Equation of Klein-Gordon

The equation of Klein-Gordon (1926), sometimes also called equation of Klein-Gordon-Fock, is a relativistic version of the equation of Schrödinger describing massive particles of Spin no one, without or with electric charge.

The equation of Klein-Gordon

Derivation

The relativistic equation giving the energy of an isolated massive particle is written:

E^2 \ = \ \ vec {p} ^ {~2} \ c^2 \ + \ m^2 \ c^4

By observing the rules of canonical quantification resulting from the quantum Mechanical not relativist, one obtains the equation known as of Klein-Gordon:

- \ \ hbar^2 \ \ frac \ = \ - \ \ hbar^2 \ c^2 \ \ Delta \ \ Psi (\ vec {R}, T) \ + \ m^2 \ c^4 \ \ Psi (\ vec {R}, T)

This equation is rewritten in the following form:

where $\ Box$ represents the Opérateur of alembertien:

\ Box \ = \ \ frac {1} {c^2} \ \ frac \ - \ \ Delta

One can also use the relativistic formalism (in natural Unités):

with convention:

\ partial_ {\ driven} = \ left (\ frac {\ partial} {\ partial T}, \ vec {\ nabla} \ right)

and

\ partial^ {\ driven} = \ left (\ frac {\ partial} {\ partial T}, - \ vec {\ nabla} \ right)

Difficulties of interpretation

The solutions of the equation of Klein-Gordon present serious difficulties of interpretation within the framework of the quantum Mécanique original, judicious theory to describe a only particle. If one seeks for example to build a Densité of probability of presence which checks the relativistic equation of continuity:

\ frac {\ partial \ rho (\ vec {R}, T)}{\ partial T} \ + \ \ vec {\ nabla} \ cdot \ vec {J} (\ vec {R}, T) \ = \ 0

the following sizes are inevitably obtained:

\ rho (\ vec {R}, T) \ = \ NR \ \ Im \ mathfrak m \ left (\ \ overline {\ Psi} (\ vec {R}, T) \ \ frac {\ partial \ Psi (\ vec {R}, T)}{\ partial T} \ \ right)

\ vec {J} (\ vec {R}, T) \ = \ NR c^2 \ \ Im \ mathfrak m \ left (\ \ overline {\ Psi} (\ vec {R}, T) \ \ vec {\ nabla} \ \ Psi (\ vec {R}, T) \ \ right)

where $\ overline {\ Psi}$ is the combined complex of $\ Psi$ , and $NR$ is an arbitrary constant. However, this density $\ rho$ is not positive everywhere, therefore cannot represent a density of probability of presence!

The relevant framework to interpret this relativistic quantum equation without difficulties is the Quantum theory of the fields

Equation of Klein-Gordon to the equation of Dirac

The fact that the density $\ rho$ is not positive everywhere comes owing to the fact that this density contains a derived first compared to time , like noticed it Dirac in 1928. This is related to the fact that the equation of Klein-Gordon contains a temporal derivative second.

Naive approach

To obtain a relativistic equation of the first order in time , one can think of quantifying the expression directly:

E \ = \ \ sqrt {p^2c^2 + m^2c^4}

The canonical procedure of quantification leads then to the equation:

I \, \ hbar \ \ frac {\ partial} {\ partial T} \ Phi (\ vec {R}, T) \ = \ \ sqrt {\ - \, c^2 \, \ Delta \, + \, m^2 \, c^4 \} \ \ Phi (\ vec {R}, T)

Because of the presence of a square root on the operator with the derivative partial space, this equation seems a priori well not very convenient to solve. One can today give a mathematically precise direction to the operator $\ sqrt {- \, c^2 \, \ Delta \, + \, m^2 \, c^4 \}$ : it is a Opérateur pseudo-differential, which has in particular the characteristic to be not-room .

The equation of Dirac

Historical aspects

The quarrel of the origins

It is amusing to notice that, according to Dirac, Schrödinger would have initially written relativistic the equation known as today of Klein-Gordon, this to try to describe the electron within the hydrogen atom. Indeed, reading of the first report of Schrödinger published in February 1926 watch that this one already tested a relativistic equation of wave, but this first report does not contain the explicitly written equation. The predictions obtained not being in conformity with the rather precise experimental results obtained by Paschen since 1916, Schrödinger would have then realized that it was the equation not relativist - said today of Schrödinger - which gave the good spectrum for hydrogen (after inclusion of the effects of spin in way Ad hoc ). Schrödinger published its relativistic equation only in the fourth memory of 1926.

Meanwhile, more precisely between September 1926 and April, not less five other articles, independently published, contained the equation known as today of Klein-Gordon. The authors of these five articles, whose references appear in the bibliography, are: Klein, Gordon, Fock, of Donder and van den Dungen, and finally Kudar.

Lastly, into its second article of 1926, Fock also opens the minimal proceedings of coupling , describing the coupling of the massive particle of electric charge $e$ to a given external electromagnetic field, represented by a quadri-potential $A_ {\ driven}$ . For more details, to see the following technical paragraph.

