aarjaneiro/planetmath huggingface dataset:
first order operators in Riemannian geometry

first order operators in Riemannian geometry

On a

pseudo-Riemannian manifold

M

, and in Euclidean space in
particular, one can express the

gradient

operator

, the divergence
operator, and the curl operator (which makes sense only if

M

is
3-dimensional) in terms of the

exterior derivative

. Let

𝒞

∞

⁢

(

M

)

denote
the ring of

smooth functions

on

M

; let

𝒳

⁢

(

M

)

denote the

𝒞

∞

⁢

(

M

)

-module of smooth vector fields, and let

Ω

1

⁢

(

M

)

denote the

𝒞

∞

⁢

(

M

)

-module of smooth 1-forms. The

contraction

with the

metric tensor

g

and its

inverse

g

-

1

, respectively, defines the

𝒞

∞

⁢

(

M

)

-module

isomorphisms

♭

:

𝒳

⁢

(

M

)

→

Ω

1

⁢

(

M

)

,

♯

:

Ω

1

⁢

(

M

)

→

𝒳

⁢

(

M

)

.

In

local coordinates

, this isomorphisms is expressed as

(

∂

∂

⁡

x

i

)

♭

=

∑

j

g

i

⁢

j

⁢

d

⁢

x

j

,

(

d

⁢

x

j

)

♯

=

∑

i

g

i

⁢

j

⁢

∂

∂

⁡

x

i

.

or as the lowering of an index. To wit, for

V

∈

𝒳

⁢

(

M

)

, we have

V

=

∑

i

=

1

n

V

i

⁢

∂

∂

⁡

x

i

,

(

V

♭

)

j

=

∑

i

=

1

n

g

i

⁢

j

⁢

V

i

,

j

=

1

,

…

,

n

.

The gradient operator, which in tensor notation is expressed as

(

grad

⁡

f

)

i

=

g

i

⁢

j

⁢

∂

⁡

f

∂

⁡

x

j

,

f

∈

𝒞

∞

⁢

(

M

)

,

can now be defined as

grad

⁡

f

=

(

d

⁢

f

)

♯

,

f

∈

𝒞

∞

⁢

(

M

)

.

Another natural

structure

on an

n

-dimensional

Riemannian manifold

is
the

volume form

,

ω

∈

Ω

n

⁢

(

M

)

, defined by

ω

=

det

⁡

g

i

⁢

j

⁢

d

⁢

x

1

∧

…

∧

d

⁢

x

n

.

Multiplication

by
the volume form defines a natural isomorphism between

functions

and

n

-forms:

f

↦

f

⁢

ω

,

f

∈

𝒞

∞

⁢

(

M

)

.

Contraction with the
volume form defines a natural isomorphism between vector fields and

(

n

-

1

)

-forms:

X

↦

X

⁢

⌋

ω

,

X

∈

𝒳

⁢

(

M

)

,

or
equivalently

∂

∂

⁡

x

i

↦

(

-

1

)

i

+

1

⁢

det

⁡

g

i

⁢

j

⁢

d

⁢

x

1

∧

…

∧

d

⁢

x

i

^

∧

…

∧

d

⁢

x

n

,

where

d

⁢

x

i

^

indicates an omitted factor. The divergence
operator, which in tensor notation is expressed as

div

⁡

X

=

∇

i

⁡

X

i

,

X

∈

𝒳

⁢

(

M

)

can be defined in a coordinate-free way by the following

relation

:

(

div

⁡

X

)

⁢

ω

=

d

⁢

(

X

⁢

⌋

ω

)

,

X

∈

𝒳

⁢

(

M

)

.

Finally, on a

3

-dimensional

manifold

we may define the curl
operator in a coordinate-free fashion by means of the following relation:

(

curl

⁡

X

)

⁢

⌋

ω

=

d

⁢

(

X

♭

)

,

X

∈

𝒳

⁢

(

M

)

.

