The concept of confined atoms has attracted significant scholarly attention as a methodology for probing atomic and molecular behavior under spatial restrictions. These systems are particularly efficient for investigating how confinement influences fundamental atomic properties such as energy spectra and ionization thresholds [1]. Atomic confinement constitutes an indispensable theoretical framework across various disciplines. Within nuclear environments, confined atoms serve as analogs for examining nuclear matter under compression, with implications for phenomena such as nuclear compressibility, shell modifications in dense media, and nucleon localization in finite nuclear systems [2]-[5]. In liquid-state physics, it forms the basis of cell models, whereas in nanoscience, it facilitates the study of quantum dots and semiconductor nanostructures [1][6]. Furthermore, the concept of confinement is pivotal for comprehending dense astrophysical conditions, such as ionized plasmas, white dwarf stars, and the interior structure of giant planets like Jupiter and Saturn [1][2][7]. Among the various paradigms investigated in numerous studies, the hydrogen atom confined within a cavity has been a subject of extensive research. Both spherical and ellipsoidal confinements have been analyzed using diverse quantum-mechanical methodologies. Research on hydrogen encapsulated within fullerene cages has also offered significant insights into endohedral complexes and nanoconfinement [2][8]-[10]. Despite the structural resemblances between Alkali atoms and hydrogen, specifically due to the presence of a single valence electron, there have been relatively fewer detailed studies on confined Alkali species [2][11]. Current research predominantly concentrates on computing total ground-state energies, for instance, for confined potassium (K) using Hartree—Fock configuration interaction methods and confined cesium (Cs) using Dirac—Fock calculations. However, comprehensive analyses of the effects of confinement on the excited-state spectra of Alkali atoms remain limited [2][3][7]. In this study, I employed a model potential approach, as outlined in [2], to characterize the valence electron of Alkali ions in specified excited states. This potential is devised to replicate the correct asymptotic behavior at large distances from the atomic core and is calibrated to align with the experimentally observed binding energy for the ground state at each angular momentum quantum number l [6][12][13]. It also affords precise forecasts for excited-state energies. To represent spatial confinement, I integrated this model potential with a spherical infinite-wall potential (impenetrable wall at finite radius R ). This composite potential framework enables the calculation of the energy spectrum of the valence electron as a function of the confining radius R , thereby providing insights into how confinement influences the electronic structure of Alkali atoms.

A brief review of relevant mathematical concepts is necessary to maintain continuity of thought and to facilitate a clear understanding of the procedure employed in this discussion.

The Schrödinger equation that governs the energy eigenfunctions of confined Alkali species modeled by a Hamiltonian comprising a system of nucleons and a single valence electron is given by:

where ( p ^ 1 , x ^ 1 ) represent the momenta and coordinates for the nucleus and electron, m 1 indicates the mass of the nucleus, m 2 that of the electron, respectively [1][7][8]. This expression depicts a typical two-body problem, where mathematics teaches separating variables as a way of demystifying into center of mass and relative motion. To this end, one introduces the center of mass frame with R ^ = ( m 1 x ^ 1 + m 2 x ^ 2 ) / M , with M = m 1 + m 2 and the total momentum P ^ = p ^ 1 + p ^ 2 . Concerning the relative motion, an obvious coordinate is the relative coordinate x ^ = x ^ 2 − x ^ 1 and the relative momentum is p ^ = m 1 M p ^ 2 − m 2 M p ^ 1 satisfying canonical conjugate relation [8]. Therefore, solving Equation (1) by the method separation of variables:

The total wave-function can be written as Ψ ( R , x ) = Φ ( R ) ψ ( x ) . The center of mass solution is a free particle problem, with the wave function Φ ( R ) = Ω e ± i k R and the energy E k = ℏ 2 k 2 / 2 M [14], where Ω is a constant, it will henceforth be neglected. Also, we can use μ = m 1 m 2 / M as the reduced mass. The relative particle wave function ψ ( x ) obeys the 3D problem:

A rearrangement and the introduction of the angular momentum term gives the momentum operator,

where L ^ is the angular momentum operator, such that we can construct functions of θ and ϕ known as spherical harmonics, which are common eigenfunctions of the operators L ^ 2 and L ^ z satisfying

with integers l and m = − l , ⋯ , + l . The next section introduces the potential model for the Alkali atoms of interest.

