Fact-checked by Grok 2 weeks ago

Parabolic coordinates

Parabolic coordinates, also known as paraboloidal coordinates, constitute an orthogonal curvilinear coordinate system in three-dimensional Euclidean space, designed to simplify the analysis of problems with axial symmetry and parabolic boundaries. The coordinates are typically denoted by u, v, and \phi, related to Cartesian coordinates by the transformation equations x = uv \cos \phi, y = uv \sin \phi, z = \frac{1}{2}(u^2 - v^2), with ranges u \geq 0, v \geq 0, and $0 \leq \phi < 2\pi.^[1] The surfaces of constant u and constant v form paraboloids of revolution about the z-axis, while constant \phi defines meridional planes containing the z-axis, making the system particularly useful for separating variables in partial differential equations involving Laplace's or the Helmholtz equation in regions bounded by such surfaces.^[2] The metric scale factors for parabolic coordinates are h_u = \sqrt{u^2 + v^2}, h_v = \sqrt{u^2 + v^2}, and h_\phi = uv, which determine the line element ds^2 = (u^2 + v^2)(du^2 + dv^2) + (uv)^2 d\phi^2 and facilitate the expression of differential operators like the gradient, divergence, and Laplacian in these coordinates.^[3] This orthogonality ensures that the coordinate vectors \mathbf{e}_u, \mathbf{e}_v, and \mathbf{e}_\phi are mutually perpendicular, simplifying computations in vector calculus. A related two-dimensional variant, parabolic cylindrical coordinates, extends to three dimensions by including a linear z-coordinate, with transformations x = \xi \eta, y = \frac{1}{2}(\eta^2 - \xi^2), z = z, and scale factors h_\xi = h_\eta = \sqrt{\xi^2 + \eta^2}, h_z = 1, applicable to problems with translational symmetry along one axis.^[3] In physics, parabolic coordinates find prominent applications in quantum mechanics, notably for solving the Schrödinger equation for the hydrogen atom in an external electric field, where they enable separation of variables to analyze the Stark effect—the linear splitting of energy levels due to the field.^[3] They also prove valuable in electrostatics for potential problems involving paraboloidal conductors and in wave propagation studies, such as acoustics or electromagnetism in parabolic domains, due to the natural alignment with confocal paraboloidal surfaces.

Fundamentals

Definition and geometry

Parabolic coordinates constitute a family of orthogonal curvilinear coordinate systems defined by level surfaces that form confocal parabolas, making them particularly suited for analyzing problems exhibiting parabolic symmetry in two or three dimensions.^[3] In this system, the coordinates parameterize space such that constant values of each coordinate trace out parabolic surfaces sharing a common focus, ensuring the coordinate curves intersect at right angles due to their confocal geometry.^[3] This orthogonality simplifies the formulation of differential equations in curvilinear systems, as the metric tensor becomes diagonal.^[4] Geometrically, parabolic coordinates arise from two interlocking families of parabolas in the plane, both sharing the same focal point but opening in perpendicular directions—one family along the positive axis and the other along the negative axis relative to the focus.^[3] Each parabola is defined as the locus of points equidistant from the focus and a corresponding directrix, a property that generates the confocal arrangement where surfaces of constant coordinate values nest around the focus without intersecting one another except at the origin.^[5] The confocal nature guarantees orthogonality, as the tangent vectors to the coordinate curves at any intersection point are perpendicular, a direct consequence of the shared focus and symmetric directrices.^[6] For visualization in two dimensions, imagine the xy-plane with the focus at the origin; one family of parabolas opens upward or downward along the y-axis, while the other opens leftward or rightward along the x-axis, creating a grid-like pattern of nested curves that fill the plane.^[4] Extending to three dimensions involves rotating these parabolic cylinders about the axis of symmetry (typically the z-axis), yielding paraboloidal surfaces that maintain the confocal and orthogonal properties, with the third coordinate often representing the azimuthal angle around this axis.^[3] This construction produces a system where coordinate surfaces are elliptic paraboloids confocal at the origin, providing a natural framework for volumetric regions bounded by parabolic shapes. As a foundational system, parabolic coordinates facilitate the separation of variables in Laplace's equation for boundary value problems involving parabolic domains, such as those with boundaries conforming to confocal paraboloids, by aligning the coordinate surfaces with the problem's inherent symmetry and enabling additive separation in the orthogonal basis.^[3]

