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Effective potential

In physics, the effective potential is a constructed potential energy function that simplifies the analysis of particle motion under central forces by incorporating the effects of angular momentum conservation into a one-dimensional radial problem. It combines the true potential energy V(r) of the central force with a centrifugal barrier term derived from the angular kinetic energy, expressed classically as V_\text{eff}(r) = V(r) + \frac{L^2}{2\mu r^2}, where L is the magnitude of the angular momentum and \mu is the reduced mass of the system.^[1] This formulation transforms the two-body central force problem into an equivalent one-body motion in an effective one-dimensional potential, allowing the use of standard techniques for solving equations of motion, identifying turning points, and determining stable orbits.^[1] In quantum mechanics, the effective potential appears in the radial Schrödinger equation as V_\text{eff}(r) = V(r) + \frac{\hbar^2 \ell (\ell + 1)}{2\mu r^2}, where \ell is the orbital angular momentum quantum number, enabling the separation of variables and the study of bound states, scattering, and energy levels in atoms like hydrogen.^[2] The concept originates from classical mechanics, where the effective potential's shape—often featuring a repulsive barrier at small radii due to the $1/r^2 term and the attractive well of V(r) at larger distances—governs qualitative behaviors such as circular orbits at minima, spiral-ins for low energies, and scattering for high energies.^[3] For instance, in gravitational or Coulomb potentials (V(r) \propto -1/r), the effective potential exhibits a minimum whose location and depth depend on L, determining the stability of planetary or atomic orbits.^[4] Quantum extensions account for wave-like behavior, with the centrifugal term preventing divergence at r=0 and imposing boundary conditions that quantize energy levels.^[5] This framework is fundamental in atomic, molecular, and nuclear physics, as well as astrophysics for modeling binary systems.^[6] Beyond central forces, the effective potential finds applications in other domains, such as quantum field theory, where it describes the vacuum energy landscape and spontaneous symmetry breaking through the minimization of V_\text{eff}(\phi) for a scalar field \phi.^[7] In plasma physics, effective potentials model interactions in strongly coupled systems by averaging pairwise forces.^[8] Its versatility stems from reducing multidimensional dynamics to effective one-dimensional problems, though interpretations vary by context—always as an auxiliary tool rather than a physical potential.^[9]

Mathematical Formulation

Definition

In central force problems, the effective potential arises as a conceptual tool to describe the radial motion of a particle or reduced two-body system under a spherically symmetric force field. It incorporates both the intrinsic interaction potential and the influence of angular momentum conservation, allowing the complex three-dimensional dynamics to be analyzed as an equivalent one-dimensional problem along the radial coordinate. The effective potential V_{\mathrm{eff}}(r) is mathematically expressed as

V_{\mathrm{eff}}(r) = V(r) + \frac{L^2}{2\mu r^2},

where V(r) is the actual central potential energy depending only on the radial distance r, L is the magnitude of the conserved angular momentum vector, and \mu is the reduced mass of the system.^[1]^[10] The second term represents the centrifugal contribution, which acts as a repulsive barrier that increases as r decreases. This formulation is particularly useful in the reduced two-body problem, where the relative motion of two interacting particles is equivalent to the motion of a single particle with mass \mu orbiting a fixed center under the central potential V(r), thereby simplifying the equations of motion to a radial form.^[3]^[11] The effective potential V_{\mathrm{eff}}(r) possesses dimensions of energy, consistent with the units of V(r) (potential energy) and the centrifugal term (kinetic energy associated with angular motion), blending true interaction effects with the fictitious centrifugal force in the non-inertial rotating frame.^[12]^[13]

Derivation

In the two-body central force problem, the motion can be reduced to that of a single particle with reduced mass \mu moving in the relative coordinate \mathbf{r} under the central potential V(r). The Lagrangian for this system in polar coordinates is

L = \frac{1}{2} \mu (\dot{r}^2 + r^2 \dot{\phi}^2) - V(r),

where r = |\mathbf{r}| and \phi is the azimuthal angle.^[14] Since the Lagrangian does not depend explicitly on \phi, the conjugate momentum p_\phi = \partial L / \partial \dot{\phi} = \mu r^2 \dot{\phi} is conserved and equals the angular momentum L.^[14] This conservation law yields

\dot{\phi} = \frac{L}{\mu r^2}.