When the valence electron in an Alkali atom is far away from the core, its dominant interaction is due to the Coulomb term and the induced dipole moment, let e = 1 [2][8].

Here, α c is the dipole polarizability of the core. At small distances from the center of the core, the Exclusion Principle gives rise to an effective strong short-range repulsion. We incorporate this effect by taking a model potential for the interaction between the core and the valence electron in a state of angular momentum l ;

Here a l is an l -dependent parameter that may be regarded as an effective hard-core radius. Taking a l to be l -dependent is reasonable because exchange effects depend on the angular momentum. This assumption is similar to the usual quantum-defect description of excited atomic states, where the quantum defect is taken to be l -dependent [2][7][8]. Employing the instrumentation of the Laplacian aids the separation of variables, the wavefunction ψ ( x ) becomes

where k is the radial quantum number, which labels the many possible solutions of the equation for ℛ k , l as:

The normalization condition is also quite different from a 1D normalization, since it reads:

Due to the non-trivial Hermitian radial momentum, we can define the radial wavefunctions as u k , l ( r ) = r ℛ k , l ( r ) with normalization

this immediately verifies that the non-standard derivative terms for ℛ k , l become completely standard when written in terms of the derivatives of u k , l :

satisfiing the half-line condition [ 0 , ∞ ) , hence Equation (10) becomes:

Let’s eliminate all the physical constants ( ℏ , μ ) by switching to dimensionless variables. Let us define a length a such that the potential energy and kinetic energy of confinement are of the same order, where we have used e = 1 ,

which defines the Bohr radius with the corresponding dimensionless energy. Henceforth, we measure length in the units of a , energies in units of E H , and mass in units of μ ; these define a system of units called atomic units, often abbreviated as “a.u.” [14]-[16]. If ρ = r / a is the dimensionless length and ℰ ˜ = E k , l / E H is the dimensionless energy, we can redefine the wavefunction as f l ( p ) = a u k , l ( r ) satisfying the dimensionless equation:

This equation is equivalent to using μ = e = ℏ = 1 in a standard Schrödinger equation in 1D on the half-line [ 0 , ∞ ) . We continue with the use of r instead ρ , and E instead ℰ ˜ such that the resulting radial equation becomes:

Next, we develop the computational finite difference approach.

There are many other numerical approaches like, spectral, finite volume, finite element, etc., that can be considered to solve Equation (17), but the chosen method is well-suited for localized potentials and works well in finite domains having boundary conditions that are well known or asymptotically approximated. Equation (17) takes the form of a linear second-order differential equation with stable boundary conditions. Method involving finite difference for solving boundary value problems replaces each of the derivatives in the differential equation with an appropriate difference quotient approximation of the problem. The particular difference quotient and step size h are chosen to maintain a specified order of truncation error. However, h cannot be chosen too small because of the general instability of the derivative approximations [14]. The numerical treatment of valence electron wave function in a confined sphere will be easily solved by transforming Equation (17) to an eigenvalue-like problem. The finite difference method for the linear second-order boundary value problem becomes,

for

which requires that the difference quotient approximations be used to estimate d 2 f l d r 2 . First, we choose an integer N > 0 and divide the interval [ a l , R ] into equal subintervals of ( N + 1 ) whose endpoints are the mesh points r i = a l + i h , for i = 0 , 1 , ⋯ , N + 1 where h = ( R − a l ) / ( N + 1 ) .

