Background

The quantum theoretic analysis of an electron in a hydrogen atom is well worked out Some would say that it p the crown jewel of the New Quantum Theory of the 1920's which replaced the Old Quantum Theory of Niels Bohr In the analysis the total energy of the hydrogen atom is negative; it is a stable system which requires an input of energy to disassocite it What is not worked out is the analysis of the case in which the electron has too much energy to be captured by the proton and passes by it going off to infinity with only a change in its direction of travel.

The analysis of this case is started with the simplified model of the electron passing through the proton as though the proton is a torus or cylinder. This keeps the analysis one dimensional.

The potential energy function is then V(r)=−α/|r| and has the shape shown below

where α is a positive constant

The Hamiltonian function for the system is

H = ½p²/m −α/|r|

where p is the momentum of the electron and m is its mass.

The Hamiltonian operator is then

H^ = ½(∂²/∂r²)/m −α/|r|

Classical Analysis

From the total energy E=½mv² −α/|r| it is found that the velocity v is given by

v = (2/m)(E+α/|r|)^½
and hence
(dr/dt)/(E+α/|r|)^½ = 2/m
or, equivalently
|r|^½(dr/dt)/(E|r|+α)^½ = 2/m
or, after factoring out an E
|r|^½(dr/dt)/(|r|+α/E)^½ = (2/m)E^½

This can be solved analytically but it is convenient to get the solution numerically i.e.,

r(t+δt) = r(t) + δt*v

Here are the results of the computation.

The electron virtually jumps where it passes the proton.

Consequently the relative probability density as a function of location is

Quantum Analysis

The time-dependent Schroedinger equation for the system is

ih(∂φ/∂t) = −h²/(2m)(∂²φ/∂r²) − (α/r)φ
which can be reduced to
(∂φ/∂t) = ih/(2m)(∂²φ/∂r²) + i (α/(hr))φ

The above equation is actually two equations. Let φ=ψ + iξ. The above equation is then

(∂ψ/∂t) = h/(2m)(∂²ξ/∂r²)
and
(∂ξ/∂t) = h/(2m)(∂²ψ/∂r²) + (α/(hr))ψ .

Consider first the case of a free electron; i.e, one in which V(r) is identically zero. The plane wave solution in that case is

φ(r, t) = C·exp(i(kr−ωt)) C

where C, k and ω are constants.

For this solution

(∂φ/∂t) = −iωC·exp(i(kr−ωt))
(∂φ/∂r) = ikC·exp(i(kr−ωt))
(∂²φ/∂r²) = −k²C·exp(i(kr−ωt))

Therefore

−iωC = i(h/(2m))[−k²C]

Thus C can be any value and

ω = (h/(2m))k²

The kinetic energy E of the plane wave, and hence its total energy, is given by

E = hω = (h²k²/(2m)

The energy E may be taken as determining the values of k and ω. Thus the higher is the energy the more rapid are the oscillation in probability density over time and space. Also the time and spatial averages of the probability density are constant over time and space, as are the time-spent probability densities of a classical free electron.

For the simulations a plane wave may be taken to be the initital condition. The simulation is then that of what happens if a proton suddenly appeared in the path of the electron.

The Simulation of the Quantum Evolution
of the Wave Function for an Electron

The dynamic equations for φ=ψ + iξ are

(∂ψ/∂t) = h/(2m)(∂²ξ/∂r²)
and
(∂ξ/∂t) = h/(2m)(∂²ψ/∂r²) + (α/(h)/r))ψ .

These equations can be expressed in matrix form as

∂Φ/∂t = (h/(2m))J∂²Φ/∂r² + (1/(h)KΦ

where Φ, J and K are, respectively, the matrices

| ψ |

| ξ |

| 0	1 |
| 1	0 |

| 0	0 |
| α/r	0 |

The second derivatives are approximated by [f(r+δr)−2f(r)+f(t−δr)]/(δr)². Once the derivatives with respect to time have been computed the next value of a function is computed as g(t+δt)=g(t)+(∂g/∂t)δt.

The simulation was computed assuming (h/(2m)) is equal to 1 and α equal to 0.1. The space and time steps were δr=0.2 and δt=0.1. Using these values the components of φ were computed one time step ahead. There values are shown below compared to their original values.

The value of the wave function, and hence also of the probability density function, is affected only in the vicinity of the proton. Thus as the energy increases the solution involves the rapid fluctuations which are eliminated by the averaging over time and consequently what is left is the time-spent probability density distribution of classical analysis. Therefore the quantum analysis does fulfill the Correspondence Principle.

(To be continued.)