The hamiltonian of a perturbed system it's given by:
a ) Find 
b ) Diagonalize the Hamiltonian 
, find 
(for the complete Hamiltonian) and expand to second order
c ) Find 
and 
and compere with subsection (b)
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The hamiltonian of a perturbed system it's given by:
a ) Find
b ) Diagonalize the...
For the Hamiltonian syste m we did in class: 2. 3 Ic (1) Show that it's a Hamiltonian system with a Hamiltonian function (2) Show that for each c > 0, {(z,y) є R2 . H(z,y) c} is a bounded inva...
For the Hamiltonian syste m we did in class: 2. 3 Ic (1) Show that it's a Hamiltonian system with a Hamiltonian function (2) Show that for each c > 0, {(z,y) є R2 . H(z,y) c} is a bounded invariant set of the dynamical system (in fact, it's also closed) (3) Find all the equilibria of this system. Show that H-() is made up of one equilibium point and two homoclinic orbits attached to it. (4) Sketch the invariant...
4. (30 points) Harmonic oscillator with perturbation Recall the Hamiltonian of an harmonic oscillator in 1D:...
4. (30 points) Harmonic oscillator with perturbation Recall the Hamiltonian of an harmonic oscillator in 1D: p21 ÃO = + mwf?, where m is the mass of the particle and w is the angular frequency. Now, let us perturb the oscillator with a quadratic potential. The perturbation is given by Î' = zgmw?h?, where g is a dimensionless constant and g <1. (a) Write down the eigen-energies of the unperturbed Hamiltonian. (b) In Lecture 3, we introduced the lowering (or...
3. Consider a system whose Hamiltonian H, admits two eigenstates y, (with eigenvalues F) and v,...
3. Consider a system whose Hamiltonian H, admits two eigenstates y, (with eigenvalues F) and v, (with eigenvalues E,). Assume E, E, and they are () orthogonal, (ifi) normalized and (ii) non-degenerate. After the perturbation is on the diagonal matrix elements become zero ie, <4, l H'l Ψ)-(4, I H'ly,)-0, while the off diagonal equals to a constant value ie. (v, l H'l%)-(wil H'ly)-c Using the 2nd order perturbation theory evaluate the energy of the perturbed system.
Diagonalize the following matrix. The real eigenvalues are given to the right of the matrix. 2...
Diagonalize the following matrix. The real eigenvalues are given to the right of the matrix. 2 2 -4 - 1 5 -4 ; 2 = 3,8 -2 7 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. 3 0 0 For P = D= 0 3 0 0 0 8 (Simplify your answer.) B. 3 00 For P = D = 0 8 0 0 0 8 (Simplify your answer.)...
Consider a three-level system where the Hamiltonian and observable A are given by the matrix Aˆ...
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...
Consider a quantum mechanical system with 4 states and an unperturbed Hamiltonian given by 1 0 0 0 Ho E0 0 2 0 a small perturbation is added to this Hamiltonian 0 0 1 0 where e is much smaller than E...
Consider a quantum mechanical system with 4 states and an unperturbed Hamiltonian given by 1 0 0 0 Ho E0 0 2 0 a small perturbation is added to this Hamiltonian 0 0 1 0 where e is much smaller than E a) [10pts] What are the energy eigenvalues of the unperturbed system of the following states? 1 o 2o 0 and which energy levels are degenerate? b) [10pts Find a good basis for degenerate perturbation theory instead of c)...
The Hamiltonian of a system in the basis In > is given by H = hw("...
The Hamiltonian of a system in the basis In > is given by H = hw(" >< 0,1 + il" >< 421-142 >< 0,1 -21°3 >< $3D Here w is a constant. Write the Hamiltonian in the form of a matrix and obtain its eigenvalues and eigenfunctions. Express the eigenfunctions in terms of the basis In > and in its eigenvalues as En = hwe If the system is initially in the state | (0) >= 10 > a. What...
Problem 8.3 - A New Two-State System Consider a new two-level system with a Hamiltonian given...
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....
(3) Take the case of 3 states ji> i-1,2, 3 which are eigenfunctions of H with...
