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Q10 The Hamiltonian of a two-state system is given by H E ( i)- I02)(2 |...
Consider a particle which is confined to move along the positive x-axis, and that has a...
Consider a particle which is confined to move along the positive x-axis, and that has a Hamiltonian where is a positive real constant having the dimensions of energy. Find the normalized wave function that corresponds to an energy eigenvalue of . The function should be finite everywhere along the positive x-axis and be square integrable. H = 8
A particle with mass m is in a one-dimensional simple harmonic oscillator potential. At time t = 0 it is described by the state where lo and l) are normalised energy eigenfunctions corresponding to e...
A particle with mass m is in a one-dimensional simple harmonic oscillator potential. At time t = 0 it is described by the state where lo and l) are normalised energy eigenfunctions corresponding to energies E and Ey and b and c are real constants. (a) Find b and c so that (x) is as large as possible. b) Write down the wavefunction of this particle at a time t later c)Caleulate (x) for the particle at time t (d)...
(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...
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 system whose Hamiltonian H and an operator A are given by the matrices...
3) Consider a system whose Hamiltonian H and an operator A are given by the matrices 71 H = 60 -1 10 -1 1 0 0 0 -1) A = a 10 4 4 0 10 1 o) 1 0 where εo has the dimensions of energy. a) What are the possible values for the measurement of the energy? (3 marks) b) Suppose that the energy is measured, giving E = - Eo. What values are obtained if we subsequently...
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,...
(3)Consider an atomic p-electron (-1) which is governed by the Hamiltonian H-Ho +Hl,where Ho=a L,.bhand H,-./2...
(3)Consider an atomic p-electron (-1) which is governed by the Hamiltonian H-Ho +Hl,where Ho=a L,.bhand H,-./2 where a,bandcare nonzero real numbers with a 굶b. (a) Determine the Hamiltonian in Matrix form for a basis | I,m > with 1-land ,n = 0,±1. You may use the formula (b)Treat H,as a perturbation of Ho. What are the energy eigenvalues and eigenfunctions of the unperturbed problem? (c)Assume as>lcl and bsslcl. Use perturbation theory to calculate eigenvalues of H to first non trivial...
Let us consider the Hamiltonians a) Determine the eigenvalues of the Hamiltonian Htot H + H1 b) Let us take the solutio...
Let us consider the Hamiltonians a) Determine the eigenvalues of the Hamiltonian Htot H + H1 b) Let us take the solutions of H to be n) (note that these are not the solutions of H1 or Htot). Calculate the matrix elements (n' HiIn). Show that for the following matrix elements we can write (0|H110) a,(2H112), and (0 H 12)-(21H10)-α/v2. Determine c) Let us now consider the situation where w w. In this limit we can take as a trial...
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...
Consider a Hamiltonian ofthe form: H=H. +AR, where 2 4) 4 -1 0-E a) Calculate the energy eigenval...
quantum mechanics
Consider a Hamiltonian ofthe form: H=H. +AR, where 2 4) 4 -1 0-E a) Calculate the energy eigenvalues of H up to the second) order b) Determine the eigenstates ofH up to the first order in.- in λ..
Consider a Hamiltonian ofthe form: H=H. +AR, where 2 4) 4 -1 0-E a) Calculate the energy eigenvalues of H up to the second) order b) Determine the eigenstates ofH up to the first order in.- in λ..
A particle with mass m is in a one-dimensional simple harmonic oscillator potential. At time t = 0 it is described by the state where lo and l) are normalised energy eigenfunctions corresponding to energies E and Ey and b and c are real constants. (a) Find b and c so that (x) is as large as possible. b) Write down the wavefunction of this particle at a time t later c)Caleulate (x) for the particle at time t (d)...
(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...
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 system whose Hamiltonian H and an operator A are given by the matrices 71 H = 60 -1 10 -1 1 0 0 0 -1) A = a 10 4 4 0 10 1 o) 1 0 where εo has the dimensions of energy. a) What are the possible values for the measurement of the energy? (3 marks) b) Suppose that the energy is measured, giving E = - Eo. What values are obtained if we subsequently...
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,...
(3)Consider an atomic p-electron (-1) which is governed by the Hamiltonian H-Ho +Hl,where Ho=a L,.bhand H,-./2 where a,bandcare nonzero real numbers with a 굶b. (a) Determine the Hamiltonian in Matrix form for a basis | I,m > with 1-land ,n = 0,±1. You may use the formula (b)Treat H,as a perturbation of Ho. What are the energy eigenvalues and eigenfunctions of the unperturbed problem? (c)Assume as>lcl and bsslcl. Use perturbation theory to calculate eigenvalues of H to first non trivial...
Let us consider the Hamiltonians a) Determine the eigenvalues of the Hamiltonian Htot H + H1 b) Let us take the solutions of H to be n) (note that these are not the solutions of H1 or Htot). Calculate the matrix elements (n' HiIn). Show that for the following matrix elements we can write (0|H110) a,(2H112), and (0 H 12)-(21H10)-α/v2. Determine c) Let us now consider the situation where w w. In this limit we can take as a trial...
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...
quantum mechanics
Consider a Hamiltonian ofthe form: H=H. +AR, where 2 4) 4 -1 0-E a) Calculate the energy eigenvalues of H up to the second) order b) Determine the eigenstates ofH up to the first order in.- in λ..
Consider a Hamiltonian ofthe form: H=H. +AR, where 2 4) 4 -1 0-E a) Calculate the energy eigenvalues of H up to the second) order b) Determine the eigenstates ofH up to the first order in.- in λ..
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