## Confined Hydrogen Atom In Spherical Cavity In N - Dimensions

ذرة الهيدروجين المحصورة في فجوة كروية في فضاء متعدد الأبعاد

##### Abstract

In this research the Schrödinger equation for a confined hydrogen atom in a spherical cavity in N dimensional spatial space has been solved for N ≥3. The eigen functions as well as the eigen values have been determined. We show that the Schrodinger equation here doesn’t differ from that of the free hydrogen atom in N dimensions; therefore they have similar wave functions namely:
(see The Equation in The PDF File)
while they differ in energy. A series solution of the Schrödinger equation is adopted here, and then, by applying the boundary conditions to the wave functions we found the energy eigen-values.The dependence of the ground state energy eigen-values of a confined hydrogen atom for l = 0 for certain values of N, on the radius of the cavity S, has been examined. We found that they depend on the radius of the cavity S, we show that for a given N, if S increases the ground state energies decreases until they approach a limiting value which approaches the energy eigen value of that N of the free hydrogen atom. While as S decreases, the ground state energy eigen values increases up until it approaches zero at a minimum value of S that is called the critical cage radius (Sc ) at which the total energy of the confined hydrogen atom equals zero. The critical values Sc are calculated for dimensions from (2-10), whose values are 0.722890, 1.835247, 3.296830, 5.088308, 7.200250, 9.617367, 12.35000, 15.36350, 18.68200 respectively, (all the values here are multiples of Bohr radius (α0)=0.529x10-10 meters It is shown here that Sc increases as N increases.
It is also shown that for a given S, the energy eigen-values for l=0 depend on the dimensionality of space N, that is, as N increases, the ground state energy eigen-values increase. The dependence of bound states of a confined H-atom, for a given S, as a function of N is investigated, and it is found that it decreases as N increases, while if we choose a larger value of S, the number of the bound states increases for each value of N. We found it interesting to compare the energy eigen-values of a confined hydrogen atom in a spherical cavity of a certain radius, with those energies of the corresponding energy eigen-states of a free hydrogen atom in the same dimension N. We found that the effect of confinement becomes more profound for larger N.
Finally, I considered the behavior of pressure on the cavity as the radius S is varied. It has been shown that the pressure exerted on the atom increases as S decreases up to a certain maximum value which occurs at a radius value called Sp max max, but then it decreases within a small range of S. In this research the Schrödinger equation for a confined hydrogen atom in a spherical cavity in N dimensional spatial space has been solved for N . The eigen functions as well as the eigen values have been determined.
We show that the Schrodinger equation here doesn’t differ from that of the free hydrogen atom in N dimensions; therefore they have similar wave functions namely
“R_λ (Ƿ) = A〖' ρl〗^l e ^((-(ρ/2))) 1F1 (l+ (N-1)/2 – λ; 2l + N – 1; ρ)”,, while they differ in energy. A series solution of the Schrödinger equation is adopted here, and then, by applying the boundary conditions to the wave functions we found the energy eigen-values.
The dependence of the ground state energy eigen-values of a confined hydrogen atom for l = 0 for certain values of N, on the radius of the cavity S, has been examined.
We found that they depend on the radius of the cavity S, we show that for a given N, if S increases the ground state energies decreases until they approach a limiting value which approaches the energy eigen value of that N of the free hydrogen atom. While as S decreases, the ground state energy eigen values increases up until it approaches zero at a minimum value of S that is called the critical cage radius (Sc ) at which the total energy of the confined hydrogen atom equals zero.
The critical values Sc are calculated for dimensions from (2-10), whose values are 0.722890, 1.835247, 3.296830, 5.088308, 7.200250, 9.617367, 12.35000, 15.36350, 18.68200 respectively, (all the values here are multiples of Bohr radius ((α0), where (α0) = 0.529x〖10〗^(-10)meters.
It is shown here that Sc increases as N increases.
It is also shown that for a given S, the energy eigen-values for l=0 depend on the dimensionality of space N, that is, as N increases, the ground state energy eigen-values increase.
The dependence of bound states of a confined H-atom, for a given S, as a function of N is investigated, and it is found that it decreases as N increases, while if we choose a larger value of S, the number of the bound states increases for each value of N.
We found it interesting to compare the energy eigen-values of a confined hydrogen atom in a spherical cavity of a certain radius, with those energies of the corresponding energy eigen-states of a free hydrogen atom in the same dimension N. We found that the effect of confinement becomes more profound for larger N.
Finally, I considered the behavior of pressure on the cavity as the radius S is varied.
It has been shown that the pressure exerted on the atom increases as S decreases up to a certain maximum value which occurs at a radius value called SP max, but then it decreases within a small range of S.