The separation of variables method for the Schrödinger equation of hydrogen in spherical coordinates

Introduction

The separation of variables method is a fundamental technique in quantum mechanics, enabling the solution of partial differential equations by reducing them to simpler ordinary differential equations. This method is particularly critical in solving the Schrödinger equation for the hydrogen atom, a cornerstone of atomic physics. The hydrogen atom's Schrödinger equation, a second-order partial differential equation, is separable into radial and angular components, allowing for the derivation of energy eigenvalues and wavefunctions. This approach not only simplifies the problem but also provides insights into the quantized nature of atomic energy levels and the behavior of electrons in hydrogen-like atoms. The method relies on the assumption that the wavefunction can be expressed as a product of functions dependent on independent variables, a principle that underpins much of modern quantum theory.

The Schrödinger Equation and Its Structure

The time-independent Schrödinger equation for the hydrogen atom is given by:
$$ -\frac{\hbar^2}{2m}\nabla^2\psi(\mathbf{r}) + V(r)\psi(\mathbf{r}) = E\psi(\mathbf{r}) $$
where $ \hbar $ is the reduced Planck constant, $ m $ is the electron mass, $ \nabla^2 $ is the Laplacian operator, $ V(r) $ is the potential energy function, and $ E $ is the energy eigenvalue. For the hydrogen atom, the potential $ V(r) $ is proportional to $ -\frac{e^2}{r} $, leading to a radial potential that decays with distance. This form of the equation is inherently separable into radial and angular components, as the Laplacian operator in spherical coordinates can be decomposed into radial and angular parts.

Separation of Variables and the Radial Equation

To apply the separation of variables method, the wavefunction $ \psi(\mathbf{r}) $ is assumed to be a product of a radial function $ R_{n,l}(r) $ and an angular function $ Y_{l,m}(\theta, \phi) $:
$$ \psi(r, \theta, \phi) = R_{n,l}(r)Y_{l,m}(\theta, \phi) $$
Substituting this into the Schrödinger equation and applying boundary conditions results in the separation of variables, yielding two independent ordinary differential equations: one for the radial part and one for the angular part. The radial equation, derived from the Schrödinger equation, is:
$$ -\frac{\hbar^2}{2m}\left[ \frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{dR}{dr}\right) + \frac{1}{2}\frac{1}{R}\frac{d^2R}{dR^2} \right] + V(r)R = E R $$
This equation is further simplified by introducing the associated Laguerre polynomials and the Bohr radius, which arise from the boundary conditions at $ r = 0 $ and $ r \to \infty $. The angular equation, derived from the Laplacian in spherical coordinates, involves the angular frequency $ \ell $ and the quantum number $ m $, leading to the solution of the Schrödinger equation in terms of spherical harmonics.

Boundary Conditions and Energy Levels

The boundary conditions for the radial equation are derived from the requirement that the wavefunction must vanish at $ r = 0 $, ensuring the physical interpretation of the electron's wavefunction being localized in the atom. This condition leads to the quantization of the radial quantum number $ n $, which is the sum of the principal quantum number $ n $ and the angular momentum quantum number $ l $. The energy levels are determined by the principal quantum number $ n $, with the energy given by:
$$ E_n = -\frac{e^2}{8\pi\epsilon_0}\frac{1}{n^2} $$
This result aligns with the Bohr model's prediction for the hydrogen atom's energy levels, demonstrating the validity of the separation of variables method in capturing the quantized nature of atomic energy. The angular part of the wavefunction, described by the spherical harmonics $ Y_{l,m}(\theta, \phi) $, determines the spatial distribution of the electron's probability density, with the quantum number $ l $ specifying the orbital angular momentum and the magnetic quantum number $ m $ specifying the orientation of the orbital in three-dimensional space.

Wavefunctions and Quantum Numbers

The full wavefunction for the hydrogen atom is the product of the radial and angular parts:
$$ \psi_{n,l,m}(r, \theta, \phi) = R_{n,l}(r)Y_{l,m}(\theta, \phi) $$
The radial function $ R_{n,l}(r) $ is determined by solving the radial equation with the boundary conditions, leading to the associated Laguerre polynomials. The energy levels are determined solely by the principal quantum number $ n $, while the angular part is governed by the quantum numbers $ l $ and $ m $. The spherical harmonics $ Y_{l,m}(\theta, \phi) $ describe the angular dependence of the wavefunction, with $ l $ determining the shape of the orbital and $ m $ specifying the magnetic quantum number. The combination of these quantum numbers defines the quantum state of the electron in the hydrogen atom, with each state corresponding to a unique energy level and spatial distribution.

Conclusion

The separation of variables method provides a powerful framework for solving the Schrödinger equation of the hydrogen atom, enabling the derivation of quantized energy levels and the determination of electron wavefunctions. By decomposing the problem into radial and angular components, the method captures the essential physics of atomic structure, including the quantization of energy and the probabilistic nature of electron positions. The resulting solutions, characterized by the principal quantum number $ n $, angular momentum quantum number $ l $, and magnetic quantum number $ m $, underpin the quantum mechanical description of hydrogen-like atoms. This approach not only validates the predictions of the Bohr model but also establishes the foundation for modern quantum theory, demonstrating the elegance and utility of separation of variables in solving complex physical problems. The method's success in the hydrogen atom case underscores its general applicability to other systems where the Schrödinger equation can be separated into independent equations for each variable.