Historical development

Parabolic coordinates were developed in the 19th century as part of the broader study of orthogonal curvilinear coordinate systems designed to facilitate the separation of variables in partial differential equations, particularly Laplace's equation. These systems arose from investigations into confocal quadrics, where parabolic coordinates represent a degenerate case of more general conic configurations.^[7] Key contributions to their formulation came from Carl Gustav Jacob Jacobi during the 1830s and 1840s, who examined elliptic and parabolic systems within confocal coordinates to enable separation of variables in integrable Hamiltonian systems. Jacobi recognized parabolic coordinates as emerging from elliptic-hyperbolic setups when one focus recedes to infinity, integrating them into the analysis of geodesic flows on quadrics and related motions.^[8]^[9] Later refinements in the late 19th century were provided by Gaston Darboux, who advanced the general theory of separation in confocal conic coordinates, excluding certain degeneracies like the parabolic case while building on Jacobi's foundations.^[7] The evolution of parabolic coordinates transitioned from two-dimensional confocal quadrics, initially explored for geometric properties, to specific adaptations for solving the Laplace equation through variable separation. This progression highlighted their utility in mathematical physics, with 20th-century classifications solidifying their role. Parabolic coordinates emerged alongside elliptic and hyperbolic systems as one of the 11 classical orthogonal coordinate systems in three-dimensional Euclidean space that admit separation of variables for the Helmholtz equation.^[10]

Two-dimensional parabolic coordinates

Coordinate transformation

The three-dimensional parabolic cylindrical coordinates, denoted as (\sigma, \tau, z), extend the two-dimensional parabolic coordinate system by incorporating the Cartesian z-coordinate without alteration. This creates a cylindrical variant suitable for describing regions with translational invariance along the z-axis. The transformation relations to Cartesian coordinates (x, y, z) are

x = \sigma \tau, \quad y = \frac{1}{2} (\tau^2 - \sigma^2), \quad z = z,

where the coordinates satisfy \sigma \geq 0, -\infty < \tau < \infty, and -\infty < z < \infty.^[11]^[12] The inverse transformation follows directly from the two-dimensional case for the (x, y) components, with z remaining unchanged; specifically, \sigma and \tau are recovered from x and y using the relations \sigma = \sqrt{-y + \sqrt{x^2 + y^2}} and \tau = x / \sigma (or equivalent forms depending on quadrant), ensuring the ranges are respected to avoid duplication. Geometrically, the constant-\sigma and constant-\tau surfaces are parabolic cylinders obtained by linearly extruding the confocal parabolic curves of the two-dimensional system along the z-axis, with the generators parallel to the z-direction. These cylinders open in opposite directions along the y-axis, intersecting orthogonally.^[12] This system spans all of \mathbb{R}^3, with the only degeneracy occurring along the focal line at the z-axis (where \sigma = 0 and \tau = 0). The confocal nature along the z-axis arises from the shared focus at the origin in every xy-plane perpendicular to the z-axis, rendering it ideal for boundary value problems invariant under translation along z or bounded by parabolic cylindrical surfaces.^[12]^[13] Unlike the rotationally symmetric paraboloidal coordinates, the parabolic cylindrical system does not possess azimuthal symmetry around the z-axis.^[14]

Scale factors

In three-dimensional parabolic cylindrical coordinates (\sigma, \tau, z), the line element is expressed as

ds^2 = h_\sigma^2 \, d\sigma^2 + h_\tau^2 \, d\tau^2 + h_z^2 \, dz^2,

where the scale factors are h_\sigma = h_\tau = \sqrt{\sigma^2 + \tau^2} and h_z = 1. These scale factors arise from the partial derivatives of the position vector with respect to each coordinate, quantifying the infinitesimal displacements along the orthogonal directions.^[12]^[15] The associated metric tensor is diagonal, with components g_{\sigma\sigma} = g_{\tau\tau} = \sigma^2 + \tau^2, g_{zz} = 1, and all off-diagonal elements zero, reflecting the orthogonality of the coordinate system. This structure follows directly from the coordinate transformation, where the z-coordinate remains unchanged from Cartesian form. The scale factors h_\sigma and h_\tau are derived by extending the two-dimensional parabolic coordinates along the z-direction without introducing distortion, thereby preserving their equality while h_z = 1 accounts for the linear mapping in z.^[16]^[15] The value h_z = 1 notably simplifies the volume element to