Here, λ ( E ) = − 2 E and ξ l ( r ) = [ 2 r − l ( l + 1 ) r 2 + α c ( r 2 + a l 2 ) 2 ] such that:

The Taylor expansion of a function f ( x ) gives f ( x 0 ) + ( x − x 0 ) d f d x | x 0 + ( x − x 0 ) 2 ! d 2 f d x 2 | x 0 + ⋯ , taking the sum of the expansions of f ( r + h ) and f ( r − h ) gives:

For convenience, I modify the notations such that, f l ( r ) = f i , f l ( r − h ) = f i − 1 , and f l ( r + h ) = f i + 1 , where i = 1 , 2 , ⋯ , N . Neglecting the negligible third-order terms, Equation (21) becomes,

then we can rearrange terms such that we have,

where I have used δ i = h 2 ξ l i − 2 . Equation (24) is easily transformed into a symmetric matrix by using the boundary conditions f l ( a l ) = f 0 = 0 and f l ( R ) = f N + 1 = 0 ,

This can simply be written in a compact form as:

Hence, restoring the expression for λ ( E ) gives

further transformation turns this to,

where [ ℳ κ ] = 1 − 2 h 2 [ ℳ ] . The symmetric matrix [ ℳ κ ] ensures physical

correctness, yields orthogonal eigenvectors that simplify analysis and guarantee completeness, enables efficient algorithms that save time and memory, enhances numerical stability, and preserves the inherent physical symmetry of the original differential system [8].

This investigation offers a comprehensive analysis explaining the confinement of valence electrons in Li, Na, K, Rb, and Cs within a potential defined by the function V ( h , r , l , α c , a l ) . Confinement is understood to denote how the potential restricts the spatial distribution and energy of the electrons, whereby a reduced radial distance r indicates a more severe confinement, and a higher angular momentum quantum number l introduces a centrifugal barrier. This research refines data originally presented in Ref. [2], utilizing similar input parameters while providing additional explanations and interpretations. The computational matrix size is on the order of several tens of thousands. Precise values for the core’s dipole polarizabilities are determined from Rydberg state energies [2][6] and rigorous calculations. The parameter a l is determined by ensuring that the minimal potential energy for a specified l corresponds with the experimentally observed lowest energy for the valence electron. Input values for the core’s dipole polarizability, the minimal energy of the valence electron for a specified l , and the deduced values of the core radius a l are presented in Table 1. The subsequent subsections elucidate the findings illustrated in Tables 2-6. The results conform with the Aufbau Principle of electronic configuration [17] as evidenced in the tables, which favors excitation into lower energy orbitals first.

Table 1. The input values of the core dipolar polarizabilities α c , the lowest energy E 0 of the valence electron for a given l = 0 , and the implied values in atomic units (a.u.) of the core radius a l [2][8].

Table 2. Energies E ( eV ) for various states of the valence electron in confined Li, for various confinement radii R ( a .u . ) .

Table 3. Energies E ( eV ) for various states of the valence electron in confined Na, for various confinement radii R ( a .u . ) .

Table 4. Energies E ( eV ) for various states of the valence electron in confined K, for various confinement radii R ( a .u . ) .

Table 5. Energies E ( eV ) for various states of the valence electron in confined Rb, for various confinement radii R ( a .u . ) .

Table 6. Energies E ( eV ) for various states of the valence electron in confined Cs, for various confinement radii R ( a .u . ) .

