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2.- The Klein-Gordon equation Utt Uxx+u= 0 arises in many areas of physics, including relativistic quantum mechanics. Let -∞ < x < ∞, t > 0, and consider the following initial value problem: the given initial values u(x, 0) and ut(x, 0) are real, and we assume that u → 0 sufficiently rapidly as |x| → ∞. (a) Show that the dispersion relation for the wave solution u = ei(kx-wt) is given by w² = 1+k². Further, show that the phase speed c(k) = w/k satisfies |c(k)| ≥ 1, while for the group velocity C(k) = we have |C(k)| ≤ 1. dw dk

2.- The Klein-Gordon equation Utt Uxx+u= 0 arises in many areas of physics, including relativistic quantum mechanics. Let -∞ < x < ∞, t > 0, and consider the following initial value problem: the given initial values u(x, 0) and ut(x, 0) are real, and we assume that u → 0 sufficiently rapidly as |x| → ∞. (a) Show that the dispersion relation for the wave solution u = ei(kx-wt) is given by w² = 1+k². Further, show that the phase speed c(k) = w/k satisfies |c(k)| ≥ 1, while for the group velocity C(k) = we have |C(k)| ≤ 1. dw dk

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 69EQ: Let x=x(t) be a twice-differentiable function and consider the second order differential equation...
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2.- The Klein-Gordon equation Utt Uxx+u= 0 arises in many areas of physics, including relativistic quantum mechanics. Let -∞ < x < ∞, t > 0, and consider the following initial value problem: the given initial values u(x, 0) and ut(x, 0) are real, and we assume that u → 0 sufficiently rapidly as |x| → ∞. (a) Show that the dispersion relation for the wave solution u = ei(kx-wt) is given by w² = 1+k². Further, show that the phase speed c(k) = w/k satisfies |c(k)| ≥ 1, while for the group velocity C(k) = we have |C(k)| ≤ 1. dw dk
Transcribed Image Text:2.- The Klein-Gordon equation Utt Uxx+u= 0 arises in many areas of physics, including relativistic quantum mechanics. Let -∞ < x < ∞, t > 0, and consider the following initial value problem: the given initial values u(x, 0) and ut(x, 0) are real, and we assume that u → 0 sufficiently rapidly as |x| → ∞. (a) Show that the dispersion relation for the wave solution u = ei(kx-wt) is given by w² = 1+k². Further, show that the phase speed c(k) = w/k satisfies |c(k)| ≥ 1, while for the group velocity C(k) = we have |C(k)| ≤ 1. dw dk
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