The Hamiltonian of a system in the basis In > is given by H = hw("...
(introduction to quantum mechanics) , the Hamiltonian matrix is H- 3. In the basis |1) - (a) Find the eigenvalues En and eigenfunctions Ion) of H. (b) The system is in state 2) initially (t 0). Find the state of the system at t in the basis n). (c) Calculate the expectation value of H. Briefly explain your result. Does it depend on time? Why? , the Hamiltonian matrix is H- 3. In the basis |1) - (a) Find the...
Consider a two-dimensional state vector space and a basis in this space lay), laz), eigenvectors of an observable A: Ala) = aja) Alaz) = azlaz) A representation of Hamiltonian operator in this basis is: H = (8 5) Find: -Eigenstates and eigenvalues of H. -If the system is in state |az) at time t=0, What is the state vector of the system at time t? -What is the probability of finding the system in the state |az) at time t?...
Q10 The Hamiltonian of a two-state system is given by H E ( i)- I02)(2 | -i | ¢1)(2 | +i | ¢2) (¢1 1) where , p2) form a complete and orthonormal basis; E is a real constant having the dimensions of energy (a) Is H Hermitian? Calculate the trace of H (b) Find the matrix representing H in the | øı), | 42) basis and calculate the eigenvalues and the eigenvectors of the matrix. Calculate the trace of...
Consider a three-level system where the Hamiltonian and observable A are given by the matrix Aˆ = µ 0 1 0 1 0 1 0 1 0 Hˆ = ¯hω 1 0 0 0 1 0 0 0 1 (a) What are the possible values obtained in a measurement of A (b) Does a state exist in which both the results of a measurement of energy E and observable A can be...
1. (25 points) The Hamiltonian operator H, for a particular molecule has a complete set of orthonormal eigenfunctions on (where n = 1,2,3,...) with corresponding eigenvalues (n-1)h. The molecule in state n is subject to measurement of the dipole moment, for which the mathematical operator is represented by M. After the measurement of the dipole moment, the wave function of the particle is: Y = 0.5 0.1 +0.7071 Q. + 0.5 Pn+1 a) Show that is normalized. Now consider a...
Problem 7.49 Problem 7.49 A hydrogen atom is placed in a uniform magnetic field Bo Bo (the Hamiltonian can be written as in Equation 4.230). Use the Feynman-Hellman theorem (Problem 7.38) to show that a En (7.114) where the electron's magnetic dipole moment10 (orbital plus spin) is Yo l-mechanical + γ S . μ The mechanical angular momentum is defined in Equation 4.231 a volume V and at 0 K (when they're all in the ground state) is41 Note: From...
Problem 8.3 - A New Two-State System Consider a new two-level system with a Hamiltonian given by i = Ti 1461 – 12) (2) (3) Also consider an observable represented by the operator Ŝ = * 11/21 - *12/11: It should (hopefully) be clear that 1) and 2) are eigenkets of the Hamiltonian. Let $1) be an eigenket of S corresponding to the smaller eigenvalue of S and let S2) be an eigenket of S corresponding to the larger eigenvalue....
Consider the dimensionless harmonic oscillator Hamiltonian, (where m = h̄ = 1). Consider the orthogonal wave functions and , which are eigenfunctions of H with eigenvalues 1/2 and 5/2, respectively. with p=_ïda 2 2 We were unable to transcribe this imageY;(r) = (1-2x2)e-r2/2 (a) Let фо(x-AgVo(x) and φ2(x) = A2V2(x) and suppose that φ。(x) and φ2(x) are normalized. Find the constants Ao and A2. (b) Suppose that, at timet0, the state of the oscillator is given by Find the constant...
2. (20 pts) Degenerate Perturbation Theory. A system with Hamiltonian H has two degenerate eigenstates l ψ )and lp : Ea h petturbationHi-h E, :}lifts the degeneracy. The matrix given is in the basis Ambatas nlist 0 Eindthe "good" states, the two eigenstates Ιψ%)-α ws> +Pr IOS)ofHL and the sorresponding eigenvalues AEF which resolve the degeneracy 2. (20 pts) Degenerate Perturbation Theory. A system with Hamiltonian H has two degenerate eigenstates l ψ )and lp : Ea h petturbationHi-h E,...
1) Compute (Stot) and (Stt tot 2 2) Show that: Stot 2 in the basis I.,W , Ih,%) , 11., , 11.,%)} 3) Suppose that the Hamiltonian for two interacting spins is given by for some interaction strength α. Express the Hamiltonian in the operator, U, in the same basis. basis, and calculate the time evolution 4) If a system begins in the state loo〉-W,%), calculate and then plot the probability for the system to be in the initial state...