The total energy E of the system, which is conserved due to time independence of the Lagrangian, is the sum of kinetic and potential energies:

E = \frac{1}{2} \mu \dot{r}^2 + \frac{1}{2} \mu (r \dot{\phi})^2 + V(r).

Substituting the expression for \dot{\phi} gives the effective kinetic energy in the radial coordinate as

\frac{1}{2} \mu \dot{r}^2 + \frac{L^2}{2 \mu r^2}.

Thus, the total energy can be rewritten as

E = \frac{1}{2} \mu \dot{r}^2 + V_{\rm eff}(r),

where the effective potential is

V_{\rm eff}(r) = V(r) + \frac{L^2}{2 \mu r^2}.

^[14] This form demonstrates that the radial motion \ddot{r} is equivalent to the one-dimensional motion of a fictitious particle of mass \mu in the effective potential V_{\rm eff}(r), with the second term representing the centrifugal contribution from angular momentum conservation.^[14]

Key Properties

Shape and Behavior

The effective potential in central force problems combines the central potential V(r) with a centrifugal term, resulting in a curve that governs radial motion. As r \to 0, the centrifugal term dominates, driving V_{\rm eff}(r) \to \infty and forming a repulsive barrier that prevents particle collapse to the origin.^[15] Conversely, as r \to \infty, the centrifugal contribution vanishes relative to V(r), so V_{\rm eff}(r) \to V(r).^[3] This asymptotic behavior ensures the effective potential retains the long-range characteristics of the original interaction while introducing short-range repulsion due to angular momentum. The overall shape of V_{\rm eff}(r) varies with the angular momentum L. For fixed L, the centrifugal barrier rises sharply near the origin, but increasing L steepens this barrier, shifts potential wells outward, and makes them shallower.^[15] In attractive cases like V(r) = -k/r (with k > 0), the effective potential acquires a characteristic pocket-like form, featuring a minimum that balances attraction and centrifugal repulsion.^[15] The centrifugal term's repulsive nature is crucial, as it stabilizes the system against singularities at r = 0 without altering the conservative force law.^[3] For power-law potentials, the effective potential's qualitative features adapt to the underlying interaction. In the inverse-square force case (yielding V(r) \propto -1/r), the curve displays a prominent barrier at small r that flattens into the attractive tail at large r, often creating a bounded well for moderate L.^[15] Similar sketches apply to other power laws, such as V(r) \propto -1/r^2 (from F \propto -1/r^3), where the barrier's influence competes more intensely with the potential's decay, potentially leading to steeper or shallower profiles depending on L.^[3] These shapes highlight how angular momentum modifies the effective landscape without specific orbital implications.

Stability and Minima

The stability of motion in a central force problem is determined by the local extrema of the effective potential V_{\text{eff}}(r) = V(r) + \frac{L^2}{2\mu r^2}, where V(r) is the central potential, L is the conserved angular momentum, \mu is the reduced mass, and r is the radial coordinate. A minimum in V_{\text{eff}}(r) occurs where the first derivative vanishes, \frac{d V_{\text{eff}}}{dr} = 0, which implies \frac{dV}{dr} = \frac{L^2}{\mu r^3}.^[16]^[15] This condition balances the radial gradient of the central potential against the centrifugal barrier, enabling equilibrium at a specific radius r_{\min}. To confirm stability at this minimum, the second derivative must satisfy \frac{d^2 V_{\text{eff}}}{dr^2} \big|_{r = r_{\min}} > 0. Substituting the form of V_{\text{eff}}(r), this test yields \frac{d^2 V}{dr^2} \big|_{r = r_{\min}} + \frac{3 L^2}{\mu r_{\min}^4} > 0.^[16]^[15]^[17] A positive second derivative indicates a local curvature that provides a restoring force for small perturbations, ensuring the orbit remains bounded near r_{\min}; otherwise, the equilibrium is unstable, leading to unbound or spiraling trajectories. Physically, such minima in the effective potential correspond to stable circular orbits, where the total energy E = V_{\text{eff}}(r_{\min}) places the system exactly at the bottom of the well.^[16]^[15] For energies slightly above this value but below any surrounding barriers, the motion is bound with small radial excursions around the circular path. The depth of the minimum relative to the asymptotic behavior of V_{\text{eff}}(r) at large r quantifies the binding energy, representing the energy required to unbind the system from the potential well.^[15] Near the minimum, small radial deviations \delta r = r - r_{\min} result in harmonic oscillations governed by the quadratic approximation of V_{\text{eff}}(r). The effective spring constant is k_{\text{eff}} = \frac{d^2 V_{\text{eff}}}{dr^2} \big|_{r = r_{\min}}, leading to an angular frequency \omega = \sqrt{\frac{k_{\text{eff}}}{\mu}}.^[16]^[17] This frequency characterizes the rate of radial libration, distinguishing stable bound motion from unstable scattering.

