Problem 9. For a system described by the Hamiltonian H = p²/2m + V(x), obtain an expression for d (p /2m) Idt. Discuss the relation of your result to the classical work-energy theorem.
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- Show the matrix representation of the operators a. position (x) b. momentum (p) for the one-dimensional harmonic oscillator problem. (Hint: Connect with operators A and At)Part bLagrangian Dynamics Ep = 0 A pendulum of length / and mass m is mounted on a block of mass M. The block can move freely without friction on a horizontal surface as shown in the adjacent figure H. 1. Find the velocity of mass m, w.r.t the origin O 2. Write the Lagrangian of the system 3. Derive the Euler Lagrange equations
- 5. Consider the two state system with basis |+) which diagonalizes the Pauli matrix 03. Generally the state of the system at time t can be written as |W(t)) = c+(t)|+) + c_(t)|-). (i) For the Hamiltonian of the system, first take H = functions c+(t) given the initial condition that at time t = 0 Eo03. Solve for the coefficient |W(0)) = |-).The Hamiltonian of an electron of mass m in a constant electric field E in one dimension can be written as Ĥ=+eEx where â and are the position and momentum operators, respectively. With initials conditions (t = 0) = 0 and p(t = 0) = 0, which one of the following gives (t) at time in the Heisenberg picture? You may use the commutator [â,p] = iħ. O a. O b. eEt2 2m O C. e Et O d. -eEt O e. eEt² m pt mThis is question 8.8 in John R. Taylor's "Classical Mechanics" textbook by the way! (ISBN: 9781891389221)
- The eigenfunctions of Hermitian Operators are orthonormal (orthogonal and normalizable). a. Prove the eigenfunctions of the Hamiltonian Operator for a particle in a box that extends from x = 0 to x = a: плх y.(x) = are orthonormal by integrating a pair of functions, y,(x) and y.(x), with n = m in one case and n m in another. b. For the ground state of a particle in a box, use the momentum and position operators to show that the expectation values are 0 for momentum and - for position, by evaluating the resulting integrals. c. Calculate the uncertainty in x and p (0x and Op) for the lowest energy state of a particle in a box by taking the standard deviation of x and p as a measure of their uncertainty: Ar =0, = (x*)-(x)* and Ap=o,=Kp*)-(p)* Is the product you obtain for OxOp consistent with the Heisenberg Uncertainty Principle?The dynamics of a particle moving one-dimensionally in a potential V (x) is governed by the Hamiltonian Ho = p²/2m + V (x), where p = is the momentuin operator. Let E, n = of Ho. Now consider a new Hamiltonian H given parameter. Given A, m and E, find the eigenvalues of H. -ih d/dx 1, 2, 3, ... , be the eigenvalues Ho + Ap/m, where A is a %3|Consider the question of finding the points on the curve xy² = 2 closest to the origin. (a) State what function is being minimized for this problem and what the constraint is. Label each. (b) Use Lagrange multipliers to find a system of equations for finding the closest point. Write this system of equations without any vectors. Include the constraint as one of the equations. (c) Solve the system of equations from part (b) to find the points closest on the curve xy² = 2 closest to the origin.