The proportion of photons, in point M is :

Which is the amplitude of the wave that describes the photon, squared.

Fondamental : Definition of the wave function

The physical state of a quantum particle is perfectly defined by its complex wave function which represents a probability amplitude.

The probability of presence of the particle, between the abscissa and is given by :

is a density of probability of presence which verifies the norm condition :

The complete description of the state of a particle of mass in space at time is possible with this wave function.

The wave function holds all of the available information : there exists no other element in quantum formalism that could let us know, before measuring, where the particle will be detected.

This probabilistic and random nature is not the result of a poor knowledge of initial conditions (as in kinetic gas theory for instance), but rather forms an integral part of quantum formalism.

The assumptions stated here for only one particle are still true for a set of particles.

A global wave function, depending on all coordinates of all particles as well as time, can be defined to describe this set of particles.

Fondamental : Schrödinger's equation (1926)

A quantum particle is put in an area of space where a potential is applied.

The potential does not depend on time.

Remark : in quantum mechanics, potential and potential energy are two equivalent terms.

Let be the wave function of this particle.

It depends on the abscissa and time .

This complex wave function verifies Schrödinger's equation :

This equation is similar to a wave equation that could be found in electromagnetism.

Fondamental : Schrödinger's time-independent equation

The solutions sought are "separated variable" solutions, also called stationary, to Schrodinger's equation.

They are expressed by :

Schrödinger's equation is verified :

Hence :

However, a function which depends on can only be equal to a function that depends on at any instant if the two functions are constant :

The time differential equation gives :

In the particular case of a free particle which has the following wave function :

It is clear that the constant must be identified to the energy of the particle, which stays constant in the potential (stationary states).

Finally :

The space differential equation gives :

The equation obtained is Schrödinger's equation for the space part of the wave function of a stationary state (or Schrödinger's equation for stationary states) :

Attention : Schrödinger's time-independent equation

The stationary states (of energy constant) of Schrodinger's equation in the case of a time-independent potential energy are :

The space part of the wave function verifies Schrodinger's time-independent equation :

The density of probability does not depend on time :

Fondamental : Mathematical properties of a wave function

The following table shows the mathematical properties of the wave function, under the mathematical form of the potential .

Mathematical properties of a wave function

Remarque : Superposition principle

It is important to notice that Schrödinger's equation is linear.

This implies that if two solutions verify this equation, then any linear combination of these two solutions is also a solution.

Hence, if two wave functions that correspond to two different physical situations can be identified, then any physical situation corresponding to a mix of these two situations is also possible.

A famous paradox, Schrodinger's cat, is presented in the following video.

Schrödinger's cat

Complément :

Is a quantum wavefunction a real thing ?
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