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Practice Problems 1. (Schrrdinger equation) A quantum mechanical particle on the line with an innite potential o outside of the interval (0, l) is...

Can you please solve question 4
Practice Problems 1. (Schr¨ordinger equation) A quantum mechanical particle on the line with an infinite potential outside of the interval (0 ,l ) is given by Schr¨ ordinger’s equation for x (0 ,l ), t R 0 , and a positive real number m R > 0 by i ∂t u ( x,t ) = - 1 2 m 2 ∂x 2 u ( x,t ) . (a) Write down the solution as infinite series for Dirichlet boundary conditions. (b) Write down the solution as infinite series for the boundary conditions u x (0 ,t ) = 0 and u ( l, t ) = 0. 2. (Eigenvalue problem) Solve the eigenvalue problem x 2 f 00 ( x ) + 3 xf 0 ( x ) + λ f ( x ) = 0 . for 1 < x < e where e is the Euler number e = 2 . 71 .. . with f (1) = f ( e ) = 0. Assume that λ > 1 and look for solutions of the form f ( x ) = x m . Determine the eigenvalues and eigenfunctions. 3. (Waves in resistant medium) Waves in a resistant medium, i.e., with friction present, are described by the PDE u tt = c 2 u xx - r u t where x (0 ,l ) and r is a constant. (a) For r = 0, i.e., no friction, find the solution explicitly in series form for the boundary conditions u (0 ,t ) = 0 ,u x ( l,t ) = 0 and the initial conditions u ( x, 0) = x,u t ( x, 0) = 0. (b) For r = 0, i.e., no friction, find the solution explicitly in series form for the inhomogenous boundary conditions u (0 ,t ) = h, u ( l,t ) = k (where h, k are constants) and the initial conditions u ( x, 0) = 0 ,u t ( x, 0) = 0. (c) For 0 < r < 2 πc l write down a series solution that satisfies Dirichlet boundary conditions and the initial conditions u ( x, 0) = 0 , u t ( x, 0) = x ( l - x ) . What happens for t → ∞ . (d) For 0 < r < 2 πc l consider boundary conditions with a periodic source at one end, i.e., u (0 ,t ) = 0 and u ( l, t ) = A e iωt for a given ω . Show that the PDE and the BC are satisfied by v ( x,t ) = A e iωt sin ( βx ) sin ( βl ) , where β 2 c 2 = ω 2 - irω . (e) In (d) show that no matter what the initial condtions for u ( x,t ) are, v ( x,t ) is the asymptotic form of the solution for t → ∞ . 4. (Complex Fourier series) (a) Show that a complex valued function, periodic, C 1 function f ( x ) is real valued if and only if its complex Fourier coefficients satisfy c * n = c - n for all n Z where * denotes complex conjugation.
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(b) Find the complex Fourier series of the square wave, i.e., the following periodic function of period 2 l f ( x ) = ± 1 for 0 < x < l 0 for - l < x < 0 . Then draw a graph of the Fourier series over ( - 2 l, 2 l ). 5. (Euler type formulas) (a) Derive the Fourier sines series of the function f ( x ) = x on (0 ,l ). It is a fact that the series can be integrated term by term. Using integration find the Fourier cosine series of x 2 . (b) Set x = 0 and recover an interesting sum identity. 6. (Convergence) ( - 1) n x 2 n is a geometric series. (a) does it converge pointwise on ( - 1 , 1)? (b) does it converge uniformly on [ - 1 , 1]? (c) does it converge in the L 2 -sense on ( - 1 , 1)? 7. (Convergence of Fourier series) Let f ( x ) be a C 1 function of period 2 π with f (0) = 0, then g ( x ) = f ( x ) / ( e ix - 1) is a continuous function of period 2 π (why?), and Z π - π h | f ( x ) | 2 + | g ( x ) | 2 i dx < . Let C n be the coefficients of the full complex Fourier series of f ( x ), and D n the coefficients of g ( x ). The Riemann-Lebesgue theorem implies that C n 0 and D n 0 as | n | → ∞ . (a) Show C n = D n - D n - 1 and N n = - N c n 0 for N → ∞ . (b) Show that the value of the Fourier series at x = 0 is 0. 2
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Question 4(a)
If f (x) is real the f (x) = f (x) We have cn (f (x)) =
Then 1
2 1
−1 f (x)e−inπx dx, 1 f (x)einπx dx cn =
−1 1 f (x)einπx dx =
−1 1 f (x)e−i(−n)πx dx = c−n =
−1...

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