Title

first order operators in Riemannian geometry

Canonical name

FirstOrderOperatorsInRiemannianGeometry

Date of creation

2013-03-22 15:28:18

Last modified on

2013-03-22 15:28:18

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

7

Author

rmilson (146)

Entry type

Definition

Classification

msc 70G45

Classification

msc 53B20

Related topic

Gradient

Related topic

Curl

Related topic

Divergence

Related topic

LeibnizNotationForVectorFields

Generated on Sat Feb 10 12:51:19 2018 by

LaTeXML
Klein-Gordon equation

Klein-Gordon equation

The

Klein-Gordon equation

is an equation of

mathematical physics

that describes spin-0 particles. It is given by:

□

⁢

ψ

=

(

m

⁢

c

ℏ

)

2

⁢

ψ

Here the

□

symbol refers to the

wave operator

, or D’Alembertian, (

□

=

∇

2

-

1

c

2

⁢

∂

t

2

)
and

ψ

is the

wave function

of a particle.
It is a Lorentz invariant

expression

.

0.1

Derivation

Like the

Dirac equation

, the Klein-Gordon equation is derived from the relativistic expression for total energy:

E

2

=

m

2

⁢

c

4

+

p

2

⁢

c

2

Instead of taking the square root (as Dirac did), we keep the equation in squared form and replace the momentum and energy with their

operator

equivalents

,

E

=

i

⁢

ℏ

⁢

∂

t

,

p

=

-

i

⁢

ℏ

⁢

∇

. This gives (in disembodied operator form)

-

ℏ

2

⁢

∂

2

∂

⁡

t

2

=

m

2

⁢

c

4

-

ℏ

2

⁢

c

2

⁢

∇

2

Rearranging:

ℏ

2

⁢

(

c

2

⁢

∇

2

-

∂

2

∂

⁡

t

2

)

=

m

2

⁢

c

4

Dividing both sides by

ℏ

2

⁢

c

2

:

(

∇

2

-

1

c

2

⁢

∂

2

∂

⁡

t

2

)

=

m

2

⁢

c

2

ℏ

2

Identifying the expression in brackets as the D’Alembertian and right-multiplying the whole expression by

ψ

, we obtain the Klein-Gordon equation:

□

⁢

ψ

=

(

m

⁢

c

ℏ

)

2

⁢

ψ

Title

Klein-Gordon equation

Canonical

name

KleinGordonEquation

Date of creation

2013-03-22 17:55:11

Last modified on

2013-03-22 17:55:11

Owner

invisiblerhino (19637)

Last modified by

invisiblerhino (19637)

Numerical id

12

Author

invisiblerhino (19637)

Entry type

Definition

Classification

msc 78A25

Classification

msc 35Q60

Synonym

Klein Gordon equation

Synonym

Klein-Gordon-Fock equation

Related topic

DiracEquation

Related topic

SchrodingersWaveEquation

Generated on Sat Feb 10 13:04:04 2018 by

LaTeXML
derivation of a definite integral formula using the method of exhaustion

derivation of a definite integral formula using the method of exhaustion

The area under an arbitrary

function

f

⁢

(

x

)

that is

piecewise continuous

on

[

a

,

b

]

can be ”exhausted” with

triangles

. The first triangle has vertices at

(

a

,

0

)

and

(

b

,

0

)

, and

intersects

f

⁢

(

x

)

at

x

=

a

+

b

-

a

2

,

yielding the estimate

A

1

=

1

2

⁢

(

b

-

a

)

⁢

f

⁢

(

a

+

b

-

a

2

)

The second approximation involves two triangles, each sharing two vertices with the original triangle, and intersecting

f

⁢

(

x

)

at

x

=

a

+

b

-

a

4

and

x

=

a

+

3

⁢

(

b

-

a

)

4

,

adding the area:

A

2

=

1

4

⁢

(

b

-

a

)

⁢

{

f

⁢

(

a

+

b

-

a

4

)

-

f

⁢

(

a

+

b

-

a

2

)

+

f

⁢

(

a

+

3

⁢

(

b

-

a

)

4

)

}

A third such approximation involves four more triangles, adding the area

A

3

=

1

8

(

b

-

a

)

{

f

(

a

+

b

-

a

8

)

-

f

(

a

+

b

-

a

4

)

+

f

⁢

(

a

+

3

⁢

(

b

-

a

)

8

)

-

f

⁢

(

a

+

b

-

a

2

)

+

f

⁢

(

a

+

5

⁢

(

b

-

a

)

8

)

-

f

(

a

+

3

⁢

(

b

-

a

)

4

)

+

f

(

a

+

7

⁢

(

b

-

a

)

8

)

}

.

