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If u(x, 0) = e-ax has a bell-shape (a Gaussian function), what is the shape of (x, t) = e-a(x-vt) ? I Already have gone through it with you in the book on page number 12, second column: Hecht 4" Edition. Look first at the Gaussian function and then keep continuing to the solution of other desired fuction given above in the same page.2. A commonly used trial function for the ground state of the hydrogen atom is a Gaussian function with a as the adjustable parameter: (20 marks) p(r) = e-arz a. Obtain an expression for the energy of as a function of a. b. Find the value of a that minimizes the energy and calculate a value for the energy- c. Compare this minimum energy with the exact value. What is the percent error? d. Do you consider this Gaussian function to be a reasonable approximation to the exact hydrogen Is atomic orbital? What is the difference between this Gaussian function and the exact hydrogen is function? Illustrate the difference with a computer-generated graph.2. A Gaussian function f(x) is given by the formula f(I) = Ae 107, where A is a constant, and the constant o, gives a measure of the spread or width of the shape. Find the Fourier transform f(k) of f(x) and show that the it is also a Gaussian function of the form f (k) = Be 3x Find the width ox and show that the produet a,ox - 1