**Exercise 4:** Determine the commutation relation: \([ \hat{L}_z, \hat{x} ]\)- **Explanation:** - This problem involves calculating the commutation relation between the \(z\)-component of the angular momentum operator \(\hat{L}_z\) and the position operator \(\hat{x}\). - In quantum mechanics, commutation relations are useful for understanding how two operators interact with each other. - This exercise is essential for students learning about the fundamental principles of quantum mechanics.

Answered: Image /qna-images/answer/42bfe724-ca48-4059-9bd4-d149a8f31307.jpg

Answered: Work out the commutation relation:

Similar questions

Show explicitly that the alpha matrices are Hermitian. Do this in Dirac-Pauli representation.
Determine the general solution of the 1-dimensional Laplace equation on the cylinder coordinates and ball coordinates!
Consider the following operators on a Hilbert space V³ (C): 0-i 0 ABAR-G , Ly i 0-i , Liz 00 √2 0 i 0 LE √2 010 101 010 What are the corresponding eigenstates of L₂? 10 00 0 0 -1 What are the normalized eigenstates and eigenvalues of L₂ in the L₂ basis?
Using the eigenvectors of the quantum harmonic oscillator Hamiltonian, i.e., n), find the matrix element (6|X² P|7).
(a) Using Dirac notation, write down the definition of a projection operator and that of a density operate and state the differences between the two.
It sometimes occurs that the generalized coordinates appear separatelyin the kinetic energy and the potential energy in such a manner that T andV maybe written in the form T =∑i fi(qi) ˙qi2 and V = ∑i Vi(qi) Show that Lagrange’s equations then separate, and that the problem canalways be reduced to quadratures.
Suppose that we want to solve Laplace’s equation inside a hollow rectangular box, with sides of length a, b and c in the x, y and z directions, respectively. Let us set up the axes so that the origin is at one corner of the box, so that the faces are located at x = 0 and x = a; at y = 0 and y = b; and at z = 0 and z = c. Suppose that the faces are all held at zero potential, except for the face atz=c,onwhichthepotentialisspecifiedtobeV(x,y,c)=V0 =const. a) Find the electrostatic potential V at a generic point inside the box.b) Find the expression for the electrostatic potential evaluated at the center of the box, i.e. deter- mine V (a/2, b/2, c/2). Simplify your answer as much as you can! c) Suppose now that a = b = c, i.e. the box is a cube. Give a simple argument which gives theexact (and simple) expression for the potential at the center of the cube. (No calculations are asked here. Use physics, wave your hands, etc. and say “the answer is such and such because ...”)
The Hamiltonian matrix has been constructed using an orthonormal basis. (1 1 0V (1 0 1) A = (2 1 0 )+(0 2 2 \2 1 4 where H = Hº + V and cis a constant. 1 2 0/ b) Use time-independent perturbation theory to determine the eigenvalues with corrections up to second order.
Provide a written answer
Prove that adding a constant to the Lagrangian L or else multiplying the Lagrangian by a constant produces a new Lagrangian L′ that is physically equivalent to L. (Physically equivalent means that the Euler-Lagrange equations for the q(t) remain the same under this change of Lagrangian).
Evaluate the commutator [Â,B̂] of the following operators.