Prove that the composite of orthogonal operators is orthogonal.
(Complex Analysis) Prove the following maximum principle for harmonic functions: Let u be harmonic in a bounded domain E and continuous in E ∪ dE. Then max(x, y)E E U dE u= sup(x, y)E E U dE u; min(x, y)E E U dE u= inf(x, y)E E U dE u. (Not the first E after the subscript (x, y) denotes element of, and the next on is the domain and the next is the derivative of the domain.)
For a harmonic oscillator confirm by explicit evaluation of the integral that the two wave functions of level 1 and 2, psi 1 and psi 2, are orthogonal (the variable x obeys: -∞ < x < +∞). Hint: Use the Hermite polynomials
3. Let (p) be a sequence of orthogonal functions on [a, b] having the property that the zero function is the only con- tinuous real-valued function f satisfying fo, dA ofor all nE N. Prove that the system (p.) is complete. (Hint: First use the hypothesis to prove that if fE P((a, b) satisfies fo, dA -0 for all n E N. then f = 0 a.e. Next use complteness of to prove that Parseval's equality holds for every fE...
Prove that u (x, y) is harmonic and find its conjugate harmonica (v (x, y)). Additionally graph both functions for different integration constants: 1)ular,y) = 2x(1 - y) 2)u(x,y) = 2.r - 3 + 3.xy? 3)(x, y) = sinhrsiny 4)u(x, y) = 72+y2
Spherical Harmonics The spherical harmonic,(on photo) is the solution that we have labelled the 3rd orbital for the hydrogen atom At what values of theta does this function have nodes? Are the nodes points, lines, planes, or other surfaces? Explain how you areived at your answer. The spherical harmonie, yr( (30sY-1) is the solution that we have labelled the 3d, orbital for the hydrogen atom for s) 1) At what values of 8 does this function have nodes? 2) Are...
Spherical harmonic at 1=1 and m=-1 is an eigen function for the hamiltonian of the rigid rotor
4) The wave functions of a one-dimensional harmonic oscillator for the states v = 0 and v = 1 are given by: V. (y) = Noe- 4; () = (47) 2ye and y = (Premu)/2 x Write the expression for the Hamiltonian eigenvalue equation for this system and show that yo satisfy the eigenvalue equation:
3. Let ū and ū be vectors. Prove that ū x ū is orthogonal to both ū and v.
*Prove that the orthogonal systems {sin nx)., and {ï, cos nxml are both complete on [0, T] 3.