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Fall 2011CHEM 760: Introductory Quantum ChemistryHomework 11a) Find the real and imaginary parts of the following quantities(2-i)3, e-2+iπ/2, and (2+2i)e-iπ/2b) Express the following complex numbers in the form reiθ4-2i, and π+ei2.If gx=Af(x), where g(x)and fx are functions and A is an operator, find g(x) for the system below. Does this system represent an eigensystem? If so, label the eigenfunction and the eigenvalue. A=d2dx2+1xddx, and fx=4x33.The Laplace transform operator L is defined as Lfx=0∞e-pxfxdx.Is this operator linear? Justify your answer.4. The operators A and B are defined as A=d2dx2, and B=x2. Use these definitions to prove AB-BA=2+4xddx.5.Writ out A2 for A=ddx+x. Hint: include f(x) before carrying out the operation.6.Determine whether the following functions are acceptable or not as wave functions over the indicated intervals.a) 1x0,∞, b) e-2xcosx 0, ∞, and c) ex-∞. ∞.7. Which of the following wave functions are normalized over the indicated two-dimensional intervals?a) e-x2+y22, 0≤x≤∞, 0≤y≤∞b)e-x+y2, 0≤x≤∞, 0≤y≤∞c)(4ab)1/2sinπxasinπyb, 0≤x≤a, 0≤y≤b Normalize those that aren’t.8.Why does * have to be everywhere real, nonnegative, finite and definite value?9.Consider the linear differential equationaxy"(x)+b(x)y'(x)+c(x)y(x)=0 where y"(x) and y'(x) are standard notation for d2y/dx2 and dy/dx, respectively. Show that if y1(x)and y2(x) are each solutions to the above differential equation, then so isyx=c1y1x+c2y2x where c1 and c2 are constants.10.Calculate the values of σE2=E2-E2 for a particle in a box in the state described byx=630a912x2(a-x)2 , 0≤x≤a11.Show that if A is Hermitian, then A-a is Hermitian.12. Given that f0(x)=a0 and f1x=a1+b1x, find the constants such that f0(x) and f1(x) are orthonormal over the interval 0≤x≤1. ................
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