This procedure eventually leads to the

formula

∫

a

b

f

⁢

(

x

)

⁢

𝑑

x

=

∑

n

=

1

∞

A

n

=

(

b

-

a

)

⁢

∑

n

=

1

∞

∑

m

=

1

2

n

-

1

(

-

1

)

m

+

1

⁢

2

-

n

⁢

f

⁢

(

a

+

m

⁢

(

b

-

a

)

/

2

n

)

References

1.

http://arxiv.org/abs/math.CA/0011078

http://arxiv.org/abs/math.CA/0011078.

2.

Int. J. Math. Math. Sci. 31, 345-351, 2002.

Title

derivation

of a

definite integral

formula using the

method of exhaustion

Canonical

name

DerivationOfADefiniteIntegralFormulaUsingTheMethodOfExhaustion

Date of creation

2013-03-22 14:56:35

Last modified on

2013-03-22 14:56:35

Owner

ruffa (7723)

Last modified by

ruffa (7723)

Numerical id

22

Author

ruffa (7723)

Entry type

Derivation

Classification

msc 78A45

Classification

msc 30B99

Classification

msc 26B15

Generated on Sat Feb 10 13:04:06 2018 by

LaTeXML
closed monoidal category

closed monoidal category

Let

𝒞

be a

monoidal category

, with

tensor product

⊗

. Then we say that

•

𝒞

is

closed

, or

left closed

, if the

functor

A

⊗

-

on

𝒞

has a

right adjoint

[

A

,

-

]

l

•

𝒞

is

right closed

if the functor

-

⊗

B

on

𝒞

has a right adjoint

[

B

,

-

]

r

•

𝒞

is

biclosed

if it is both left closed and right closed.

A biclosed

symmetric monoidal category

is also known as a

symmetric monoidal closed

category

. In a symmetric monoidal closed category,

A

⊗

B

≅

B

⊗

A

, so

[

A

,

B

]

l

≅

[

A

,

B

]

r

. In this case, we denote the right adjoint by

[

A

,

B

]

.

Some examples:

•

Any

cartesian closed category

is symmetric monoidal closed.

•

In particular, as a category with finite

products

is

symmetric

monoidal

, it is biclosed iff it is cartesian closed.

•

An example of a biclosed monoidal category that is not symmetric monoidal is the category of

bimodules

over a

non-commutative

ring. The right adjoint of

A

×

-

is

[

A

,

-

]

l

, where

[

A

,

B

]

l

is the collection of all left

R

-linear

bimodule homomorphisms

from

A

to

B

, while the right adjoint of

-

×

A

is

[

A

,

-

]

r

, where

[

A

,

B

]

r

is the collection of all right

R

-linear bimodule homomorphisms from

A

to

B

. Unless

R

is

commutative

,

[

A

,

B

]

l

≠

[

A

,

B

]

r

in general.

more to come…

Title

closed monoidal category

Canonical name

ClosedMonoidalCategory

Date of creation

2013-03-22 18:30:25

Last modified on

2013-03-22 18:30:25

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 81-00

Classification

msc 18-00

Classification

msc 18D10

Related topic

IndexOfCategories

Defines

left closed

Defines

right closed

Defines

biclosed

Defines

symmetric monoidal closed

Generated on Sat Feb 10 13:04:38 2018 by

LaTeXML
representation of a C_c⁢(𝖦) *- topological algebra

representation of a

C

c

⁢

(

𝖦

)

*- topological algebra

Definition 0.1

.

