I Klein-Gordon Equation with boundary conditions

dsaun777
Messages
296
Reaction score
39
I am trying to find solutions for the Klien-Gordon equations in 1-d particle in a box. The difference here is the box itself oscillating and has boundary conditions that are time dependent, something like this L(t)=L0+ΔLsin(ωt). My initial approach is to use a homogeneous solution and use Eigenfunction expansion to get a solution. But I can't make progress. Are there other methods that are easier? maybe numerical methods...
 
Physics news on Phys.org
Do you mean that you have homogenous BCs at x = 0 and x = L(t)?

Set \epsilon = \Delta L/L_0 so that the boundary condition is applied at x = L_0(1 + \epsilon \sin \omega t). Then for small \epsilon, you can seek an asymptotic expansion \psi = \psi_0 + \epsilon \psi_1 + \dots where the boundary conditions on the \psi_n are shifted to x = L_0 by expanding \psi(L_0(1 + \epsilon \sin \omega t),t) in Taylor series in x about x = L_0 and comparing coefficients of powers of \epsilon. This gives you a series of problems which can be solved by eigenfunction expansion.

EDIT: It might be better to solve for each \psi_n using a Laplace transform in time. The change of variable \tilde x = x/L_0 may help to simplify the algebra. Also note that if \omega is a natural frequency of any of these problems then the resulting resonance will cause the asymptotic assumption to break down, since eventually the amplitude will exceed \epsilon^{-1}.
 
Last edited:
  • Like
Likes dsaun777 and ergospherical
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. Towards the end of the first lecture for the Qiskit Global Summer School 2025, Foundations of Quantum Mechanics, Olivia Lanes (Global Lead, Content and Education IBM) stated... Source: https://www.physicsforums.com/insights/quantum-entanglement-is-a-kinematic-fact-not-a-dynamical-effect/ by @RUTA
If we release an electron around a positively charged sphere, the initial state of electron is a linear combination of Hydrogen-like states. According to quantum mechanics, evolution of time would not change this initial state because the potential is time independent. However, classically we expect the electron to collide with the sphere. So, it seems that the quantum and classics predict different behaviours!
Back
Top