dV = h_\sigma h_\tau h_z \, d\sigma \, d\tau \, dz = (\sigma^2 + \tau^2) \, d\sigma \, d\tau \, dz,

which integrates over regions bounded by parabolic cylinders more tractably than in distorted systems. Furthermore, the product h_\sigma h_\tau h_z = \sigma^2 + \tau^2 plays a key role in enabling the separation of variables for partial differential equations in cylindrical parabolic geometries, such as the Helmholtz equation.^[12]^[16] As an illustration, the gradient operator in these coordinates incorporates the scale factors as

\nabla \phi = \frac{1}{h_\sigma} \frac{\partial \phi}{\partial \sigma} \hat{\sigma} + \frac{1}{h_\tau} \frac{\partial \phi}{\partial \tau} \hat{\tau} + \frac{1}{h_z} \frac{\partial \phi}{\partial z} \hat{z} = \frac{1}{\sqrt{\sigma^2 + \tau^2}} \left( \frac{\partial \phi}{\partial \sigma} \hat{\sigma} + \frac{\partial \phi}{\partial \tau} \hat{\tau} \right) + \frac{\partial \phi}{\partial z} \hat{z},

highlighting how the identical h_\sigma and h_\tau symmetrize the in-plane components relative to the unmodified z-direction.^[15]

Three-dimensional parabolic coordinates

Coordinate transformation

The three-dimensional parabolic coordinates, also known as confocal paraboloidal coordinates and denoted as (u, v, \phi), are obtained by rotating the two-dimensional parabolic system about its axis of symmetry (the z-axis), introducing azimuthal symmetry. This system is suitable for problems with rotational invariance around the z-axis and boundaries formed by paraboloids of revolution. The transformation relations to Cartesian coordinates (x, y, z) are

x = uv \cos \phi, \quad y = uv \sin \phi, \quad z = \frac{1}{2}(u^2 - v^2),

where the coordinates satisfy u \geq 0, v \geq 0, and $0 \leq \phi < 2\pi.^[17]^[1] The inverse transformation is given by \phi = \atan2(y, x), and

u = \sqrt{\sqrt{x^2 + y^2 + z^2} + z}, \quad v = \sqrt{\sqrt{x^2 + y^2 + z^2} - z},

ensuring the non-negative ranges for u and v. These expressions follow from solving the system using \rho = \sqrt{x^2 + y^2} = uv and the relations u^2 + v^2 = 2\sqrt{\rho^2 + z^2} and u^2 - v^2 = 2z.^[17] Geometrically, the surfaces of constant u form paraboloids of revolution opening upward along the positive z-direction, given by z = \frac{1}{2}u^2 - \frac{\rho^2}{2u^2}, while constant v surfaces are paraboloids opening downward along the negative z-direction, z = -\frac{1}{2}v^2 + \frac{\rho^2}{2v^2}. The constant \phi surfaces are meridional half-planes containing the z-axis. These confocal paraboloids intersect orthogonally, sharing a common focus at the origin.^[17]^[2] This coordinate system covers all of \mathbb{R}^3, with degeneracy occurring along the z-axis where u = v = 0. The rotational symmetry around the z-axis makes it ideal for boundary value problems bounded by paraboloidal surfaces or with axial symmetry, such as in quantum mechanics for the hydrogen atom in electric fields.^[17]

Scale factors

In three-dimensional parabolic coordinates (u, v, \phi), the line element is expressed as

ds^2 = h_u^2 \, du^2 + h_v^2 \, dv^2 + h_\phi^2 \, d\phi^2,

where the scale factors are h_u = h_v = \sqrt{u^2 + v^2} and h_\phi = uv. These scale factors are derived from the partial derivatives of the position vector with respect to each coordinate, quantifying the infinitesimal arc lengths along the orthogonal directions.^[1]^[2] The associated metric tensor is diagonal, with components g_{uu} = g_{vv} = u^2 + v^2, g_{\phi\phi} = (uv)^2, and all off-diagonal elements zero, reflecting the orthogonality of the system. This structure arises directly from the rotational extension of the two-dimensional parabolic coordinates, preserving the equality of h_u and h_v while h_\phi accounts for the azimuthal variation.^[17] The volume element is

dV = h_u h_v h_\phi \, du \, dv \, d\phi = uv (u^2 + v^2) \, du \, dv \, d\phi,

which facilitates integration over regions bounded by confocal paraboloids. The product of scale factors h_u h_v h_\phi = uv (u^2 + v^2) is crucial for separating variables in partial differential equations, such as the Laplace or Helmholtz equation, in paraboloidal geometries.^[1] As an illustration, the gradient operator in these coordinates is

