Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Excited State and Product Wavefunctions for the One Dimensional Dirac Equations

Francesco R. Ruggeri

Published Aug 27, 2019• 0 citations
Publisher

Abstract

In a previous note, the nonrelativistic bound state wavefunction W(x) was written as Wo(x)W1(x) with Wo(x) being the ground state wavefunction. W1(x) satisfies an eigenvalue equation, similar to the Schrodinger equation, but with the potential term replaced with a coupling term: -1/2m d/dx d/dx W1(x) - 1/m d/dx W1(x) [ d/dx Wo(x)/Wo(x) ] = (E-Eo) W1 (x) ((1)) Mathematically, ((1)) may be compared to the hypergeometric differential equation and solutions found for the Schrodinger if coefficients meet certain criteria. A similar approach applies to the Klein Gordon and Dirac equations In this note, we wish to examine possible physical significance of Wo(x)W1(x). In addition, we wish to obtain an equation, analogous to ((1)) for both the Klein Gordon and Dirac equations to compare the coupling terms.

DOI
10.5281/zenodo.3378853

Related papers

Decoupling the One Dimensional Dirac Equations With a Scalar Potential

Francesco R. Ruggeri

In a previous note (1), it was noted one could write a solution of the bound state Schrodinger equation as W(x)=Wo(x)Dn(x) where W(x) is the wavefunction and Wo(x), the ground state wavefunction. Inserting into the Schrodinger equation one may obtain the Schrodinger equation for Wo(x) and an equation coupling Wo(x) and Dn(x). In this note, we suggest such an approach may be applied to the one dimensional bound Dirac equations. This allows one to obtain decoupled equations for u1(x) and v1(x) where u(x)=uo(x)u1(x) and v(x)=v1(x)vo(x) are the full solutions. Thus, if one is able to obtain a solution for uo(x) and vo(x) by inspection, the decoupled equations allow one to find the general solution. We apply this approach to the case of a scalar potential g|x|. The Dirac problem for this potential has been solved in the literature (2) with decoupling occurring through the taking of spatial derivatives. Here we wish to suggest a general method.

•Aug 10, 2019•0 citations

The Dirac Equation and Time Flux

Francesco R. Ruggeri

It was argued in a number of previous notes that for the nonrelativistic case, W(x,t) transforms under x->x+a, t-> t+b (with a and b infinitesimal) as exp(ipx - iEt), except with p replaced with = - i d/dx W(x,t) and E replaced with i d/dt W(x,t). It was suggested this was a driving condition of nonrelativistic quantum mechanics, and if this is the case, one might expect these ideas to carry over into relativistic quantum mechanics. For relativistic quantum mechanics, however, one has the Einstein energy-momentum relationship E*E= p*p + mo*mo (c=1) which leads to -d/dt d/dt W(x,t) and -d/dx d/dx W(x,t) being used. The momentum term is fine because it is related to < p*p conditional >, but the second derivative of time seems to be a little confusing as it represents < E*E conditional > and not <E conditional >. If <E conditional> has physical significance, as it seems to be the case in nonrelativistic quantum mechanics, one might expect an equation with this time flux i.e. i d/dt W(x,t) / W(x,t) = < E conditional > and not < E*E conditional >. Dirac wrote such an equation, but obtaining a single time derivative comes at a price. The Dirac equation, in one dimension, contains matrices, d/dx and d/dt in addition to a two vector wavefunction (U(x,t), V(x,t)) instead of W(x,t). Thus, the Dirac equation in one dimension leads to coupled equations with U(x,t) and V(x,t). The single derivative d/dx is deceptive because U(x,t) can be written in terms of d/dx V(x,t) and vice versa, so ultimately the Dirac equations contain d/dx d/dx which is needed to obtain < p*p conditional >. The question is: How can one obtain < E conditional >? This may be done by using a (d/dt + constant ) -1 operator i,e. almost an inverse time operator, in other words either (E+m)-1 or (E-m) -1 where E can be thought of as id/dt. Thus, the RHS of the Dirac equations is i d/dt U(x,t) or i d/dt V(x,t) which suggests an infinitesimal step forward in time, while the LHS side contains (id/dt +m)-1 or (-id/dt-m) -1 which seems to be representative of undoing such a step. It should be noted that one may take Einstein’s energy-momentum equation E*E = p*p + mo*mo, without any knowledge of quantum mechanics, and cast it as 2x2 matrix equation, linear in E, namely H (U,V) = E (U,V). The matrix H serves as an evolution operator and another matrix serves as a velocity operator. One may derive the Schrodinger zitterbewegung equation for the velocity matrix without any use of wavefunctions and derivatives which are key to quantum mechanics. Nevertheless, the idea of noncommuting matrices, vectors and zitterbewegung is present. There are several choices for the matrix H, but one leads to the one dimensional Dirac coupled equations Thus, the “spin” features may be obtained directly from E*E= p*p+mo*mo without any quantum mechanics. (1), (2). The purpose of this note is to examine some of these ideas.

