A NEW SIMPLE METHOD FOR HYDROGEN ATOM AND MOLECULE ION IN MOMENTUM SPACE WITH USING DIRAC DELTA FUNCTIONS

The momentum-space wave functions have been obtained for the one-dimensional hydrogen atom and the symmetric and antisymmetric states of the one-dimensional hydrogen molecule ion with δ-function interactions between the particles. The uncertainty relation has been evaluated as a function of the distance between the nuclei in the molecule ion. For the symmetric state this is finite for the collapsed molecule. For the antisymmetric state the energy is positive for a nuclear separation less than a0=ℏ2/me2 and the uncertainty in the position of the electron is infinite.The momentum-space wave functions have been obtained for the one-dimensional hydrogen atom and the symmetric and antisymmetric states of the one-dimensional hydrogen molecule ion with δ-function interactions between the particles. The uncertainty relation has been evaluated as a function of the distance between the nuclei in the molecule ion. For the symmetric state this is finite for the collapsed molecule. For the antisymmetric state the energy is positive for a nuclear separation less than a0=ℏ2/me2 and the uncertainty in the position of the electron is infinite.

American Journal of PhysicsJul 1, 19837 citations

Hydrogen atom in the momentum representation

John R. Lombardi

The wave functions for the hydrogen atom are determined in the momentum representation, using momentum variables ${p}_{r}$ , ${p}_{\ensuremath{\theta}}$, ${p}_{\ensuremath{\varphi}}$ which are associated with the spherical polar variables $r$ , $t(={e}^{i\ensuremath{\pi}cos\ensuremath{\theta}})$ and $v({e}^{i\ensuremath{\varphi}})$. The functions may be written as ${{\ensuremath{\varphi}}_{n,l,m}({p}_{r},{p}_{\ensuremath{\theta}},{p}_{\ensuremath{\varphi}})={\ensuremath{\alpha}}_{n,l}({p}_{r}){\ensuremath{\beta}}_{l}^{m}({p}_{\ensuremath{\theta}})\ensuremath{\delta}({p}_{\ensuremath{\varphi}}+m),}{{\ensuremath{\alpha}}_{n,l}({p}_{r})={\left(\frac{n}{i{p}_{0}}\right)}^{3l+z}\ensuremath{\Sigma}\stackrel{n\ensuremath{-}l\ensuremath{-}1}{k=0}\left({n\ensuremath{-}l\ensuremath{-}1}{k}\right)\frac{{2}^{k\ensuremath{-}l+1}}{(2l+k+1)(2l+k)\ensuremath{\cdots}(l+k+2)}{\left(\frac{\frac{i{p}_{0}}{n}}{{p}_{r}\ensuremath{-}\frac{i{p}_{0}}{n}}\right)}^{l+k+2},}{{\ensuremath{\beta}}_{l}^{m}({p}_{\ensuremath{\theta}})=\ensuremath{\Sigma}\stackrel{E(\frac{l}{2})}{j=E(\frac{m}{2})}{{a}_{j}{p}_{\ensuremath{\theta}}}^{\ensuremath{-}\frac{(m+1)}{2}}{J}_{\frac{j\ensuremath{-}(m\ensuremath{-}1)}{2}}({p}_{\ensuremath{\theta}}),}$ where $\ensuremath{\delta}$ is the Dirac delta function and ${p}_{0}=Z{a}_{0}^{\ensuremath{-}1}$ ( ${a}_{0}$ is the Bohr radius). ${J}_{\ensuremath{\nu}}({p}_{0})$ are Bessel functions of integer or half-integer order. The radial functions are shown to have poles up to order $n+1$ at ${p}_{r}=i\frac{{p}_{0}}{n}$ .The wave functions for the hydrogen atom are determined in the momentum representation, using momentum variables ${p}_{r}$, ${p}_{\ensuremath{\theta}}$, ${p}_{\ensuremath{\varphi}}$ which are associated with the spherical polar variables $r$, $t(={e}^{i\ensuremath{\pi}cos\ensuremath{\theta}})$ and $v({e}^{i\ensuremath{\varphi}})$. The functions may be written as ${{\ensuremath{\varphi}}_{n,l,m}({p}_{r},{p}_{\ensuremath{\theta}},{p}_{\ensuremath{\varphi}})={\ensuremath{\alpha}}_{n,l}({p}_{r}){\ensuremath{\beta}}_{l}^{m}({p}_{\ensuremath{\theta}})\ensuremath{\delta}({p}_{\ensuremath{\varphi}}+m),}{{\ensuremath{\alpha}}_{n,l}({p}_{r})={\left(\frac{n}{i{p}_{0}}\right)}^{3l+z}\ensuremath{\Sigma}\stackrel{n\ensuremath{-}l\ensuremath{-}1}{k=0}\left({n\ensuremath{-}l\ensuremath{-}1}{k}\right)\frac{{2}^{k\ensuremath{-}l+1}}{(2l+k+1)(2l+k)\ensuremath{\cdots}(l+k+2)}{\left(\frac{\frac{i{p}_{0}}{n}}{{p}_{r}\ensuremath{-}\frac{i{p}_{0}}{n}}\right)}^{l+k+2},}{{\ensuremath{\beta}}_{l}^{m}({p}_{\ensuremath{\theta}})=\ensuremath{\Sigma}\stackrel{E(\frac{l}{2})}{j=E(\frac{m}{2})}{{a}_{j}{p}_{\ensuremath{\theta}}}^{\ensuremath{-}\frac{(m+1)}{2}}{J}_{\frac{j\ensuremath{-}(m\ensuremath{-}1)}{2}}({p}_{\ensuremath{\theta}}),}$ where $\ensuremath{\delta}$ is the Dirac delta function and ${p}_{0}=Z{a}_{0}^{\ensuremath{-}1}$ (${a}_{0}$ is the Bohr radius). ${J}_{\ensuremath{\nu}}({p}_{0})$ are Bessel functions of integer or half-integer order. The radial functions are shown to have poles up to order $n+1$ at ${p}_{r}=i\frac{{p}_{0}}{n}$.