\nabla f = \frac{1}{h_u} \frac{\partial f}{\partial u} \hat{u} + \frac{1}{h_v} \frac{\partial f}{\partial v} \hat{v} + \frac{1}{h_\phi} \frac{\partial f}{\partial \phi} \hat{\phi} = \frac{1}{\sqrt{u^2 + v^2}} \left( \frac{\partial f}{\partial u} \hat{u} + \frac{\partial f}{\partial v} \hat{v} \right) + \frac{1}{uv} \frac{\partial f}{\partial \phi} \hat{\phi},

demonstrating the symmetry between the u and v directions and the distinct azimuthal component.^[2]

Applications

In electrostatics and gravitation

Parabolic coordinates prove particularly useful in electrostatics for solving Laplace's equation in geometries featuring confocal parabolic boundaries, such as a conducting parabolic cylinder. Consider the potential due to an infinite line charge parallel to a grounded conducting parabolic cylinder; the boundary conditions align naturally with constant-σ or constant-τ surfaces, allowing separation of variables in two-dimensional parabolic coordinates (σ, τ). The resulting solutions take the form Φ(σ, τ) = Σ [A_n f_n(σ) g_n(τ) + B_n f_n(τ) g_n(σ)], where f_n and g_n satisfy ordinary differential equations derived from the separated Laplace operator.^[18] In these coordinates, Laplace's equation for the electrostatic potential Φ in two dimensions assumes the form

\frac{1}{h_\sigma h_\tau} \left[ \frac{\partial}{\partial \sigma} \left( \frac{h_\tau}{h_\sigma} \frac{\partial \Phi}{\partial \sigma} \right) + \frac{\partial}{\partial \tau} \left( \frac{h_\sigma}{h_\tau} \frac{\partial \Phi}{\partial \tau} \right) \right] = 0,

where h_σ and h_τ are the scale factors. Assuming a separated solution Φ(σ, τ) = S(σ) T(τ) leads to two independent ordinary differential equations: one for S(σ) involving parabolic cylinder functions and one for T(τ), enabling exact analytic solutions for the potential and field.^[19] The gravitational analog follows identically, as Poisson's equation for the gravitational potential in vacuum reduces to Laplace's equation outside sources; thus, parabolic coordinates yield exact solutions for the potential in parabolic trough geometries, such as an infinite mass distribution along a parabolic cylinder, with the same separation yielding products of functions of σ and τ.^[19] In three dimensions, parabolic cylindrical coordinates extend this approach to infinite trough potentials, where the z-coordinate is independent, and the problem separates into a two-dimensional parabolic part along with trivial z-dependence, facilitating solutions for axisymmetric or uniform configurations like charged or massive troughs.^[12] This coordinate system's confocal nature enables exact closed-form solutions for potentials bounded by parabolic surfaces, a capability absent in Cartesian coordinates where such boundaries lead to non-separable equations.^[20] Early applications of parabolic coordinates in such potential problems trace to 19th-century physicists, including Jacobi's work on separable systems in the mid-1800s, which laid groundwork for these electrostatic and gravitational analyses.^[21]

In quantum mechanics and wave equations

In quantum mechanics, parabolic coordinates are particularly valuable for solving the time-independent Schrödinger equation for the hydrogen atom subjected to a uniform electric field, known as the Stark effect. The potential in this case is V(\mathbf{r}) = -\frac{e^2}{r} + e F z, where F is the electric field strength along the z-axis. By transforming to parabolic coordinates (\xi, \eta, \phi), defined such that z = \frac{1}{2} (\xi - \eta) and r = \frac{1}{2} (\xi + \eta), the Schrödinger equation