Classical Applications

Central Force Problems

In central force problems, the motion of a reduced mass \mu under a central potential V(r) can be analyzed by separating the radial and angular coordinates, leveraging conservation of angular momentum L = \mu r^2 \dot{\phi}. The total energy E is conserved and expressed as E = \frac{1}{2} \mu \dot{r}^2 + V_{\text{eff}}(r), where the effective potential is V_{\text{eff}}(r) = V(r) + \frac{L^2}{2\mu r^2}.^[15]^[3] This formulation reduces the three-dimensional problem to an equivalent one-dimensional motion in r, with the centrifugal term acting as a barrier.^[15] Classical turning points occur where \dot{r} = 0, so V_{\text{eff}}(r) = E, marking the inner (r_{\min}) and outer (r_{\max}) limits of radial excursion.^[15]^[3] Between these points, the particle undergoes oscillatory radial motion if E lies within the well of V_{\text{eff}}(r); specifically, the motion is bound if E exceeds the global minimum of V_{\text{eff}}(r) but is less than the asymptotic value at infinity (often 0 for attractive potentials), resulting in closed or precessing orbits confined to finite r.^[15] For E above the minimum but still negative (assuming an attractive V(r)), the orbits remain bound but elliptical-like; if E > 0, the motion is unbound, leading to scattering trajectories where the particle approaches from infinity, interacts, and recedes.^[15]^[3] The period of radial oscillation T_r quantifies the time for one full cycle between turning points and is given by T_r = 2 \sqrt{\frac{\mu}{2}} \int_{r_{\min}}^{r_{\max}} \frac{dr}{\sqrt{E - V_{\text{eff}}(r)}}.^[15] This integral captures the time-averaged radial dynamics, independent of the angular motion. In contrast, the orbital angular period T_\phi, which corresponds to the time for the azimuthal angle \phi to advance by $2\pi, is T_\phi = \frac{2\pi \mu r^2}{L} for circular orbits at radius r.^[18] For non-inverse-square central forces, T_r \neq T_\phi, causing the orbit to precess: the angular advance per radial period \Delta \phi = \int_0^{T_r} \dot{\phi} \, dt = \frac{L}{\mu} \int_{r_{\min}}^{r_{\max}} \frac{dr}{r^2 \sqrt{\frac{2}{\mu} (E - V_{\text{eff}}(r))}} deviates from $2\pi.^[19]^[20] If \Delta \phi / 2\pi is not a rational number, the trajectory traces a rosette pattern, densely filling an annular region rather than closing after finite revolutions.^[18]^[20] Circular orbits, corresponding to minima of V_{\text{eff}}(r), represent fixed points where T_r \to \infty and precession vanishes.^[15]