•Jul 24, 2019•0 citations

Quantum Excited States and Energy Eigenvalues

Francesco R. Ruggeri

In bound state quantum mechanics, one has statistical observables (based on P(x) where P is probability) such as spatial density d(x), potential energy density V(x)d(x) and kinetic energy density .5mv(x)v(x) d(x), where v(x) is the classical velocity obtained from .5mv(x)v(x) + V(x) = E. (Kinetic energy density can also be written as -1/2m W(x) d/dx d/dx W(x) where W(x) is the wavefunction). The energy of a state can be obtained by integrating these statistical observables i.e. Integral dx d(x) V(x) + Integral dx d(x) .5mv(x)v(x) = E. If one performs this integration for two different states, one may calculate the energy difference. This begs the question: why does quantum mechanics deal with conditional probabilities such as P(p/x) directly related to W(x)? The issue perhaps begins with the appearance of V(x) and E in the time independent Schrodinger equation. Kinetic energy (not kinetic energy density) should appear and be somehow linked to some motion. A question arises as to why the energy density is proportional to the spatial density i.e. Energy density = E d(x)? This is directly linked to the time-independent Schrodinger equation, but one may try to argue that classical energy conservation .5mv(x)v(x) + V(x) = E at each point x should hold in quantum mechanics with .5mv(x)v(x) equal to Sum over p p*p/2m P(p/x) where P(p/x) = fp exp(ipx)/W(x). If one, however, considers excited states written as Wn(x)= Wo(x)Hn(x) where Wo(x) is the ground state wavefunction, one finds that: -1/2m d/dx d/dx Hn(x) - 1/m [d/dx Wo(x)]/Wo(x) [d/dx Hn(x)] = E1 Hn(x) ((1)) In other words, an eigenvalue equation appears again with no V(x) present. If one assumes Hn(x) has physical meaning (i.e. is related to a phonon or photon), then ((1)) may suggest there is a physical principle in place, i.e. the idea that one must have a “constant” energy at each point x. Here E1 is strictly kinetic energy (although not all of it). This “principle” has consequences as it restricts the values of E1 (i.e. discrete eigenvalues) and leads often to a periodic Hn(x) which has direct effect on kinetic energy density and potential energy density. Thus, having Hn(x) interact with Wo(x) to produce a constant energy E1 times Hn(x) does affect kinetic and energy densities at each x point, just not kinetic energy itself (it is constant i.e. E1). The purpose of this note is to examine this possible principle behind quantum mechanics.

•Jul 5, 2019•0 citations

The hydrogen atom in one dimension

R.E. Moss

The Schrödinger, Klein–Gordon, and Dirac equations for the hydrogen atom are solved in D dimensions, where D may be noninteger. For D=1 the nondegenerate ground state with infinite binding energy is obtained as a limit as D tends to 1 for the Schrödinger equation. The corresponding solution of the Klein–Gordon equation is shown to be unacceptable, contradicting previous work. Unexpectedly, no bound state solutions of the Dirac equation with D=1 are found at all. A few comments are made on the D=3 and 2 cases.

American Journal of Physics•May 1, 1987•57 citations

Bound states of Klein–Gordon equation and Dirac equation for ring-shaped non-spherical oscillator scalar and vector potentials

Zhang Xue-Ao, Ke Chen, Duan Zheng-Lu

Solving the Klein–Gordon equation and Dirac equation with ring-shaped non-spherical oscillator gives the exact bound state wavefunction and energy equation and the relations between non-relativistic Schrödinger equation, Klein–Gordon equation and Dirac equation with equal scalar and vector potentials.

Chinese Physics•Dec 23, 2004•51 citations

Relativistic Harmonic Oscillator and the Klein Gordon Equation

Francesco R. Ruggeri

The Klein Gordon equation for the harmonic oscillator may be written as (E-.5kx2)2=p2+mo2 ((1)) where mo is the rest mass and p the relativistic momentum. In the literature, it is argued that the resulting x4 term dominates for large x and prevents the Klein Gordon equation from being solved for bound states. As a result, it is replaced with an equation of the form: (E2-mo2) = (p-imwx)(p+imw) ((2)) where p=-ihbar d/dx in one dimension which may be solved exactly. Recently, the authors of (1) have used perturbation theory to calculate energy changes to ((1)) by using -.5V2 - e2 +2eV as a perturbative potential where V=.5kx2 and E=e+mo. They utilise the unperturbed wavefunction exp(-ax2) so large x4 values are damped. Their results are close to the exact calculations of ((2)) for the case of k=1 and m=1. In this note, we argue that for an oscillator there are classical turning points at which the quantum average kinetic energy -1/2m d/dx d/dx W should be zero and small nearby even if the average kinetic energy near x=0 corresponds to relativistic momenta. Thus, it is argued there are different spatial regions, each with its own quantum equation. The Schrodinger equation may be applied to regions near the classical turning point moving to infinite. In such a case, there is no issue with an x4 term. Near x=0, the full Klein Gordon equation ((1)) may be applied. There are questions of then matching the two sets of solutions. In this note, we attempt to investigate this scenario and compare results with those of (1) and (2). (1) and (2) suggest similar relativistic corrections to the nonrelativistic ground state oscillator energy of more than 20%. In this note, we suggest a smaller value.