-\frac{\hbar^2}{2m} \nabla^2 \psi + V \psi = E \psi

separates into three independent ordinary differential equations: one azimuthal equation yielding e^{i m \phi} with integer m, and two radial-like equations in \xi and \eta.^[22] The \eta-equation resembles the radial Coulomb problem and is solved by associated Laguerre polynomials, while the \xi-equation, incorporating the linear Stark term, is addressed using confluent hypergeometric functions of the first kind, {}_1F_1(a; b; z).^[23] This separation enables exact analytical solutions for the energy levels and wave functions, revealing a hidden dynamical symmetry akin to an SO(2,1) \times SO(2,1) structure that persists for specific field strengths, allowing closed-form expressions without perturbation theory. The resulting Stark states are parabolic in character, with quantum numbers (n_\xi, n_\eta, m) where the principal quantum number n = n_\xi + n_\eta + |m| + 1, and energy shifts linear in F for degenerate levels, scaling as \Delta E \propto 3 n (n_1 - n_2) F / 2Z in atomic units, with n_1, n_2 as electric quantum numbers.^[22] This exact solvability contrasts with spherical coordinates, where the Stark perturbation mixes states, and has been foundational for understanding field-induced level splittings and tunneling ionization in Rydberg atoms. Parabolic coordinates also facilitate solutions to the Helmholtz equation \nabla^2 \psi + k^2 \psi = 0 for wave propagation in parabolic geometries, relevant to both quantum scattering and acoustics. In two dimensions, separation yields ordinary differential equations whose solutions involve parabolic cylinder functions D_\nu(\zeta), which describe modes confined within parabolic boundaries, such as sound waves in trough-like domains. For instance, in acoustics, this applies to propagation in parabolic reflectors or channels, where boundary conditions (Dirichlet for rigid walls) lead to quantized wavenumbers and evanescent or propagating modes depending on frequency. In three dimensions with rotational symmetry (parabolic rotational coordinates), the azimuthal part decouples, and the meridional equations again invoke parabolic cylinder functions, enabling exact eigenmode expansions for acoustic pressure fields in paraboloidal cavities. In quantum contexts, three-dimensional parabolic (cylindrical) coordinates prove useful for modeling particle motion in parabolic waveguides, such as quantum wires with confining potentials mimicking optical fibers. The Schrödinger equation separates, yielding guided modes expressed via parabolic cylinder functions, which capture bound states and dispersion relations for electrons or photons in curved nanostructures. This framework reveals quantization along the waveguide axis and transverse confinement, with applications to polariton propagation in two-dimensional parabolic channels formed by bent graphene or similar materials. Recent applications include quantum modeling of field emission from graphene blades, where parabolic coordinates describe the wavefunction in the vacuum region to compute emission currents under strong fields (as of 2023).^[24] In optics, parabolic coordinates enable the design of metasurfaces for generating non-diffracting parabolic beams at microwave frequencies, enhancing wireless power transfer efficiency (as of 2024).^[25] Modern extensions leverage these analytical separations for numerical methods, enhancing accuracy in high-field regimes. For example, basis expansions in parabolic functions combined with perturbation or variational techniques yield highly precise resonance positions in the Stark spectrum, converging to 20+ decimal places for low-lying states. Such approaches build on the exact separability to initialize spectral methods or accelerate convergence in time-dependent simulations of field-dressed atoms.