Gravitational Orbits

In the context of gravitational orbits, the two-body problem can be reduced to an equivalent one-body problem with reduced mass \mu = \frac{m M}{m + M}, where m and M are the masses of the orbiting body and central body, respectively. For planetary motion where M \gg m, \mu \approx m. The gravitational potential energy is V(r) = -\frac{[G](/page/G) M m}{r}, leading to the effective potential V_{\text{eff}}(r) = -\frac{G M m}{r} + \frac{L^2}{2 \mu r^2}, where L is the conserved angular momentum.^[4] This effective potential governs the radial motion, analogous to a one-dimensional particle in a potential well.^[1] Circular orbits occur at the minimum of V_{\text{eff}}(r), found by setting the derivative to zero: \frac{d V_{\text{eff}}}{dr} = \frac{G M m}{r^2} - \frac{L^2}{\mu r^3} = 0. Solving yields the circular orbit radius r_c = \frac{L^2}{G M m \mu}.^[4] At this radius, the centripetal force is balanced by gravity, and the total energy is E = -\frac{G M m}{2 r_c}. For planetary systems with M \gg m, this simplifies to r_c = \frac{L^2}{G M m^2}, consistent with Keplerian motion.^[21] For bound orbits with total energy E < 0, the motion is confined between turning points where E = V_{\text{eff}}(r), resulting in closed elliptical paths. The semi-major axis a of the ellipse is given by a = -\frac{G M m}{2 E}, relating the energy to the orbit's size.^[4] The eccentricity e, which determines the orbit's shape (with $0 \leq e < 1 for ellipses), is e = \sqrt{1 + \frac{2 E L^2}{\mu (G M m)^2}}. As e approaches 0, the orbit becomes circular; higher e values produce more elongated ellipses.^[4] Unbound orbits occur when E \geq 0, allowing the body to escape to infinity. The threshold case E = 0 corresponds to parabolic trajectories with e = 1, defining the escape condition from the gravitational well. For E > 0, hyperbolic orbits (e > 1) result, with the body approaching from infinity and receding without bound. This escape threshold implies a minimum launch velocity from radius r, known as the escape velocity v_{\text{esc}} = \sqrt{\frac{2 G M}{r}} (for M \gg m), derived from setting E = 0.^[21] The analysis of the effective potential provides a modern framework for deriving Kepler's laws of planetary motion, originally empirical observations later explained by Newton's law of universal gravitation in his Philosophiæ Naturalis Principia Mathematica (1687). Specifically, the elliptical orbits (first law), equal areas in equal times (second law, from angular momentum conservation), and harmonic period law (third law, from the energy and r_c relations) emerge directly from the inverse-square force and effective potential structure.^[22]

Quantum Applications

Radial Schrödinger Equation

In quantum mechanics, the time-independent Schrödinger equation for a particle in a central potential V(r) in three dimensions is solved by separating variables in spherical coordinates. The wave function is expressed as \psi(\mathbf{r}) = R(r) Y_{l m}(\theta, \phi), where Y_{l m} are the spherical harmonics that satisfy the angular part of the equation, and R(r) is the radial wave function depending only on the radial distance r./04%3A_Energy_Levels/4.03%3A_Solutions_to_the_Schrodinger_Equation_in_3D) This separation leads to the radial Schrödinger equation for the function u(r) = r R(r), which takes the form

-\frac{\hbar^2}{2\mu} \frac{d^2 u}{dr^2} + V_{\text{eff}}^q(r) u(r) = E u(r),

where \mu is the reduced mass, E is the energy eigenvalue, and the quantum effective potential is

V_{\text{eff}}^q(r) = V(r) + \frac{\hbar^2 l(l+1)}{2 \mu r^2}.

Here, l is the orbital angular momentum quantum number, an integer replacing the classical angular momentum magnitude L via L = \hbar \sqrt{l(l+1)}.^[2]^[23] The centrifugal term \frac{\hbar^2 l(l+1)}{2 \mu r^2} arises from the angular components of the Laplacian operator in spherical coordinates, specifically the \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial}{\partial r} \right) - \frac{l(l+1)}{r^2} structure after separation, which confines the motion to a one-dimensional radial problem with this additional barrier-like potential./04%3A_Energy_Levels/4.03%3A_Solutions_to_the_Schrodinger_Equation_in_3D) This term is quantum analogous to the classical effective potential's centrifugal contribution, as outlined in the Mathematical Formulation. The radial equation requires boundary conditions ensuring physical regularity: u(0) = 0 at r = 0, imposed by the centrifugal barrier that repels the wave function from the origin for l > 0, and u(r) \to 0 as r \to \infty for bound states to normalize the probability density.^[2]^[23]