•Dec 31, 2019•0 citations
In a previous note, the nonrelativistic bound state wavefunction W(x) was written as Wo(x)W1(x) with Wo(x) being the ground state wavefunction. W1(x) satisfies an eigenvalue equation, similar to the Schrodinger equation, but with the potential term replaced with a coupling term: -1/2m d/dx d/dx W1(x) - 1/m d/dx W1(x) [ d/dx Wo(x)/Wo(x) ] = (E-Eo) W1 (x) ((1)) Mathematically, ((1)) may be compared to the hypergeometric differential equation and solutions found for the Schrodinger if coefficients meet certain criteria. A similar approach applies to the Klein Gordon and Dirac equations In this note, we wish to examine possible physical significance of Wo(x)W1(x). In addition, we wish to obtain an equation, analogous to ((1)) for both the Klein Gordon and Dirac equations to compare the coupling terms.
In a previous note (1), it was noted one could write a solution of the bound state Schrodinger equation as W(x)=Wo(x)Dn(x) where W(x) is the wavefunction and Wo(x), the ground state wavefunction. Inserting into the Schrodinger equation one may obtain the Schrodinger equation for Wo(x) and an equation coupling Wo(x) and Dn(x). In this note, we suggest such an approach may be applied to the one dimensional bound Dirac equations. This allows one to obtain decoupled equations for u1(x) and v1(x) where u(x)=uo(x)u1(x) and v(x)=v1(x)vo(x) are the full solutions. Thus, if one is able to obtain a solution for uo(x) and vo(x) by inspection, the decoupled equations allow one to find the general solution. We apply this approach to the case of a scalar potential g|x|. The Dirac problem for this potential has been solved in the literature (2) with decoupling occurring through the taking of spatial derivatives. Here we wish to suggest a general method.
It was argued in a number of previous notes that for the nonrelativistic case, W(x,t) transforms under x->x+a, t-> t+b (with a and b infinitesimal) as exp(ipx - iEt), except with p replaced with = - i d/dx W(x,t) and E replaced with i d/dt W(x,t). It was suggested this was a driving condition of nonrelativistic quantum mechanics, and if this is the case, one might expect these ideas to carry over into relativistic quantum mechanics. For relativistic quantum mechanics, however, one has the Einstein energy-momentum relationship E*E= p*p + mo*mo (c=1) which leads to -d/dt d/dt W(x,t) and -d/dx d/dx W(x,t) being used. The momentum term is fine because it is related to < p*p conditional >, but the second derivative of time seems to be a little confusing as it represents < E*E conditional > and not <E conditional >. If <E conditional> has physical significance, as it seems to be the case in nonrelativistic quantum mechanics, one might expect an equation with this time flux i.e. i d/dt W(x,t) / W(x,t) = < E conditional > and not < E*E conditional >. Dirac wrote such an equation, but obtaining a single time derivative comes at a price. The Dirac equation, in one dimension, contains matrices, d/dx and d/dt in addition to a two vector wavefunction (U(x,t), V(x,t)) instead of W(x,t). Thus, the Dirac equation in one dimension leads to coupled equations with U(x,t) and V(x,t). The single derivative d/dx is deceptive because U(x,t) can be written in terms of d/dx V(x,t) and vice versa, so ultimately the Dirac equations contain d/dx d/dx which is needed to obtain < p*p conditional >. The question is: How can one obtain < E conditional >? This may be done by using a (d/dt + constant ) -1 operator i,e. almost an inverse time operator, in other words either (E+m)-1 or (E-m) -1 where E can be thought of as id/dt. Thus, the RHS of the Dirac equations is i d/dt U(x,t) or i d/dt V(x,t) which suggests an infinitesimal step forward in time, while the LHS side contains (id/dt +m)-1 or (-id/dt-m) -1 which seems to be representative of undoing such a step. It should be noted that one may take Einstein’s energy-momentum equation E*E = p*p + mo*mo, without any knowledge of quantum mechanics, and cast it as 2x2 matrix equation, linear in E, namely H (U,V) = E (U,V). The matrix H serves as an evolution operator and another matrix serves as a velocity operator. One may derive the Schrodinger zitterbewegung equation for the velocity matrix without any use of wavefunctions and derivatives which are key to quantum mechanics. Nevertheless, the idea of noncommuting matrices, vectors and zitterbewegung is present. There are several choices for the matrix H, but one leads to the one dimensional Dirac coupled equations Thus, the “spin” features may be obtained directly from E*E= p*p+mo*mo without any quantum mechanics. (1), (2). The purpose of this note is to examine some of these ideas.
