Schrödinger’s wave equation

Schrödinger’s wave equation

\PMlinkescapephrase

wave equation

The

Schr

ö

dinger wave equation

is considered to be the most basic equation of non-relativistic quantum mechanics. In three spatial dimensions (that is, in

ℝ

3

) and for a single particle of mass

m

, moving in a field of potential energy

V

, the equation is

i

⁢

ℏ

⁢

∂

∂

⁡

t

⁢

Ψ

⁢

(

𝒓

,

t

)

=

-

ℏ

2

2

⁢

m

⋅

△

⁢

Ψ

⁢

(

𝒓

,

t

)

+

V

⁢

(

𝒓

,

t

)

⁢

Ψ

⁢

(

𝒓

,

t

)

,

where

𝒓

:=

(

x

,

y

,

z

)

is the position vector,

ℏ

=

h

⁢

(

2

⁢

π

)

-

1

,

h

is Planck’s

constant

,

△

denotes the

Laplacian

and

V

⁢

(

𝒓

,

t

)

is the value of the potential energy at point

𝒓

and time

t

.
This equation is a

second order

homogeneous

partial differential equation

which is used to determine

Ψ

, the so-called

time-dependent

wave function

, a complex function which describes the state of a physical system at a certain point

𝒓

and a time

t

(

Ψ

is thus a

function

of 4 variables:

x

,

y

,

z

and

t

). The right hand side of the equation represents in fact the

Hamiltonian operator

(

http://planetmath.org/HamiltonianOperatorOfAQuantumSystem

) (or energy operator)

H

⁢

Ψ

⁢

(

𝒓

,

t

)

, which is represented here as the sum of the kinetic energy and potential energy

operators

. Informally, a wave function encodes all the information that can be known about a certain quantum mechanical system (such as a particle). The function’s main

interpretation

is that of a

position probability density

for the particle

1

1

This is in fact a little imprecise since the wave function is, in a way, a statistical tool: it describes a large number of identical and identically prepared systems. We speak of the wave function of one particle for convenience.

(or system) it describes, that is, if

P

⁢

(

𝒓

,

t

)

is the probability that the particle is at position

𝒓

at time

t

,

then an important

postulate

of M. Born states that

P

⁢

(

𝒓

,

t

)

=

|

Ψ

⁢

(

𝒓

,

t

)

|

2

.

An example of a (relatively simple) solution of the equation is given by the wave function of an arbitrary (non-relativistic) free

2

2

By free particle, we imply that the field of potential energy

V

is everywhere

0

.

particle (described by a

wave packet

which is obtained by superposition of fixed momentum solutions of the equation). This wave function is given by:

Ψ

⁢

(

𝒓

,

t

)

=

∫

𝒦

A

⁢

(

𝒌

)

⁢

e

i

⁢

(

𝒌

⋅

𝒓

-

ℏ

⁢

𝒌

2

⁢

(

2

⁢

m

)

-

1

⁢

t

)

⁢

𝑑

𝒌

,

where

𝒌

is the

wave vector

and

𝒦

is the set of all values taken by

𝒌

.

For a free particle, the equation becomes

i

⁢

ℏ

⁢

∂

∂

⁡

t

⁢

Ψ

⁢

(

𝒓

,

t

)

=

-

ℏ

2

2

⁢

m

⋅

△

⁢

Ψ

⁢

(

𝒓

,

t

)

and it is easy to check that the aforementioned wave function is a solution.

An important special case is that when the energy

E

of the system

does not depend on time

, i.e.

H

⁢

Ψ

=

E

⁢

Ψ

, which gives rise to the

time-independent Schrödinger equation

:

E

⁢

Ψ

⁢

(

𝒓

)

=

-

ℏ

2

2

⁢

m

⋅

△

⁢

Ψ

⁢

(

𝒓

)

+

V

⁢

(

𝒓

)

⁢

Ψ

⁢

(

𝒓

)

.

There are a number of

generalizations

of the Schrödinger equation, mostly in order to take into account special relativity, such as the

Dirac equation

(which describes a spin-

1

2

particle with mass) or the

Klein-Gordon equation

(describing spin-

0

particles).

Title

Schrödinger’s wave equation

Canonical name

SchrodingersWaveEquation

Date of creation

2013-03-22 15:02:31

Last modified on

2013-03-22 15:02:31

Owner

Cosmin (8605)

Last modified by

Cosmin (8605)

Numerical id

28

Author

Cosmin (8605)

Entry type

Topic

Classification

msc 81Q05

Classification

msc 35Q40

Synonym

Schrödinger’s equation

Synonym

time-independent Schrödinger wave equation

Related topic

SchrodingerOperator

Related topic

HamiltonianOperatorOfAQuantumSystem

Related topic

Quantization

Related topic

DiracEquation

Related topic

KleinGordonEquation

Related topic

PauliMatrices

Related topic

DAlembertAndDBernoulliSolutionsOfWaveEquation

Defines

wave function

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