(c) Consider any linear bounded operator B : H → H i. Show that B - † is anti-Hermitian and + B is Hermitian. ii. Show that B can be expressed as a linear combination of a Hermitian and an anti-Hermitian operator.
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- Let V = (xy, xy, (x−y)z). V is solenoidal. Use the homotopy or cone operator method to find a vector potential for it. Is your answer unique? If not, what is the most general vector potential for V?Under what conditions will a linear operator L̑ on a function space be Hermitian?(d) Show that the position operator (f = x) and the hamiltonian operator (H = -(n2/2m)d² /dx2 + V(x)) are hermitian. %3D
- WOw  is a Hermitian operator. lµ) is an eigenvector to Å with „cnvalue 1. [ø) is also an eigenvector with eigenvalue µ. Both |4) and lø) are normalized. µ # 1. Compute the following: a. ¡µ) = 4 > AY b. (plå = = 1 FOperty of Hermation Cvator ofe. Compute (9|Ã\µ} – (w\Â\µ) to show that |b) and |9) are orthogonal to each other. 入# MAPN>-0Two given vectors à & bo in the plane of a → modules alb. let 2= a + b Modulo of 2: is always less than a +b b- is always equal to atb C-less than or equal to atb1find the hamiltonian operator H. in a differential form
- In the operator eigenvalue equation, Af(x) =a f(x), which of the following statements is not true? the effect of the operator, A, on f(x) is to increase its magnitude by a factor of a Omultiples of f(x) would be eigenfunctions of the operator, A Of(x) is an eigenfunction of the operator, A the number, a, must be equal to 0 or 1 OOO OFor each of the following vector fields F , decide whether it is conservative or not by computing curl F . Type in a potential function f (that is, ∇f=F). F(x,y)=(−3siny)i+(10y−3xcosy)j