American Journal of Physics

One-dimensional hydrogen atom and hydrogen molecule ion in momentum space

I. Richard Lapidus

Published Jul 1, 1983• 7 citations

Abstract

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.

DOI: 10.1119/1.13176

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American Journal of Physics

One-dimensional hydrogen atom and hydrogen molecule ion in momentum space

I. Richard Lapidus

Published Jul 1, 1983• 7 citations

Publisher

Abstract

DOI: 10.1119/1.13176

Related papers

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Jan 1, 20060 citations

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Juan F. P. Mosquera

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The coordinate-space wave function

ψ(x)

of quasi-one-dimensional atoms is defined in the

x\geq 0

region only. This poses a typical problem to write a physically acceptable momentum-space wave function

ϕ(p)

from the Fourier transform of

ψ(x)

. We resolve the problem with special attention to the behavior of real and imaginary parts of the complex-valued function

ϕ(p)

as a function of

p

and confirm that

ϕ_i(p)

(the imaginary part of

ϕ(p)

) represents the correct momentum-space wave function. We make use of the results for

ψ(x)

and

ϕ_i(p)

to express the position- and momentum-space Fisher information in terms of the principal quantum number and energy eigen value of the system and provide some useful checks on the result presented with particular attention on the information theoretic uncertainty relation.