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By M. Bocher

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Let u(x, t) be as in 1(a) and let v(x, t) be as in 1(e), and suppose w(x, t) = 3u(x, t) − 2v(x, t). Conclude that w also satisfies the Wave Equation, without explicitly computing any derivatives of w. 4. Suppose u(x, t) and v(x, t) are both solutions to the Wave equation, and w(x, t) = 5u(x, t) + 2v(x, t). Conclude that w also satisfies the Wave Equation. x+t 5. Let u(x, t) = x−t cos(y) dy = sin(x + t) − sin(x − t). Show that u satisfies the (one-dimensional) Wave Equation ∂t2 u = △u. 6. By explicitly computing derivatives, show that the following functions satisfy the (twodimensional) Wave Equation ∂t2 u = △u.

By now, these analogies and guesses have been overwhelmingly vindicated by experimental evidence. The best justification for quantum mechanics is that it ‘works’, by which we mean that its theoretical predictions match all available empirical data with astonishing accuracy. 2, we cannot derive quantum theory from ‘first principles’, because the postulates of quantum mechanics are the first principles. Instead, we will simply state the main assumptions of the theory, which are far from self-evident, but which we hope you will accept because of the weight of empirical evidence in their favour.

Hint: u(x, y, z) is sometimes called the Coulomb potential. Remark: Observe that ∇ u(x, y, z) = − x, y 2 1 −x2 − y 2 1 = exp exp be the (two-dimensional) 4πt 4t 4πt 4t Gauss-Weierstrass Kernel. Show that u satisfies the (two-dimensional) Heat equation, ∂t u = △u. 5. Let u(x, y; t) = 6. Let α and β be real numbers, and let h(x, y) = sinh(αx) · sin(βy). (a) Compute △ h(x, y). (b) Suppose h is harmonic. Write an equation describing the relationship between α and β. 3 Recall that a function h : RD −→ R is harmonic if △u ≡ 0.

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An Introduction to the Study of Integral Equations by M. Bocher


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