A

representation

of a

C

c

⁢

(

G

)

(

http://planetmath.org/C_cG

) topological

*

–

algebra

is defined as

a

continuous

*

–

morphism

from

C

c

⁢

(

G

)

to

B

⁢

(

H

)

, where

G

is a

topological
groupoid

, (usually a locally compact groupoid,

G

l

⁢

c

),

H

is a

Hilbert space

,
and

B

⁢

(

ℋ

)

is the

C

*

-algebra of

bounded linear operators

on the Hilbert space

ℋ

;
of course, one considers the

inductive limit

(strong)

topology

to be defined on

C

c

⁢

(

𝖦

)

,
and then also an operator

weak

topology to be defined on

B

⁢

(

ℋ

)

.

Title

representation of a

C

c

⁢

(

𝖦

)

*-

topological algebra

Canonical name

RepresentationOfACcmathsfGTopologicalAlgebra

Date of creation

2013-03-22 18:16:24

Last modified on

2013-03-22 18:16:24

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

21

Author

bci1 (20947)

Entry type

Definition

Classification

msc 81-00

Classification

msc 18D05

Classification

msc 55N33

Classification

msc 55N20

Classification

msc 55P10

Classification

msc 55U40

Synonym

groupoid

C*-algebra representations

Related topic

C_cG

Related topic

BoundedOperatorsOnAHilbertSpaceFormACAlgebra

Related topic

GelfandNaimarkSegalConstruction

Defines

representation of a topological *- algebra

Generated on Sat Feb 10 13:04:48 2018 by

LaTeXML
supercategory of variable molecular sets

supercategory of variable molecular sets

Definition 0.1

.

A

supercategory of variable molecular sets

(or

molecular set variables

)

,

§

m

⁢

s

⁢

v

is defined
as the

supercategory

whose class of objects is the class of all possible

variable molecular sets

,

m

⁢

s

⁢

v

’s, (or molecular set variables), and whose class of

morphisms

is the class of all molecular transformations,

η

t

, of

m

⁢

s

⁢

v

’s.

Remark

All such

m

⁢

s

⁢

v

–supercategories are

semantic

interpretations

of the

ETAS axioms

(syntax).
This is a

mathematical representation

of chemical reaction systems in terms of molecular sets that vary with time (or

m

⁢

s

⁢

v

’s, and their

transformations

as a result of diffusion, collisions, and chemical reactions.

References

1

Bartholomay, A. F.: 1960.

Molecular Set Theory

. A mathematical representation for chemical reaction mechanisms.

Bull. Math. Biophys.

,

22

: 285-307.

2

Bartholomay, A. F.: 1965. Molecular Set Theory: II. An aspect of biomathematical

theory of sets

.,

Bull. Math. Biophys.

27

: 235-251.

3

Bartholomay, A.: 1971. Molecular Set Theory: III. The Wide-Sense Kinetics of Molecular Sets .,

Bulletin of

Mathematical Biophysics

,

33

: 355-372.

4

Baianu, I. C.: 1983,

Natural Transformation

Models in Molecular Biology., in

Proceedings of the SIAM Natl. Meet

., Denver, CO.; Eprint No. 3675 at cogprints.org/3675/01 as “Naturaltransfmolbionu6.pdf”.

5

I.C. Baianu: 1970,

Organismic Supercategories

: II. On Multistable Systems.

Bulletin of Mathematical Biophysics

,

32

: 539-561.

6

I.C. Baianu, Brown R., J. F. Glazebrook, and Georgescu G.: 2006, Complex Nonlinear Biodynamics in

Categories

,

Higher Dimensional Algebra

and Łukasiewicz–Moisil Topos: Transformations of
Neuronal, Genetic and Neoplastic

networks

,

Axiomathes

16

Nos. 1–2, 65–122.

7

I.C. Baianu: 1980,

Natural Transformations of Organismic Structures

.

Bulletin of Mathematical
Biophysics

42

: 431-446.

8

I.C. Baianu: 1984, A Molecular-Set-Variable Model of Structural and Regulatory Activities in Metabolic and

Genetic Networks

.,

FASEB Proceedings

43

, 917.

