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3) Consider a system whose Hamiltonian H and an operator A are given by the matrices...
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...
Exercise 1: Consider a physical system whose state space, which is three-dimensional is spanned b...
Exercise 1: Consider a physical system whose state space, which is three-dimensional is spanned by the orthonormal basis formed by three kets |ф11ф2) and IP2). I- In this basis, the Hamiltonian operator H of the system and the observable A are written as: H- ho 0 2 0 A h0 01 where o is real constant And the state ofthe system att-os: ΙΨ(0))siip)+1P2》怡1%) 1- Calculate the commutator [H. A] 2- Determine the energies of the system. 3- Determine the eigen-values...
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. Consider a rigid rotor whose Hamiltonian is given by H L2(21) where L is the angular momentum operator and I is the...
3. Consider a rigid rotor whose Hamiltonian is given by H L2(21) where L is the angular momentum operator and I is the moment of inertia of the rotator. Its rotation is described by a wave function: (0, N{Yo0(0,6)(1 3i) Y1-1(0,6) 2 Y21(0.0) Y20(0.) Find the normalization constant, N. (i) Find the probability to occupy state Yo0- (ii Find the expectation value of L2 of this state (iii Find the expectation value of L2 of this state (iv) Find (L2L2/21...
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.
Q10 The Hamiltonian of a two-state system is given by H E ( i)- I02)(2 |...
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...
3. Ehrenfest's Theorem states that dA i for an observable A and a time-independent Hamiltonian H....
3. Ehrenfest's Theorem states that dA i for an observable A and a time-independent Hamiltonian H. Consider a QM system in 1D with time-independent Hamiltonian H 2 V(x). Use Ehrenfest's Theorem to determine Jlx) and p). What is 듦(z)? 4. A projection operator Pn is defined by where there is no summation Ση implied, and the states n> are a complete set of oth onormal states (a) Show that Pa satisfies 2 (b) Show that Pn acting on an arbitrary...
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...
Exercisel: Consider a physical system whose state space, which is three-dimensional is spanned by...
Exercisel: Consider a physical system whose state space, which is three-dimensional is spanned by the orthonormal basis formed by three kets lu, lu2) and lu). 1- In this basis, the Hamltonian operator H of the system and the observable A are written as H-h 1 0 0A where w is real constant. And the state of the system at tu0 is: 19(0)--lu:) + luz) + lus) 1- Calculate the commutator [H, A]. 2- Determine (H)s(Y(0)[H1Ψ(0) 3- Calculate ΔH,[H-hy-VIP-R2 = ((H2)-(HPF...
Question A2: Coherent states of the harmonic oscillator Consider a one-dimensional harmonic oscillator with the Hamiltonian...
Question A2: Coherent states of the harmonic oscillator Consider a one-dimensional harmonic oscillator with the Hamiltonian 12 12 m2 H = -2m d. 2+ 2 Here m and w are the mass and frequency, respectively. Consider a time-dependent wave function of the form <(x,t) = C'exp (-a(x – 9(t)+ ik(t)z +io(t)), where a and C are positive constants, and g(t), k(t), and o(t) are real functions of time t. 1. Express C in terms of a. [2 marks] 2. By...
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...
Exercise 1: Consider a physical system whose state space, which is three-dimensional is spanned by the orthonormal basis formed by three kets |ф11ф2) and IP2). I- In this basis, the Hamiltonian operator H of the system and the observable A are written as: H- ho 0 2 0 A h0 01 where o is real constant And the state ofthe system att-os: ΙΨ(0))siip)+1P2》怡1%) 1- Calculate the commutator [H. A] 2- Determine the energies of the system. 3- Determine the eigen-values...
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. Consider a rigid rotor whose Hamiltonian is given by H L2(21) where L is the angular momentum operator and I is the moment of inertia of the rotator. Its rotation is described by a wave function: (0, N{Yo0(0,6)(1 3i) Y1-1(0,6) 2 Y21(0.0) Y20(0.) Find the normalization constant, N. (i) Find the probability to occupy state Yo0- (ii Find the expectation value of L2 of this state (iii Find the expectation value of L2 of this state (iv) Find (L2L2/21...
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.
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...
3. Ehrenfest's Theorem states that dA i for an observable A and a time-independent Hamiltonian H. Consider a QM system in 1D with time-independent Hamiltonian H 2 V(x). Use Ehrenfest's Theorem to determine Jlx) and p). What is 듦(z)? 4. A projection operator Pn is defined by where there is no summation Ση implied, and the states n> are a complete set of oth onormal states (a) Show that Pa satisfies 2 (b) Show that Pn acting on an arbitrary...
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...
Exercisel: Consider a physical system whose state space, which is three-dimensional is spanned by the orthonormal basis formed by three kets lu, lu2) and lu). 1- In this basis, the Hamltonian operator H of the system and the observable A are written as H-h 1 0 0A where w is real constant. And the state of the system at tu0 is: 19(0)--lu:) + luz) + lus) 1- Calculate the commutator [H, A]. 2- Determine (H)s(Y(0)[H1Ψ(0) 3- Calculate ΔH,[H-hy-VIP-R2 = ((H2)-(HPF...
Question A2: Coherent states of the harmonic oscillator Consider a one-dimensional harmonic oscillator with the Hamiltonian 12 12 m2 H = -2m d. 2+ 2 Here m and w are the mass and frequency, respectively. Consider a time-dependent wave function of the form <(x,t) = C'exp (-a(x – 9(t)+ ik(t)z +io(t)), where a and C are positive constants, and g(t), k(t), and o(t) are real functions of time t. 1. Express C in terms of a. [2 marks] 2. By...
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