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Let 0 E1 0 A 0 E/ Be the matrix representation of the Hamiltonian for a...
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....
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...
(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). F...
(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...
2. (20 pts) Degenerate Perturbation Theory. A system with Hamiltonian H has two degenerate eigenstates l ψ )and lp : Ea...
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. We begin with a two state system with states labeled by |1) and [2). This...
1. We begin with a two state system with states labeled by |1) and [2). This may seem unphysical; however, there are many two state systems in quantum mechanics such spin 1/2 particles. The Hamiltonian we consider is (a) Compute the eigenvalues of H (b) Compute the eigenvectors of H, normalize them, and express them both as column vectors and in terms of | 1〉 and |2) (c) Denoting the two eigenvectors as lva) and |Vb), compute l/a) <>a and...
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...
13. Let W = {ī E R4 : Ai = 0} for some constant matrix A....
13. Let W = {ī E R4 : Ai = 0} for some constant matrix A. Suppose all solutions are 1 ES1 lo 1 +r , where t,s,r can be any real numbers. Let S = 0 1 'lo (a) (3 pts) What must the dimensions of the matrix A be? Justify briefly. (b) (8 pts) Show directly from the definition that S is a linearly independent set. (c) (6 pts) Without doing any further) computations, explain why S is...
3. Let B2 be the linear operator B2f (x):- f(0)2 2 (1f (1)2, which maps functions f defined at 0,...
With explanation!
3. Let B2 be the linear operator B2f (x):- f(0)2 2 (1f (1)2, which maps functions f defined at 0, 1 to the quadratic polynomials Pa. This is the Bernstein operator of degree 2, Let T = B21Py be the restriction of B2 to the quadratics. (a) Find the matrix representation of T with respect to the basis B = [1,2,2 (b) Find the matrix representation of T with respect to the basis C = (1-x)2, 22(1-2),X2]. (c)...
2 Two-level system Consider the time-dependent tion ihub = Hub Hamiltonian Schrödinger equa- for ...
2 Two-level system Consider the time-dependent tion ihub = Hub Hamiltonian Schrödinger equa- for a two-level system with a (13) Use the ansatz ψ-ee(t)e-iwt/21e) + cg(t)ewt/21g) (14) for the state a) Derive the (exact) differential equations for ce(t),cg(t) b) Use a Fourier-series ansatz, ce- en einwptan ,eg Ση einWptbn. Show that the equations hold. Find m (consider the case separately) wWp c) Find an iterative procedure to solve these equations to higher and higher accuracy. Calcu- late the leading order...
Let T be a linear operator on a finite dimensional vector space with a matrix representation...
Let T be a linear operator on a finite dimensional vector space with a matrix representation A = 1 1 0 0] 16 3 2 1-3 -1 0 a. (3 pts) Find the characteristic polynomial for A. b. (3 pts) Find the eigenvalues of A. C. (2 pts) Find the dimension of each eigenspace of A. d. (2 pts) Using part (c), explain why the operator T is diagonalizable. e. (3 pts) Find a matrix P and diagonal matrix D...
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 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...
(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...
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. We begin with a two state system with states labeled by |1) and [2). This may seem unphysical; however, there are many two state systems in quantum mechanics such spin 1/2 particles. The Hamiltonian we consider is (a) Compute the eigenvalues of H (b) Compute the eigenvectors of H, normalize them, and express them both as column vectors and in terms of | 1〉 and |2) (c) Denoting the two eigenvectors as lva) and |Vb), compute l/a) <>a and...
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...
13. Let W = {ī E R4 : Ai = 0} for some constant matrix A. Suppose all solutions are 1 ES1 lo 1 +r , where t,s,r can be any real numbers. Let S = 0 1 'lo (a) (3 pts) What must the dimensions of the matrix A be? Justify briefly. (b) (8 pts) Show directly from the definition that S is a linearly independent set. (c) (6 pts) Without doing any further) computations, explain why S is...
With explanation!
3. Let B2 be the linear operator B2f (x):- f(0)2 2 (1f (1)2, which maps functions f defined at 0, 1 to the quadratic polynomials Pa. This is the Bernstein operator of degree 2, Let T = B21Py be the restriction of B2 to the quadratics. (a) Find the matrix representation of T with respect to the basis B = [1,2,2 (b) Find the matrix representation of T with respect to the basis C = (1-x)2, 22(1-2),X2]. (c)...
2 Two-level system Consider the time-dependent tion ihub = Hub Hamiltonian Schrödinger equa- for a two-level system with a (13) Use the ansatz ψ-ee(t)e-iwt/21e) + cg(t)ewt/21g) (14) for the state a) Derive the (exact) differential equations for ce(t),cg(t) b) Use a Fourier-series ansatz, ce- en einwptan ,eg Ση einWptbn. Show that the equations hold. Find m (consider the case separately) wWp c) Find an iterative procedure to solve these equations to higher and higher accuracy. Calcu- late the leading order...
Let T be a linear operator on a finite dimensional vector space with a matrix representation A = 1 1 0 0] 16 3 2 1-3 -1 0 a. (3 pts) Find the characteristic polynomial for A. b. (3 pts) Find the eigenvalues of A. C. (2 pts) Find the dimension of each eigenspace of A. d. (2 pts) Using part (c), explain why the operator T is diagonalizable. e. (3 pts) Find a matrix P and diagonal matrix D...
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