Title

supercategory of variable molecular sets

Canonical name

SupercategoryOfVariableMolecularSets

Date of creation

2013-03-22 18:30:08

Last modified on

2013-03-22 18:30:08

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

5

Author

bci1 (20947)

Entry type

Definition

Classification

msc 81-00

Classification

msc 92B05

Classification

msc 18-00

Synonym

supercategory of molecular set variables

Synonym

m

⁢

s

⁢

v

-supercategory

Related topic

CategoryOfMolecularSets

Related topic

ETAS

Related topic

MolecularSetVariable

Related topic

MolecularSetTheory

Related topic

SupercategoriesOfComplexSystems

Related topic

ComplexSystemsBiology

Related topic

AbstractRelationalBiology

Defines

m

⁢

s

⁢

v

-supercategory

Defines

classes of molecular set variables and their chemical transformations with time

Generated on Sat Feb 10 13:04:50 2018 by

LaTeXML
symmetric monoidal category

symmetric monoidal category

A

monoidal category

𝒞

with

tensor product

⊗

is said to be

symmetric

if for every pair

A

,

B

of objects in

𝒞

, there is an

isomorphism

s

A

⁢

B

:

A

⊗

B

≅

B

⊗

A

that is natural in both

A

and

B

such that the following

diagrams

are

commutative

1.

(

unit coherence for

s

):

\xymatrix@

+

=

2

⁢

c

⁢

m

⁢

A

⊗

I

⁢

\xy@@ix@

⁢

Generated on Sat Feb 10 13:05:01 2018 by

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Hamiltonian operator

Hamiltonian operator

Definition 0.1

The

Hamiltonian operator

H

introduced in quantum mechanics (QM) by Schroedinger (and thus sometimes also called the

Schroedinger

operator

) on the Hilbert space

L

2

⁢

(

ℝ

n

)

is given by the action:

ψ

↦

[

-

∇

2

+

V

⁢

(

x

)

]

⁢

ψ

,

ψ

∈

L

2

⁢

(

ℝ

n

)

,

The operator defined above

[

-

∇

2

+

V

⁢

(

x

)

]

, for a potential function

V

⁢

(

x

)

specified as the real-valued function

V

:

ℝ

n

→

ℝ

is called the

Hamiltonian operator

,

ℍ

, and only very rarely the

Schrödinger operator

.

0.1

Schroedinger formulation of QM

The energy conservation (quantum) law written with the operator

ℍ

as the
Schrödinger equation is fundamental in quantum mechanics and is perhaps the most utilized, mathematical

computation

device in quantum mechanics of systems with a finite number of degrees of freedom. There is also, however, the alternative approach in the Heisenberg picture, or formulation, in which the observable and other operators are time-dependent whereas the state vectors

ψ

are time-independent, which reverses the time dependences betwen operators and state vectors from the more popular Schrödinger formulation. Other formulations of

quantum theories

occur in

quantum field theories

(QFT), such as QED (

quantum electrodynamics

) and QCD (quantum chromodynamics).

0.2

Heisenberg formulation of QM

Although the two formulations, or pictures, are unitarily (or mathematically)

equivalent

, however, sometimes the claim is made that the Heisenberg picture is “more natural and fundamental than the Schrödinger” formulation because the Lorentz invariance from General Relativity is also encountered in the Heisenberg picture,
and also because there is a ‘correspondence’ between the

commutator

of an observable operator with the Hamiltonian operator, and the

Poisson bracket

formulation of classical mechanics. If the state vector

ψ

, or

|

ψ

⟩

does not change with time as in the Heisenberg picture, then the ‘equation of motion’
of a (quantum) observable operator is :

d

d

⁢

t

⁢

A

q

⁢

u

⁢

a

⁢

n

⁢

t

⁢

u

⁢

m

=

(

i

⁢

ℏ

)

-

1

⁢

[

A

,

H

]