Physical Review ASep 1, 198033 citations

The hydrogen atom in the momentum representation; a critique of the variables comprising the momentum representation

John R. Lombardi, John F. Ogilvie

Chemical PhysicsJul 11, 20204 citations

Path integrals for hydrogen atom

Nikom Choosiri, Virulh Sa-yakanit

Jan 1, 19830 citations

Momentum Representation

S. Rajasekar, R. Velusamy

This chapter discusses the momentum eigenfunctions and momentum probability distribution for some simple time-independent quantum systems. The position-space distribution is useful when we perform measurements of position of a particle. Photo-cells and photographic plates are useful for position measurements for photons. The momentum representation is helpful in the case of measurement of momentum of a particle. The momentum representation is completely equivalent to the coordinate representation. In spite of their complete equivalence, the momentum representation could throw much insight on certain features that may remain hidden in the coordinate representation. A method using delta function interactions has been introduced for constructing momentum wave function for hydrogen atom and hydrogen molecule ion. Momentum space is shown to be better suited for exploring electronic wave functions of quasiperiodic systems.This chapter discusses the momentum eigenfunctions and momentum probability distribution for some simple time-independent quantum systems. The position-space distribution is useful when we perform measurements of position of a particle. Photo-cells and photographic plates are useful for position measurements for photons. The momentum representation is helpful in the case of measurement of momentum of a particle. The momentum representation is completely equivalent to the coordinate representation. In spite of their complete equivalence, the momentum representation could throw much insight on certain features that may remain hidden in the coordinate representation. A method using delta function interactions has been introduced for constructing momentum wave function for hydrogen atom and hydrogen molecule ion. Momentum space is shown to be better suited for exploring electronic wave functions of quasiperiodic systems.

CRC Press eBooksSep 29, 20220 citations

The wave functions for the hydrogen atom are determined in the momentum representation, using momentum variables

{p}_{r}

, ${p}_{\ensuremath{\theta}}$, ${p}_{\ensuremath{\varphi}}$ which are associated with the spherical polar variables

r

, $t(={e}^{i\ensuremath{\pi}cos\ensuremath{\theta}})$ and $v({e}^{i\ensuremath{\varphi}})$. The functions may be written as ${{\ensuremath{\varphi}}_{n,l,m}({p}_{r},{p}_{\ensuremath{\theta}},{p}_{\ensuremath{\varphi}})={\ensuremath{\alpha}}_{n,l}({p}_{r}){\ensuremath{\beta}}_{l}^{m}({p}_{\ensuremath{\theta}})\ensuremath{\delta}({p}_{\ensuremath{\varphi}}+m),}{{\ensuremath{\alpha}}_{n,l}({p}_{r})={\left(\frac{n}{i{p}_{0}}\right)}^{3l+z}\ensuremath{\Sigma}\stackrel{n\ensuremath{-}l\ensuremath{-}1}{k=0}\left({n\ensuremath{-}l\ensuremath{-}1}{k}\right)\frac{{2}^{k\ensuremath{-}l+1}}{(2l+k+1)(2l+k)\ensuremath{\cdots}(l+k+2)}{\left(\frac{\frac{i{p}_{0}}{n}}{{p}_{r}\ensuremath{-}\frac{i{p}_{0}}{n}}\right)}^{l+k+2},}{{\ensuremath{\beta}}_{l}^{m}({p}_{\ensuremath{\theta}})=\ensuremath{\Sigma}\stackrel{E(\frac{l}{2})}{j=E(\frac{m}{2})}{{a}_{j}{p}_{\ensuremath{\theta}}}^{\ensuremath{-}\frac{(m+1)}{2}}{J}_{\frac{j\ensuremath{-}(m\ensuremath{-}1)}{2}}({p}_{\ensuremath{\theta}}),}$ where $\ensuremath{\delta}$ is the Dirac delta function and ${p}_{0}=Z{a}_{0}^{\ensuremath{-}1}$ (

{a}_{0}

is the Bohr radius). ${J}_{\ensuremath{\nu}}({p}_{0})$ are Bessel functions of integer or half-integer order. The radial functions are shown to have poles up to order

n+1

{p}_{r}=i\frac{{p}_{0}}{n}