References

[1]
[PDF] [gex72] Paraboloidal coordinates
Whereas parabolic cylindrical coordinates [gex71] have an axis of continuous translational symme- try, paraboloidal coordinates with ranges u ≥ 0, v ≥ 0, ...
[2]
[PDF] Solutions
6) Paraboloidal (also called 3D parabolic) coordinates u, v, φ are related to Cartesian coordinates by. (. ) 2. 2. 1 cos , sin , . 2 x uv y uv z. u v φ φ.
[3]
[PDF] COORDINATE SYSTEMS , ox ox - LSU Math
Other applications of parabolic coordinates are included in the problems. EXERCISES. Find ~l, h;, and h~ if the parabolic coordinates ({;, 'Y}. 0/) are ...
[4]
3.8 Optional— Integrals in General Coordinates
for Parabolic Coordinates. Parabolic 5. The name comes from the fact that both the curves of constant u and the curves of constant v are parabolas ...
[5]
[PDF] Math 321 Vector and Complex Calculus for the Physical Sciences
Jan 25, 2018 · Curvilinear coordinates, orthogonal coordinates. The ... provides orthogonal, confocal parabolic coordinates x, y for the (u, v) plane.
[6]
Parabolas | CK-12 Foundation
A parabola is a curve such that any point on the curve is equidistant from another point, called the focus, and a line called the directrix.
[7]
Are two confocal parabolas always orthogonal?
Feb 25, 2021 · For them to intersect orthogonally, they do have to share the same axis of symmetry, and face opposite directions. Jaap Scherphuis. – Jaap ...Parabolic Cylinder Coordinates - Mathematics Stack ExchangeHow many intersections are there for two confocal parabolas with ...More results from math.stackexchange.com
[8]
[PDF] The classical Bertrand-Darboux problem - arXiv
May 7, 2006 · Another important feature (known already to Jacobi) of the four systems of coordinates is that the Carte- sian, parabolic and polar coordinate ...
[9]
[PDF] 1014-Z1-180 - Joint Mathematics Meetings
Properties of confocal quadrics were established by the beginning of the XIXth century (Dupin, Ivory, Monge). However, as shown by Jacobi ... quadrics ...
[10]
None
### Summary of Historical Information on Parabolic Coordinates, Jacobi's Contribution, Dates, and Role in Separation of Variables for the Stark Problem or Laplace Equation
[11]
[PDF] Separation of Variables in the Special Diagonal Hamilton-Jacobi ...
Diagrams in the "Results" section of this document are from Methods of Theoretical Physics,. Part I, by Morse and Feshbach ... Parabolic Coordinates (7) .50.
[12]
12.17 Physical Applications ‣ Applications ‣ Chapter 12 Parabolic ...
The main applications of PCFs in mathematical physics arise when solving the Helmholtz equation where k is a constant, and ∇ 2 is the Laplacian in Cartesian ...
[13]
Parabolic Cylindrical Coordinates -- from Wolfram MathWorld
... Morse and Feshbach ... See also. Confocal Paraboloidal Coordinates, Helmholtz Differential Equation--Parabolic Cylindrical Coordinates, Parabolic Coordinates ...
[14]
[PDF] arXiv:0903.0819v3 [quant-ph] 20 May 2009
May 20, 2009 · Parabolic-cylindrical coordinates. 2. Page 3. The parabolic-cylindrical coordinate system (u, v, z) is defined by the transformations [9] x + ...
[15]
https://dslavsk.sites.luc.edu/courses/phys301/classnotes/scalefactorscomplete.pdf
[16]
[PDF] AN INTRODUCTION TO CURVILINEAR ORTHOGONAL ...
These dot product results prove that parabolic cyclindrical coordinates are an orthogonal system. Question : Suppose you are presented with a coordinate system ...
[17]
[PDF] arXiv:1103.5942v1 [quant-ph] 30 Mar 2011
Mar 30, 2011 · The scale factors (metric coefficients) are hλ = hµ = pλ2 + µ2, hz = 1. FIG. 1: Coordinate curves in parabolic cylinder coordinates. z - k2 - 1 ...Missing: three- | Show results with:three-<|control11|><|separator|>
[18]
http://www.fis.puc.cl/~jalfaro/Fim8530/apoyo/William%20R.%20Smythe-Static%20and%20Dynamic%20Electricity%20(with%20Solutions%20Manual)%20%20-Taylor%20&%20Francis%20(1989).pdf
[19]
[PDF] methods of theoretical physics
Separable Coordinates for Laplace's Equation in Two. Dimensions. Separation of the Wave Equation. Rectangular and Para- bolic Coordinates. Polar and Elliptic ...
[20]
[PDF] methods of theoretical physics - astrosen.unam.mx
Morse and Feshbach Methods of Theoretical Physics. Muskat Physical Principles ... Parabolic Coordinates. Bispherical Coordinates. Toroidal Coordinates ...
[21]
Integrability and separation of variables in Calogero-Coulomb-Stark ...
Feb 23, 2016 · ... parabolic coordinates. (2) The two-center Coulomb problem, which admits a separation of variables in elliptic coordinates. These systems ...
[22]
Quadratic Stark Effect - Richard Fitzpatrick
It is possible to obtain this result, without recourse to perturbation theory, by solving Schrödinger's equation in parabolic coordinates [115,41]. next · up ...Missing: separation | Show results with:separation
[23]
[PDF] The Stark effect in hydrogen
Jan 8, 2016 · We will show that the Schrödinger equation can be separated in parabolic coordinates and we will use this to find the wave functions for the ...