Question Three a) The Chebyshev polynomials of the first kind can be obtained from the recurrence...

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Question Three a) The Chebyshev polynomials of the first kind can be obtained from the recurrence relation Tn+1(x) = 2xTn (x) - Tn-1(x) with To(x) = 1 and T₁(x) = x. i) Show that any two Chebyshev polynomials are orthogonal with respect to the weighting factor (12²) in the closed interval [-1, 1]. [6] ii) Pick the first three Chebyshev polynomials and use Gram-Schmidt orthonormalization procedure to form an orthonormal set in the closed interval [-1, 1]. b) Given an eigenvector (or wave function) n(x) = √√2/L sin(nx/L)in the interval [0, 1], find the position-momentum uncertainty relation (Ar)n (Ap)n and plot it as function of n. [8] c) Give a statement of Heisenberg uncertainty principle and out line four examples where it is demonstrated in everyday life. [7] Question Four k 7L Question Three a) The Chebyshev polynomials of the first kind can be obtained from the recurrence relation Tn+1(x) = 2xTn (x) - Tn-1(x) with To(x) = 1 and T₁(x) = x. i) Show that any two Chebyshev polynomials are orthogonal with respect to the weighting factor (12²) in the closed interval [-1, 1]. [6] ii) Pick the first three Chebyshev polynomials and use Gram-Schmidt orthonormalization procedure to form an orthonormal set in the closed interval [-1, 1]. b) Given an eigenvector (or wave function) n(x) = √√2/L sin(nx/L)in the interval [0, 1], find the position-momentum uncertainty relation (Ar)n (Ap)n and plot it as function of n. [8] c) Give a statement of Heisenberg uncertainty principle and out line four examples where it is demonstrated in everyday life. [7] Question Four k 7L

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