+

(

∂

⁡

A

∂

⁡

t

)

c

⁢

l

⁢

a

⁢

s

⁢

s

⁢

i

⁢

c

⁢

a

⁢

l

Title

Hamiltonian operator

Canonical name

HamiltonianOperator

Date of creation

2013-03-22 18:11:32

Last modified on

2013-03-22 18:11:32

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

34

Author

bci1 (20947)

Entry type

Definition

Classification

msc 81P05

Classification

msc 81Q05

Classification

msc 81-00

Synonym

Schrödinger operator

Synonym

Schrödinger formulation of QM

Related topic

QuantumSystem

Related topic

HamiltonianAlgebroids

Related topic

LieAlgebroids

Related topic

MathematicalBiology

Related topic

Quantization

Related topic

SchrodingersWaveEquation

Related topic

SchrodingerOperator

Related topic

CanonicalQuantization

Related topic

QuantumSpaceTimes

Defines

QM

Defines

energy conservation in finite quantum systems

Defines

Heisenberg formulation of QM

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Lie algebroids

Lie algebroids

0.1

Topic on Lie algebroids

This is a topic entry on Lie algebroids that focuses on their quantum applications and

extensions

of current algebraic theories.

Lie algebroids

generalize

Lie algebras

, and in certain quantum systems they represent

extended quantum (algebroid)

symmetries

. One can think of a

Lie algebroid

as generalizing the idea of a

tangent bundle

where the

tangent space

at a point is effectively the

equivalence class

of curves meeting at that point (thus
suggesting a

groupoid

approach), as well as serving as a site on which to study

infinitesimal

geometry (see, for example, ref.

[

Mackenzie2005

]

). The formal definition of a

Lie algebroid

is presented next.

Definition 0.1

Let

M

be a

manifold

and let

𝔛

⁢

(

M

)

denote the set of vector fields on

M

. Then, a

Lie algebroid

over

M

consists of a

vector bundle

E

⟶

M

,
equipped with a Lie bracket

[

,

]

on the space of sections

γ

⁢

(

E

)

,
and a

bundle map

Υ

:

E

⟶

T

⁢

M

, usually called the

anchor

.
Furthermore, there is an

induced map

Υ

:

γ

⁢

(

E

)

⟶

𝔛

⁢

(

M

)

,
which is required to be a map of Lie algebras, such that given

sections

α

,

β

∈

γ

⁢

(

E

)

and a

differentiable function

f

, the following

Leibniz rule

is satisfied :

[

α

,

f

⁢

β

]

=

f

⁢

[

α

,

β

]

+

(

Υ

⁢

(

α

)

)

⁢

β

.

(0.1)

Example 0.1

.

A typical example of a Lie algebroid is obtained when

M

is a

Poisson
manifold

and

E

=

T

*

⁢

M

, that is

E

is the

cotangent bundle

of

M

.

Now suppose we have a

Lie groupoid

𝖦

:

r

,

s

:

(0.2)

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quantum logic toposes

quantum logic toposes

This is a topic entry on

extensions

of standard and

elementary

toposes to

quantum topoi

founded upon

many-valued logics

.

Definition 0.1

.

A

quantum logic

topos

(

QLT

) is defined as an

extension of the

concept

of a topos

in which the

Heyting algebra

or

subobject classifier

of the standard

elementary topos

is replaced by a

quantum logic

that is axiomatically defined by a

non-commutative

lattice

structure

such as that of a many valued,

L

⁢

M

n

-

logic algebra

, modified to a non-distributive lattice structure corresponding to that of the quantum
physics events.

Remark 0.1

.

Quantum logics topoi

are thus

generalizations

of the Birkhoff and von Neumann definition of

quantum state spaces

based on their definition of a quantum logic (lattice), as well as a

non-Abelian

, higher dimensional extension of the recently proposed concept of a ‘quantum’ topos which employs the (

commutative

) Heyting logic algebra as a subobject classifier.

Some specific examples are considered in the following two recent references.

References

1

Butterfield, J. and C. J. Isham: 2001, space-time and the
philosophical challenges of

quantum gravity

., in C. Callender and
N. Hugget (eds. )

Physics Meets Philosophy at the Planck
scale.

