7. Demonstrate that p -iha/ox is an Hermitian operator. Find the Hermitian conjugate of 7. Demonstrate that p -iha/ox is an Hermitian operator. Find the Hermitian conjugate of
An operator A is Hermitian if it satisfies | dx V AV = dr (AW)*v for all v. (a) Show that pl is not Hermitian, where I, k are positive integers. (Hint: First, show that p and are Hermitian. Then show that if A is Hermitian, A" is Hermitian. The remaining piece involves commutators of rand p.] (b) Show that the symmetric combination (c'p + x)/2 is Hermitian.
qm 09.3 3. An operator  is Hermitian if it satisfies the condition $ $(y) dx = (Ap) u dx, for any wavefunctions $(x) and y(x). (i) The time dependent Schrödinger equation is ih au = fu, at where the Hamiltonian operator is Hermitian. Show that the equation of mo- tion for the expectation value of any Hermitian operator  is given by d(A) IH, Â]), dt ħi i = where the operator  does not depend explicitly on time....
both pls 1) Which of the following operator(s) is/are Hermitian? a) id/dy? b) d/dy2 c) id/dy You may assume that the functions on which these operators operate are appropriately well behaved at infinity. (Hint #1: .. P dy = f. y pudy where the integral hudu = Uv - Sudv. Hint #2: Use y = e) 2) In each case below show (in the space provided directly) that F(y) is an eigen- function of the operator A and find the...
please answer question 1) to 3) fully step by step 7 marks (4 marks) (3 marks) (a) Is the operator P(v) y'+2ty linear? (Show workings) (b) Find the null space of the operator in (a) above. Question 2 5 marks Let P be a linear operator. Suppose that y is a particular solution to the equation Ply) = b. Prove that any solution to this equation can be written as y = Yo + yi for yo au clement of...
2. Schrodinger equation In quantum mechanics, physical quantities cor- respond to Hermitian operators. In particular, the total energy of the system corresponds to the Hamiltonian operator H, which is a hermitian operator The 'state of the system' is a time dependent vector in an inner product space, l(t)). The state of the system obeys the Schrodinger equation We assume that there are no time-varying external forces on the system, so that the Hamiltonian operator H is not itself time-dependent a)...
6. Let T: P, – P, be the linear operator defined as T(p(x)) = p(5x), and let B = {1,x,x?} be the standard basis for Pz. a.) (5 points) Find [7]s, the matrix for T relative to B.
OX 1 P(x) 0.3 0.2 0.15 0.35 2 3 Find the standard deviation of this probability distribution. Give your answer to at least 2 decimal places
7. For each linear operator T find a formula for T*. (a) Find T*(91: y2) for T(21, 12) = (2x1 + (1 + i).12, -21 +3112) on C. (b) Find T*(y1, y2) for T(11, 12) = (211 – 5i72,5in1 + 7x2) on C2
6. Let :P - P be the linear operator defined as (p(x)) - (5x), and let B = (1.x.x) be the standard basis for P2 a.) (5 points) Find the matrix for T relative to B. b.) (4 points) Let p(x) = x + 6x Determine (px)then find (p(x)) using (Tle from parta c.) (1 point) Check your answer to part b by evaluating T(x+6x) directly
Consider the linear operator, L. on Pdefined by L(P) = p(3)x3 + p(2)x2 + P(1)ą + p0). Find the matrix representation of L with respect to the standard basis of P {1, 2, 2, 23).