1) Alkali Spectra

The eigenvalue spectra provide a visualization of the energies of the Alkali systems under the confining potential. Figure 1 displays the first five energy levels ( E 1 , E 2 , E 3 , E 4 , E 5 ) in (eV) as a function of the quantum number n (ranging from 1 to 5) for each quantum number of angular momentum l = 0 , 1 , 2 and the radial extent r ( ∞ , 20 , 15 , 10 , 7 , 5 , 4 , 3 ) in (a.u.). This section explains the spectra significance in the context of the radial expression in Equation (17). The spacing

Figure 1. Illustration of the energy spectra for each angular momentum value l = 0 , 1 , 2 , these plots show the first five eigenvalues ( E 1 , E 2 , E 3 , E 4 , E 5 ) in eV against their index n = 1 , 2 , 3 , 4 , 5 with separate curves for each r . Each curve represents the energy spectrum at a fixed r , l . Each plot’s row represents the Alkali species, i.e., Li, Na, K, Rb, and Cs; and each column gives l = 0 , 1 , 2 dependence, respectively. Angular momentum values l = 1 , 2 include further extrapolation of eigen energies.

Figure 2. The energy difference plot of Alkali species, such that for each l = 0 , 1 , 2 , these plots depict Δ E n = E n + 1 − E n in eV against r , with separate curves for each n = 1 , 2 , 3 , 4 . Each plot’s row represents the Alkali species, i.e., Li, Na, K, Rb, and Cs; and each column gives l = 0 , 1 , 2 dependence, respectively. Angular momentum values l = 1 , 2 include further extrapolation of eigen energies.

between energy levels ( E n + 1 − E n ) indicates the degree of confinement. Smaller r results in larger energy level spacings, as the electron is confined to a smaller region, increasing quantization. Larger r allows the wavefunction to spread, reducing spacings and approaching a continuum-like spectrum. In terms of the angular momentum, the higher l increases the centrifugal barrier ( l ( l + 1 ) / r 2 ), raising all energy levels and potentially altering the confinement. For l = 0 , the absence of the centrifugal term leads to lower energies indicating less confinement resistance. The plots show how the energy spectra become more discrete at smaller r , reflecting stronger confinement. For example, at r = 3 , the levels are widely spaced, while at r → ∞ , they are closer, resembling a less confined system.

2) Energy Difference Predicting Confinement

The energy differences are computed based on the energies obtained from solving Equation (17). The larger Δ E n at smaller r means stronger confinement, as the potential compresses the wavefunction, increasing energy level distances. As r becomes bigger, Δ E n becomes smaller, indicating a less confined system where levels are closer together. Figure 2 gives these energy differences. The higher l indicates a centrifugal barrier, which may reduce Δ E n for some n , as the effective potential becomes shallower at larger r . For l = 0 , the Coulomb term dominates, this leads to a larger Δ E n at small r .

3) Alkali Radial Probability Density and Wave Functions

The radial probability density facilitates a comprehensive understanding of the spatial distribution of an electron’s probability within a potential’s confines. These plots display | x ψ 1 ( x ) | 2 , the radial probability density. The computation of the radial probability density utilizes the ground state wavefunction ψ 1 ( x ) , corresponding to the first eigenfunction obtained from the solution of Equation (17). The plots in Figure 3 quantitatively assess the strength of confinement. For example, a pronounced increase in Δ E 1 as r decreases from ∞ to 3 indicates a transition to a state characterized by pronounced confinement. Variations across n demonstrate the distinct effects of confinement on excited states. The radial probability density | x ψ 1 ( x ) | 2 encapsulates the spherical symmetry of the system by incorporating the squared wavefunction together with the surface area element 4 π x 2 . This conjunction is integrated across angles within a three-dimensional framework, approximated in this context by x 2 for the radial component. This dimension serves as a more physically significant metric than the wavefunction itself, reflecting the likelihood per unit radial distance of finding the electron. In the case of l = 0 , the peak of | x ψ 1 ( x ) | 2 in close proximity to x = r 1 denotes the most probable radius, predominantly dictated by Coulomb attraction. Conversely, for l > 0 , the centrifugal barrier induces an outward displacement of this peak. The higher l shifts the peak of | x ψ 1 ( x ) | 2 to larger x , due to the centrifugal barrier pushing the electron away from the origin. For l = 0 , the peak is closest to x = r 1 . The plots show where the electrons are most likely located. A sharp peak at small r indicates strong localization, while a broad distribution at large r suggests a delocalized electron - the shift for higher l highlights angular