Atomic and Molecular Systems

In the hydrogen atom, the quantum effective potential governs the radial motion of the electron in a central Coulomb field, given by V(r) = -\frac{e^2}{4\pi \epsilon_0 r}, combined with the centrifugal term to form V_{\rm eff}^q(r) = -\frac{e^2}{4\pi \epsilon_0 r} + \frac{l(l+1)\hbar^2}{2 \mu r^2}, where \mu is the reduced mass and l is the orbital angular momentum quantum number.^[6] The exact solutions to the radial Schrödinger equation using this effective potential yield bound-state energy levels E_n = -\frac{\mu e^4}{2 \hbar^2 n^2 (4\pi \epsilon_0)^2}, where the principal quantum number n = n_r + l + 1, with n_r denoting the radial quantum number.^[6] These levels are independent of l for a given n, reflecting the degeneracy in the pure Coulomb problem, and the ground-state energy is -13.6 eV.^[6] The centrifugal barrier in V_{\rm eff}^q(r) plays a crucial role in determining the radial wave function behavior. For l = 0 (s-states), the absence of this barrier allows the electron wave function to penetrate closer to the nucleus, reaching non-zero probability at r = 0, which maximizes the attractive Coulomb interaction.^[24] In contrast, for higher l, the repulsive \frac{l(l+1)\hbar^2}{2 \mu r^2} term dominates at small r, creating a barrier that confines the wave function away from the origin (scaling as r^{l+1} near r = 0) and effectively reduces the depth of the potential well, leading to higher (less negative) energies for states with the same n but larger l in perturbed systems.^[25] In multi-electron atoms, the effective potential for outer radial orbitals approximates the screened Coulomb interaction due to the inner electron cloud shielding the nuclear charge. A common model is the Yukawa potential, V(r) \approx -\frac{(Z - \sigma) e^2}{4\pi \epsilon_0 r} e^{-r/D}, where Z is the atomic number, \sigma accounts for screening by core electrons, and D is a screening length, enabling variational or numerical solutions for approximate energy levels and radial distributions.^[26] This effective potential captures the reduced attraction for valence electrons, facilitating mean-field treatments like Hartree-Fock for orbital energies. For diatomic molecules, the quantum effective potential describes vibrational-rotational levels by combining the anharmonic Morse potential, V(r) = D_e \left(1 - e^{-\alpha (r - r_e)}\right)^2, with the centrifugal term \frac{\hbar^2 l(l+1)}{2 \mu r^2} (where D_e is the dissociation energy, \alpha a scaling parameter, r_e the equilibrium bond length, \mu the reduced mass, and l the rotational quantum number), forming V_{\rm eff}^q(r).^[27] Solving the radial Schrödinger equation with this potential yields quantized energies that account for coupled vibrations and rotations, with the rotational constant B = \frac{\hbar^2}{2 \mu r_e^2} setting the scale for rotational spacing in the rigid-rotor limit.^[28] In alkali atoms, the quantum defect introduces a slight deviation from the pure hydrogenic energy levels due to core penetration by the valence electron, modifying the effective potential near the nucleus. The observed levels follow E_n \approx -\frac{\rm Ry}{(n - \delta_l)^2}, where Ry is the Rydberg constant and \delta_l (typically 0.1–1 for low l) quantifies the phase shift from short-range core interactions, most pronounced for s- and p-states where penetration allows overlap with the ionic core.^[29] This effect breaks the exact l-degeneracy, with \delta_l decreasing for higher l as the centrifugal barrier limits core access.^[29]

