(a) Define the commutator of the operators and Ω2. (2 marks) (b) Determine the commutators of the...
Derive the following commutators: [x, P_x] [x, P_y] [P_x, P_x] [x, T_x] [P_x, T_x] Can both the position and momentum of the particle in the same direction be known precisely? Can both the position and kinetic energy of the particle in the same direction be known precisely? Determine whether the following operators are Hermitian Operators. You may assume that the functions on which these operators operate behave properly in the limit of plusminus x rightarrow infinity. [see McQuarrie section 4-5]...
b) Explain the function of commutator switch in a DC motor. (2 marks)
2. (9 points total) Uncertainty relations. a) (1 point) Compute the commutator of the operators of coordinate and momentum in one dimension. b) (1 point) Two Hermitian operators A and B satisfy the relation [A, B] = iſ, where I is a number. Prove that I' is real. c) (1 point) Give the definition of the uncertainties A A and A B. d) (2 points) In this and subsequent parts of the question, we consider a normalized quantum stately) with...
(2.) Consider the orbital angular momentum operator defined in terms of the position and momentum operators as p. Define the angular momentum raising and lowering operators as L± = LztiLy. Use the commutation relations for the position and m omentum operators and find the commutators for: (a.) Lx, Lz and Ly, Lz; (b.) L2, Lz; (c.) L+,L
(a) (6 marks) Determine the values of A, B and C such that 1 x(x
2 + 1) = A x + Bx + C x 2 + 1 . Hence evaluate Z 1 x(x 2 + 1)dx.
(b) (6 marks) Find the area under the curve y = xe−x for x ≥ 0.
3. (a) (6 marks) Determine the values of A, B and C such that 1 А + Bir + 12+1 (22+1) 1 Hence evaluate dr. (2+1)...
Exercise 2:Commutators Given (AB, C) - ABC - ACB + ACB-CAB - ABC] + [A, CJB. 1- Show that the commutator[L.L]is equal to zero. 2-Computethecommutator(Ly, Lx]. 3-Compute the commutator 3.Lx]. 4-Compute the commutator[LLY.Lx]. 5- Add the results from (2)-(4) to compute [LLyLx]. Lxercise 3: Matrix elements The angular momentum operators acting on the angular momentum eigenstates. Il determined by L-11.m) = (1 + 1) - m(m + 1)2.m +1) L_11.m) = /(1+1) - mm - 1)/1.m-1) L211,m) = hm|1,m) 1-...
a) Discuss why the de Broglie wavelength λ corresponding to a momentum p (p wavenumber given by k # 2n/A) leads to a representation of p by the operator p as (h/) (d/dx) hk, where k is the b) Using theoperao orm of p given in part a, show that,pih c) The total energy of a simple harmonic oscillator of mass M and spring constant K can be written as H- p2/M + ke . If the mass is displaced...
Linearity is an important concept to be familiar with. For the following operators, L.), determine if they are linear or nonlinear a. L (x) = ax2 + bx b. L (x) = Vax C. L = m + ad, so that L(f)= *+af d. L (f) = $x f(x)dx
(4) The Pauli spin matrices are a set of 3 complex 2 x 2 matrices that are used in quantum mechanics to take into account the interaction of the spin of a particle with an external electromagnetic field. σ2 10), (a) Find the eigenvalues and corresponding eigenvectors for all three Pauli spin matrices. Show all of vour work (b) In Linear Algebra, two matrices A and B are said to commute if AB BA and their commutator defined as [A,...
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2h Question 20 marks Soalan 2 [10 markah) Harmonic oscillator problem can be solved based on operator and algebraic method where operators are given as a = (x + and a = (x 2) (a, and a is raising and lowering operator respectively) a) Prove that the commutator (a.a.) = 1. (Hints: Use commutator identities, lx.pl = ih and (x,x) = p.pl=0) [3 marks) b) Show that the...