, Cambridge University Press,pp.33–89.

2

Butterfield, J. and C. J. Isham: 1998, 1999, 2000–2002, A topos
perspective on the Kochen–Specker

theorem

I - IV,

Int. J.
Theor. Phys

,

37

No 11., 2669–2733

38

No 3.,
827–859,

39

No 6., 1413–1436,

41

No 4.,
613–639.

Title

quantum logic toposes

Canonical name

QuantumLogicToposes

Date of creation

2013-03-22 18:13:18

Last modified on

2013-03-22 18:13:18

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

29

Author

bci1 (20947)

Entry type

Topic

Classification

msc 81P68

Classification

msc 18B25

Classification

msc 81P15

Classification

msc 81P10

Synonym

extended quantum topos

Related topic

AxiomsOfTopoi

Related topic

GeneralizedToposesTopoiWithManyValuedLogicSubobjectClassifiers

Related topic

AlgebraicCategoryOfLMnLogicAlgebras

Related topic

QuantumLogic

Related topic

QuantumOperatorAlgebrasInQuantumFieldTheories

Related topic

Topos

Defines

quantum logics topoi

Defines

quantum logics toposes

Defines

toposes with quantum logics subobject classifiers

Generated on Sat Feb 10 13:05:17 2018 by

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Schrödinger operator

Schrödinger operator

Let

V

:

ℝ

n

→

ℝ

be a real-valued function.
The

Schroedinger operator

H

on the

Hilbert space

L

2

⁢

(

ℝ

n

)

is given by the action

ψ

↦

-

∇

2

⁡

ψ

+

V

⁢

(

x

)

⁢

ψ

,

ψ

∈

L

2

⁢

(

ℝ

n

)

.

This can be obviously re-written as:

ψ

↦

[

-

∇

2

+

V

⁢

(

x

)

]

⁢

ψ

,

ψ

∈

L

2

⁢

(

ℝ

n

)

,

where

[

-

∇

2

+

V

⁢

(

x

)

]

is the

Schrödinger

operator, which is now
called the

Hamiltonian operator

(

http://planetmath.org/HamiltonianOperatorOfAQuantumSystem

),

H

.

For stationary

quantum systems

such as electrons in ‘stable’ atoms the

Schrödinger equation

takes the very simple form :

𝐇

⁢

ψ

=

E

⁢

ψ

, where

E

stands for energy

eigenvalues

of the stationary quantum states. Thus, in quantum mechanics of systems with finite degrees of freedom that are ‘stationary’, the Schrödinger operator is used to calculate the (time-independent) energy states of a quantum system with potential energy

V

⁢

(

x

)

. Schrödinger called this operator the ‘Hamilton’ operator (

http://planetmath.org/HamiltonianOperatorOfAQuantumSystem

), or the

Hamiltonian

(

http://planetmath.org/HamiltonianOperatorOfAQuantumSystem

), and the latter name is currently used in almost all of quantum physics publications, etc. The eigenvalues give the energy levels, and the wavefunctions are given by the

eigenfunctions

.
In the more general, non-stationary, or ‘dynamic’ case, the Schrödinger equation takes the general form:

𝐇

⁢

ψ

=

(

-

i

)

⁢

∂

⁡

ψ

/

∂

⁡

t

.

Title

Schrödinger operator

Canonical name

SchrodingerOperator

Date of creation

2013-03-22 14:02:08

Last modified on

2013-03-22 14:02:08

Owner

mhale (572)

Last modified by

mhale (572)

Numerical id

30

Author

mhale (572)

Entry type

Definition

Classification

msc 81Q10

Synonym

Hamiltonian operator

Related topic

HamiltonianOperatorOfAQuantumSystem

Related topic

SchrodingersWaveEquation

Related topic

CanonicalQuantization

Related topic

QuantumOperatorAlgebrasInQuantumFieldTheories

Related topic

QuantumSpaceTimes

Related topic

SchrodingerOperator

Defines

quantum system dynamics and eigenvalues

Generated on Sat Feb 10 13:05:27 2018 by

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