2.A scalar (r, y, t) solves the massive Klein-Gordon equation in two spatial di-8² 8² m²c²əx² Əy² h²Ꭷ +mensions18²c² Ət²-=0.Here e (the speed of light), ħ (Planck's constant), and m (the mass of the scalar) are allconstants.Find the general solution to this equation using separation of variables.

Answered: 2. A scalar (r, y, t) solves the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

2.
A scalar (r, y, t) solves the massive Klein-Gordon equation in two spatial di-
8² 8² m²c²
əx² Əy² h²
Ꭷ +
mensions
18²
c² Ət²
-=0.
Here e (the speed of light), ħ (Planck's constant), and m (the mass of the scalar) are all
constants.
Find the general solution to this equation using separation of variables.

Expert Solution

Step 1: introduction

Given that:

The massive Klein-Cordon equation in two spatial dimensions:

$1 over c squared fraction numerator partial differential squared psi over denominator partial differential t squared end fraction minus fraction numerator partial differential squared psi over denominator partial differential x squared end fraction minus fraction numerator partial differential squared psi over denominator partial differential y squared end fraction plus fraction numerator m squared c squared over denominator h squared end fraction psi equals 0$

where c is the speed of the car.

h is the Planck's constant.

m is the mass of the scalar.

To find: The genera; solution to this equation using separation of variables.

Concept:

To find the general solution to the massive Klein-Gordon equation in two spatial dimensions using the separation of variables, assume that the solution can be written as a product of functions, each depending on a single variable.

Use the method of separation of variables to achieve this.