References

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11.3 Effective Potential - BOOKS
The sum of the ordinary potential, and the angular part of the kinetic energy, − ℓ 2 2 μ r 2 is called the effective potential . 🔗
[2]
[PDF] 22. The Radial Equation - Physics
which we can group with the potential energy to form the effective potential energy function,. Veff(r) = ¯h2l(l + 1). 2mr2. + V (r),. (8) just as we often do in ...
[3]
[PDF] 8 Central-Force Motion
The effective potential is the sum of these two potentials and has a characteristic dip where the potential energy has a minimum. The result of this dip is ...<|control11|><|separator|>
[4]
[PDF] The Two-Body Problem - UCSB Physics
Smaller values of angular mo- mentum will result in effective potentials with much deeper potential minima, while larger values of the angular momentum will ...<|control11|><|separator|>
[5]
[PDF] Schrödinger Equation - JILA
The effective potential Veff(r) is the sum of the potential V (r) with a ... orbiting in such a potential can be separated into radial and angular parts.
[6]
Energy Levels of Hydrogen Atom - Richard Fitzpatrick
The effective potential has a simple physical interpretation. The first part is the attractive Coulomb potential, and the second part corresponds to the ...
[7]
[PDF] Vacuum Energy and Effective Potentials - UT Physics
The zero point energies of the fields depend on their masses, and the net vacuum energy density acts as an effective potential for the ϕ field; it is this ...Missing: physics | Show results with:physics
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[PDF] Effective Potential Kinetic Theory for Strongly Coupled Plasmas
Abstract. The effective potential theory (EPT) is a recently proposed method for extending traditional plasma kinetic and transport theory into the strongly ...
[9]
16.1 The Radial Equation - BOOKS
The term ℏ 2 ℓ ( ℓ + 1 ) / 2 μ r 2 is the centrifugal contribution to the effective potential energy. It behaves like a repulsive force, and it increases with ℓ ...
[10]
[PDF] Classical Mechanics - Rutgers Physics
Oct 5, 2010 · ... central force field about the origin, with a potential U(r) = U(|r ... effective potential. Ueff(r) = U(r) +. L2. 2µr2 . Thus we have ...
[11]
[PDF] classical-mechanics-by-kibble-and-berkshire.pdf - Physica Educator
for ... effective potential energy function. U(r) = J2. 2mr2. + V (r). (4.13). It is easy ...
[12]
[PDF] Classical Dynamics - DAMTP
This is the definition of a central force. An example is given by the ... where the effective potential is defined to be. Veff(θ) = − g l cosθ +. J2.
[13]
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The effective potential is the sum of the centrifugal potential (the first ... equation with potential V (r). Since there is a well we expect bound ...
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None
### Summary of Effective Potential Derivation for Central Force Problems
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[PDF] 4. Central Forces - DAMTP
The minimum of the effective potential occurs at r* = ml2/k and takes the value. Veff(r*) = k2/2ml2. The possible forms of the motion can be characterised by ...Missing: behavior asymptotic limits
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effective potential' Ueff = l^2/(2mr^2) + U(r) For a circular orbit
The effective potential approach fits right into what we did before for frequency of small oscillations near a potential minimum · The expansion r = ro+x is good ...
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[PDF] Gravitation and Kepler's Laws - University of Southampton
Right: rosette orbit of a particle with this effective potential. Such a potential describes, for example, the force of attraction between nucleons in an ...<|control11|><|separator|>
[19]
Perihelion precession in power-law potentials: Hénon's theorem
Aug 1, 2022 · The shift of the angle of the perihelion, ϕ ⁠, during one radial period will be called the precession angle Φ = Δ ϕ = ϕ ( T r ) − ϕ ( 0 ) ⁠. In ...Missing: rosette | Show results with:rosette
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The radial and azimuthal oscillations now have different periods, and the orbits are no longer closed ellipses. The periastron precesses. One of the early ...
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disproportionate in magnitude. 9.4.5 Escape velocity. The threshold for escape from a gravitational potential occurs at E = 0. Since E = T + U is conserved ...
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We present here a calculus-based derivation of Kepler's Laws. You should be familiar with the results, but need not worry about the details of the derivation.
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Indeed, if we define the effective potential by. ( )( )( ). rV r ll. rV eff. +. +. = 2. 2. 1. Page 7. then this is a 1D Schrödinger equation, with the effective ...
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[PDF] HYDROGEN ATOM
Sep 25, 2013 · Veff = -e2/r + l(l + 1)¯h2/(2mr2). The second term, called the centrifugal barrier, arises as a result of the orbital motion of the electron.
[26]
[PDF] Ground-state stability and criticality of two-electron atoms ... - FaMAF
A simple model to describe the effect of the screening in the Coulomb potential is the Yukawa potential, where an exponential decay is introduced, 1/r → exp (−r ...Missing: multi- | Show results with:multi-
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None
### Summary: Use of Effective Potential for Diatomic Molecules with Morse Potential Plus Centrifugal Term
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Mordad 15, 1404 AP · The notion of the quantum defect is important in atomic and molecular spectroscopy and also in unifying spectroscopy with collision theory.