Figure 3. The radial probability density | x ψ ( x ) | 2 for each of the angular momentum proportional to the probability of finding the electron at radius x , with curves for each r . Each plot’s row represents the Alkali species, i.e., Li, Na, K, Rb, and Cs; and each column gives l = 0 , 1 , 2 dependence, respectively. Angular momentum values l = 1 , 2 include further extrapolation of eigen energies.

momentum’s role in reducing confinement near the core. Figure 4 shows the 2D illustration of the wavefunctions resulting from the solution to Equation (17).

Figure 4. Spatial visualization of the probability density | ψ ( x , y ) | 2 for the ground state wavefunction of Li for r = 10 .

4) Effective Potential of Alkali Species

The form of the potential function signifies confinement. The Coulomb term ( − 2 / x 2 ) serves to attract the electron towards the nucleus, while the core term ( α c / ( x 2 + a l 2 ) 2 ) introduces a short-range repulsion. In contrast, the centrifugal term ( l ( l + 1 ) / x 2 ) effectuates repulsion when l > 0 . A reduction in r truncates the potential at a lesser value of R , thus elevating the energy levels by diminishing the electron’s available space.

As illustrated in Figure 5, when considering l = 0 , the potential is devoid of the centrifugal term, thereby forming a profound well predominantly influenced by Coulomb attraction, resulting in substantial confinement. Conversely, for l = 1 , 2 , the centrifugal barrier increases the potential near the nucleus, thereby diminishing confinement by displacing the electron outward.

The graphical representations elucidate why energy levels increase (become less negative) for lower values of r : the truncated potential well becomes both shallower and narrower. The centrifugal barrier at higher l values demonstrates why E 1 is elevated, as the electron experiences less confinement near the nucleus.

Figure 5. For each l = 0 , 1 , 2 , these plots show the effective potential V ( x ) = l ( l + 1 ) x 2 − 2 x + α c ( x 2 + a l 2 ) 2 in atomic units against x . Each plot’s row represents the Alkali species, i.e., Li, Na, K, Rb, and Cs; and each column gives l = 0 , 1 , 2 dependence respectively.

This study offers an in-depth analysis of the quantum confinement phenomena observed in Alkali elements within a custom l dependent potential. The eigenvalue spectrum plots exhibit discrete energy levels that ascend with an increase in the quantum number, attributing higher energies and increased spacing to smaller radial extents owing to enhanced confinement; conversely, larger radial extents correspond to reduced energies and diminished spacing. The centrifugal component raises energy levels compared to l = 0 , indicative of its repulsive nature. Analysis of energy difference plots corroborates that Δ E n increases as r decreases, with l = 0 demonstrating maximal spacing at minimal r , which is tempered by the centrifugal barrier at elevated l . The core potential ( α c , α l ) governs these variations, inferring the presence of a wider, albeit slightly weaker, potential well that significantly influences the electronic configuration. The radial probability density plots display a concentrated distribution proximal to x = r 1 for l = 0 , shifting outward for l = 1 , 2 , with more acute peaks arising from heightened confinement at reduced r , whereas the 2D wavefunction density plot showcases a centrally symmetric peak for l = 0 and r that attenuates towards the periphery, moderated by the core’s repulsive force. The high grid resolution employed in n guarantees accuracy, meticulously capturing the detailed variations across the parameters.

I express my deepest gratitude to the East African Institute for Fundamental Research (ICTP-EAIFR) for providing exceptional training that has significantly enhanced my research capabilities. Their commitment to fostering scientific excellence in the region is truly commendable. I am also sincerely thankful to Prof. Steven Ndengue for his generous advice and invaluable guidance, which have been instrumental in shaping this work. This study received no external funding from any organization or agency.