In bound state quantum mechanics, one has statistical observables (based on P(x) where P is probability) such as spatial density d(x), potential energy density V(x)d(x) and kinetic energy density .5mv(x)v(x) d(x), where v(x) is the classical velocity obtained from .5mv(x)v(x) + V(x) = E. (Kinetic energy density can also be written as -1/2m W(x) d/dx d/dx W(x) where W(x) is the wavefunction). The energy of a state can be obtained by integrating these statistical observables i.e. Integral dx d(x) V(x) + Integral dx d(x) .5mv(x)v(x) = E. If one performs this integration for two different states, one may calculate the energy difference. This begs the question: why does quantum mechanics deal with conditional probabilities such as P(p/x) directly related to W(x)? The issue perhaps begins with the appearance of V(x) and E in the time independent Schrodinger equation. Kinetic energy (not kinetic energy density) should appear and be somehow linked to some motion. A question arises as to why the energy density is proportional to the spatial density i.e. Energy density = E d(x)? This is directly linked to the time-independent Schrodinger equation, but one may try to argue that classical energy conservation .5mv(x)v(x) + V(x) = E at each point x should hold in quantum mechanics with .5mv(x)v(x) equal to Sum over p p*p/2m P(p/x) where P(p/x) = fp exp(ipx)/W(x). If one, however, considers excited states written as Wn(x)= Wo(x)Hn(x) where Wo(x) is the ground state wavefunction, one finds that: -1/2m d/dx d/dx Hn(x) - 1/m [d/dx Wo(x)]/Wo(x) [d/dx Hn(x)] = E1 Hn(x) ((1)) In other words, an eigenvalue equation appears again with no V(x) present. If one assumes Hn(x) has physical meaning (i.e. is related to a phonon or photon), then ((1)) may suggest there is a physical principle in place, i.e. the idea that one must have a “constant” energy at each point x. Here E1 is strictly kinetic energy (although not all of it). This “principle” has consequences as it restricts the values of E1 (i.e. discrete eigenvalues) and leads often to a periodic Hn(x) which has direct effect on kinetic energy density and potential energy density. Thus, having Hn(x) interact with Wo(x) to produce a constant energy E1 times Hn(x) does affect kinetic and energy densities at each x point, just not kinetic energy itself (it is constant i.e. E1). The purpose of this note is to examine this possible principle behind quantum mechanics.
The Schrödinger, Klein–Gordon, and Dirac equations for the hydrogen atom are solved in D dimensions, where D may be noninteger. For D=1 the nondegenerate ground state with infinite binding energy is obtained as a limit as D tends to 1 for the Schrödinger equation. The corresponding solution of the Klein–Gordon equation is shown to be unacceptable, contradicting previous work. Unexpectedly, no bound state solutions of the Dirac equation with D=1 are found at all. A few comments are made on the D=3 and 2 cases.
The Klein Gordon equation for the harmonic oscillator may be written as (E-.5kx2)2=p2+mo2 ((1)) where mo is the rest mass and p the relativistic momentum. In the literature, it is argued that the resulting x4 term dominates for large x and prevents the Klein Gordon equation from being solved for bound states. As a result, it is replaced with an equation of the form: (E2-mo2) = (p-imwx)(p+imw) ((2)) where p=-ihbar d/dx in one dimension which may be solved exactly. Recently, the authors of (1) have used perturbation theory to calculate energy changes to ((1)) by using -.5V2 - e2 +2eV as a perturbative potential where V=.5kx2 and E=e+mo. They utilise the unperturbed wavefunction exp(-ax2) so large x4 values are damped. Their results are close to the exact calculations of ((2)) for the case of k=1 and m=1. In this note, we argue that for an oscillator there are classical turning points at which the quantum average kinetic energy -1/2m d/dx d/dx W should be zero and small nearby even if the average kinetic energy near x=0 corresponds to relativistic momenta. Thus, it is argued there are different spatial regions, each with its own quantum equation. The Schrodinger equation may be applied to regions near the classical turning point moving to infinite. In such a case, there is no issue with an x4 term. Near x=0, the full Klein Gordon equation ((1)) may be applied. There are questions of then matching the two sets of solutions. In this note, we attempt to investigate this scenario and compare results with those of (1) and (2). (1) and (2) suggest similar relativistic corrections to the nonrelativistic ground state oscillator energy of more than 20%. In this note, we suggest a smaller value.