![]() ![]() At the same time, \( \Theta(\theta) \) and \( \Phi(\varphi) \) together describe the angular distribution of the particle.ĭemonstrating Separation of Variables through Schrödinger Equation Example The radial part, \( R(r) \), may illustrate the behaviour of the particle in the radial direction. The function \( \psi \) has been separated into variables for radius, polar angle, and azimuthal angle-leading us to the term 'separation of variables'.Įach function represents a piece of the solution and is determined by solving the Schrödinger Equation. Hence, the wave function for a central potential case can be written as: In these cases, the solution for the wave function \( \psi(r, \theta, \varphi) \) can be expressed as a product of functions, each of which depends only on one of the coordinates. However, if the potential \( V(r) \) only depends on the distance \( r \) and not the angular variables, the equation simplifies considerably. Under certain circumstances, the Schrödinger Equation can be quite complex and challenging to solve directly. Now, regarding the separation of variables, this is a common mathematical technique used to simplify certain types of differential equations, including the Schrödinger Equation. Essentially, it exhibits spherical symmetry. When talking about a 'central potential', you're referring to a potential that solely depends on the radial distance from a given point, usually the origin. What does Separation of Variables mean in Central Potentials? ![]() The mathematical technique applied in solving the Schrödinger Equation for a particle in such a field is known as the separation of variables. It provides insights into numerous physical situations, such as the movement of electrons in atoms. In Quantum Mechanics, understanding the behaviour of a particle in a central potential is a fundamental topic. Solving this equation will not only show you how to find the eigenfunctions and eigenvalues, but more importantly, demonstrate how Quantum Physics applies to more complex systems.Įxploring Separation of Variables for Central Potentials This equation can't be solved directly, but we can turn to a set of solutions, known as the Spherical Harmonics and the associated Laguerre polynomials. Suppose \( H \) is the operator \( -\frac m \omega^2 r^2 \psi = E \psi \] In other words, we’re looking for the wave function of the system. Our aim is to find \( \psi \) given \( H \). ![]() Solving the Schrödinger Equation: An Exercise The energy \( E \) is constant because it's time-independent. The wave function \( \psi \) provides information about the probability distribution of the position of a particle within a system. It's the operator we apply to the wave function \( \psi \). The Hamiltonian operator \( H \) corresponds to the total energy of the system, including kinetic and potential energy. Let's explain these components in more detail.
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