(3) Take the case of 3 states ji> i-1,2, 3 which are eigenfunctions of H with degenerate eigenenergies, represented in matrix form. 10 0 0 H(0) 10 10 01,Ιψο > 12 A perturbation Hamiltonian is applied to the original system H 0 0 10 of matrix 3 0 0] a-1 Fl("-Volo a , with the off-diagonal elements given by a and numerically 1 (a) Calculate the energies of the three states to first-order perturbation theory represented by H'- HHby adding...
8. Consider the system where the performance of the system is given by V -d a. Find the state function of Pontryagin H. b. Using the state function determine the optimal input c. Determine the eq...
8. Consider the system where the performance of the system is given by V -d a. Find the state function of Pontryagin H. b. Using the state function determine the optimal input c. Determine the equations governing λ and x. d. Determine u°()given x(0)-0 and x(1) I
8. Consider the system where the performance of the system is given by V -d a. Find the state function of Pontryagin H. b. Using the state function determine the optimal input c....
For the Hamiltonian syste m we did in class: 2. 3 Ic (1) Show that it's a Hamiltonian system with a Hamiltonian function (2) Show that for each c > 0, {(z,y) є R2 . H(z,y) c} is a bounded invariant set of the dynamical system (in fact, it's also closed) (3) Find all the equilibria of this system. Show that H-() is made up of one equilibium point and two homoclinic orbits attached to it. (4) Sketch the invariant...
4. (30 points) Harmonic oscillator with perturbation Recall the Hamiltonian of an harmonic oscillator in 1D: p21 ÃO = + mwf?, where m is the mass of the particle and w is the angular frequency. Now, let us perturb the oscillator with a quadratic potential. The perturbation is given by Î' = zgmw?h?, where g is a dimensionless constant and g <1. (a) Write down the eigen-energies of the unperturbed Hamiltonian. (b) In Lecture 3, we introduced the lowering (or...
3. Consider a system whose Hamiltonian H, admits two eigenstates y, (with eigenvalues F) and v, (with eigenvalues E,). Assume E, E, and they are () orthogonal, (ifi) normalized and (ii) non-degenerate. After the perturbation is on the diagonal matrix elements become zero ie, <4, l H'l Ψ)-(4, I H'ly,)-0, while the off diagonal equals to a constant value ie. (v, l H'l%)-(wil H'ly)-c Using the 2nd order perturbation theory evaluate the energy of the perturbed system.
Diagonalize the following matrix. The real eigenvalues are given to the right of the matrix. 2 2 -4 - 1 5 -4 ; 2 = 3,8 -2 7 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. 3 0 0 For P = D= 0 3 0 0 0 8 (Simplify your answer.) B. 3 00 For P = D = 0 8 0 0 0 8 (Simplify your answer.)...
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...
Consider a quantum mechanical system with 4 states and an unperturbed Hamiltonian given by 1 0 0 0 Ho E0 0 2 0 a small perturbation is added to this Hamiltonian 0 0 1 0 where e is much smaller than E a) [10pts] What are the energy eigenvalues of the unperturbed system of the following states? 1 o 2o 0 and which energy levels are degenerate? b) [10pts Find a good basis for degenerate perturbation theory instead of c)...
The Hamiltonian of a system in the basis In > is given by H = hw(" >< 0,1 + il" >< 421-142 >< 0,1 -21°3 >< $3D Here w is a constant. Write the Hamiltonian in the form of a matrix and obtain its eigenvalues and eigenfunctions. Express the eigenfunctions in terms of the basis In > and in its eigenvalues as En = hwe If the system is initially in the state | (0) >= 10 > a. What...
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....
(3) Take the case of 3 states ji> i-1,2, 3 which are eigenfunctions of H with degenerate eigenenergies, represented in matrix form. 10 0 0 H(0) 10 10 01,Ιψο > 12 A perturbation Hamiltonian is applied to the original system H 0 0 10 of matrix 3 0 0] a-1 Fl("-Volo a , with the off-diagonal elements given by a and numerically 1 (a) Calculate the energies of the three states to first-order perturbation theory represented by H'- HHby adding...
8. Consider the system where the performance of the system is given by V -d a. Find the state function of Pontryagin H. b. Using the state function determine the optimal input c. Determine the equations governing λ and x. d. Determine u°()given x(0)-0 and x(1) I
8. Consider the system where the performance of the system is given by V -d a. Find the state function of Pontryagin H. b. Using the state function determine the optimal input c....
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