Technical details

Minimal coupling

For a load $e$ in the presence of a given external electromagnetic field, represented by the quadri-potential $A_ {\ driven}$ , regulation of minimal coupling of Fock led to substitute for the quadri-impulse $p_ {\ driven}$ following quantity:

p_ {\ driven} \ quad \ longrightarrow \ quad p_ {\ driven} \, - \, E \, A_ {\ driven}

Explicitly let us introduce the component temporal and space of the quadri-impulse:

p^ {\ driven} \ = \ \ begin {pmatrix} \ frac {E} {C} \ \ \ vec {p} \ end {pmatrix}

and

p_ {\ driven} \ = \ \ begin {pmatrix} \ frac {E} {C} \ \ - \ \ vec {p} \ end {pmatrix}

and of the quadri-potential:

A^ {\ driven} \ = \ \ begin {pmatrix} \ frac {V} {C} \ \ \ vec {has} \ end {pmatrix}

and

A_ {\ driven} \ = \ \ begin {pmatrix} \ frac {V} {C} \ \ - \ \ vec {has} \ end {pmatrix}

One obtains then explicicitement:

p^ {\ driven} \, - \, E \, A^ {\ driven} \ = \ \ begin {pmatrix} \ frac {1} {C} \, \ left (\, E \, - \, E \, V \, \ right) \ \ \ vec {p} \, - \, E \, \ vec {have} \ end {pmatrix}

Derivation of the equation

One sets out again of the relativistic equation of dispersion of an isolated massive particle:

E^2 \ = \ \ vec {p} ^ {~2} \ c^2 \ + \ m^2 \ c^4 \ quad \ Longleftrightarrow \ quad p_ {\ driven} \, p^ {\ driven} \ = \ m^2 \, c^2

One introduced the minimal coupling of Fock there:

\ left (p_ {\ driven} \, - \, E \, A_ {\ driven} \ right) \, \ left (p^ {\ driven} \, - \, E \, A^ {\ driven} \ right) \ = \ m^2 \, c^2

what explicitly gives in substituent the components:

\ left (E \, - \, E \, V \ right) ^2 \ - \ \ left (\ vec {p} \, - \, E \, \ vec {have} \ right) ^2 \ = \ m^2 \, c^4

In the presence of a static Coulomb potential describing the interaction of the electron with a proton (supposed infinitely heavy), the potential vector is null: $\ vec {has} = 0$ , and one a:

\ left (\, E \, + \, \ frac {e^2} {4 \ pi \ epsilon_0 R} \, \ right) ^2 \ - \ \ vec {p} ^ {~2} \ = \ m^2 \, c^4

One applies the canonical quantification to this traditional equation, which becomes an operator with the derivative partial:

\ left (\, I \, \ hbar \, \ frac {\ partial ~~} {\ partial T} \, + \, \ frac {e^2} {4 \ pi \ epsilon_0 R} \, \ right) ^2 \ - \ \ left (\, - \, I \, \ hbar \, \ vec {\ nabla} \, \ right) ^2 \ = \ m^2 \, c^4

from where the equation dependant on time:

\ left (\, I \, \ hbar \, \ frac {\ partial ~~} {\ partial T} \, + \, \ frac {e^2} {4 \ pi \ epsilon_0 R} \, \ right) ^2 \ \ Psi (\ vec {R}, T) \ + \ \ hbar^2 \, \ Delta \ Psi (\ vec {R}, T) \ = \ m^2 \, c^4 \ \ Psi (\ vec {R}, T)

One seeks finally the stationary states of energy $E$ constante in the form of a purely space function multiplied by exponential oscillating in time:

\ Psi (\ vec {R}, T) \ = \ \ psi (\ vec {R}) \ e^ {- iEt/\ hbar}

One obtains then the equation with the eigenvalues:

\ left (E \, + \, \ frac {e^2} {4 \ pi \ epsilon_0 R} \, \ right) ^2 \ \ psi (\ vec {R}) \ + \ \ hbar^2 \, \ Delta \ psi (\ vec {R}) \ = \ m^2 \, c^4 \ \ psi (\ vec {R})

Schrödinger was discouraged by the fact that this equation does not give the correct spectrum of the hydrogen atom. The following energy levels indeed are obtained:

E_ {N, L} \ = \ mc^2 \ \ left \, 1 \, - \, \ frac {\ alpha^2} {2 \, n^2} \, - \, \ frac {\ alpha^4} {2 \, n^4} \ \ left (\ frac {N} {L + \ frac12} \, - \, \ frac {3} {4} \ right) \, \ right \ + \ O (\ alpha^6)

where the principal quantum number $n$ is a strictly positive integer, the orbital quantum number $l$ is a positive integer ranging between 0 and $n - 1$ , and $\ alpha$ are the Constante of fine structure:

\ alpha \ = \ \ frac {e^2} {4 \ pi \ epsilon_0 \ hbar C} \ \ simeq \ \ frac {1} {137,04}

The term of order $\ alpha^2$ is correct, but the following term of order $\ alpha^4$ , which describes the fine structure, is not in conformity with the experimental results obtained by Paschen since 1916. The correct expression with this order is indeed:

E_ {N, L} \ = \ mc^2 \ \ left \, 1 \, - \, \ frac {\ alpha^2} {2 \, n^2} \, - \, \ frac {\ alpha^4} {2 \, n^4} \ \ left (\ frac {N} {J + \ frac12} \, - \, \ frac {3} {4} \ right) \, \ right \ + \ O (\ alpha^6)

where the quantum number $j = L + 1/2$ , the additional half being related to the Spin of the electron, which is not included in the equation of Klein-Gordon.

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