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Please finish this question with step-by-step details, thx! Consider one electron in a 1D box of...
8. Consider one electron in a 1D box of side L. Its wavefunction is given by...
8. Consider one electron in a 1D box of side L. Its wavefunction is given by V3 V3 2V3i where ф1(x), фг(x), and фз(x) are the first 3 eigenfunctions of the Hamiltonian, A, of a particle in a 1D box, h2 d2 a) Is Ψ(x) normalized? If it is not normalized it, normalize it! b) Is ų (x) an eigenfunction of A? If it is an eigenfunction, what is the eigenvalue?
8. Consider one electron in a 1D box of side L. Its wavefunction is given by...
8. Consider one electron in a 1D box of side L. Its wavefunction is given by из where ф1(x), фг(x), and фз(x) are the first 3 eigenfunctions of the Hamiltonian, H, of a particle in a 1D box, 2m dx2 a) Is Ψ(x) normalized? If it is not normalized it, normalize it! b) Is Ψ(x) an eigenfunction of A? If it is an eigenfunction, what is the 9. A linear polyene contains 8 -electrons, and absorbs light with412 nm. b)...
Problem 3: Time-Independent Perturbation Theory Consider the particle in a 1D box of size L, as...
Problem 3: Time-Independent Perturbation Theory Consider the particle in a 1D box of size L, as in Fig. 3. A perturbation of the form. V,δ ((x-2)2-a2) with a < L is applied to the unperturbed Hamiltonian of the 1D particle in a box (solutions on the equation sheet). Here V is a constant with units of energy. Remember the following propertics of the Dirac delta function m,f(x)6(x-a)dx f(a) 6(az) が(z) = = ds( dz E, or Ψ(x)-En 10 0.0 0.2...
Consider a particle with mass m described by the Hamilton operator for a one-dimensional harmonic oscillator...
Consider a particle with mass m described by the Hamilton operator for a one-dimensional harmonic oscillator 2 Zm 2 The normalized eigenfunctions for Hare φη (x) with energies E,,-(n + 2) ha. At time t-0 the wavefunction of the particle is given by у(x,0)- (V3іфі (x) + ф3(x)). Now let H' be an operator given by where k is a positive constant. 1) Show that H' is Hermitian. 2) Express H' by the step-operators a+ and a 3) Calculate the...
1. (50 points) Consider the particle in a one-dimensional box (0 s x S L). Assume a term is added to the Hamiltonian of the form: πχ V(x)g sin Sketch the potential and the expected eigenfunction (...
1. (50 points) Consider the particle in a one-dimensional box (0 s x S L). Assume a term is added to the Hamiltonian of the form: πχ V(x)g sin Sketch the potential and the expected eigenfunction (small g). In the limit of small g, find the second order correction to the ground state energy 2. (50 points) For a diatomic molecule rotating in free space, the Hamiltonian may be written: 12 21 Where L is the total angular momentum operator,...
Consider the finite difference matrix operator for the 1D model problem u(/d2- f(x) on domain [0,...
Consider the finite difference matrix operator for the 1D model problem u(/d2- f(x) on domain [0, 1] with boundary conditions u(0) = 0 and u(1) = 0, given by [-2 1 1-2 1 E RnXn h2 1 -2 1 This matrix can be considered a discrete version of the continuous operator d/da2 that acts upon a function(r). (a) Show that the n eigenvectors of A are given by the vectors ) (p-1,... , n) with components and with eigenvalues h2...
Question #2: 6 pts] Find the eigenvalues and the normalized eigenvectors of the matrix 21 2...
Question #2: 6 pts] Find the eigenvalues and the normalized eigenvectors of the matrix 21 2 -1 2 Question #3: 10 pts] The electron in a hydrogen atom is a linear combination of eigenstates. Let us assume a limited linear combination to provide some sample calculations $(r, θ, φ) 2 ,1,0,0 + '2,1,0 (a) Normalize the above equation. (b) What are the possible results of individual measurements of energy, angular momentum, and the z-component of angular momentum? (c) What are...
8. Consider one electron in a 1D box of side L. Its wavefunction is given by V3 V3 2V3i where ф1(x), фг(x), and фз(x) are the first 3 eigenfunctions of the Hamiltonian, A, of a particle in a 1D box, h2 d2 a) Is Ψ(x) normalized? If it is not normalized it, normalize it! b) Is ų (x) an eigenfunction of A? If it is an eigenfunction, what is the eigenvalue?
8. Consider one electron in a 1D box of side L. Its wavefunction is given by из where ф1(x), фг(x), and фз(x) are the first 3 eigenfunctions of the Hamiltonian, H, of a particle in a 1D box, 2m dx2 a) Is Ψ(x) normalized? If it is not normalized it, normalize it! b) Is Ψ(x) an eigenfunction of A? If it is an eigenfunction, what is the 9. A linear polyene contains 8 -electrons, and absorbs light with412 nm. b)...
Problem 3: Time-Independent Perturbation Theory Consider the particle in a 1D box of size L, as in Fig. 3. A perturbation of the form. V,δ ((x-2)2-a2) with a < L is applied to the unperturbed Hamiltonian of the 1D particle in a box (solutions on the equation sheet). Here V is a constant with units of energy. Remember the following propertics of the Dirac delta function m,f(x)6(x-a)dx f(a) 6(az) が(z) = = ds( dz E, or Ψ(x)-En 10 0.0 0.2...
Consider a particle with mass m described by the Hamilton operator for a one-dimensional harmonic oscillator 2 Zm 2 The normalized eigenfunctions for Hare φη (x) with energies E,,-(n + 2) ha. At time t-0 the wavefunction of the particle is given by у(x,0)- (V3іфі (x) + ф3(x)). Now let H' be an operator given by where k is a positive constant. 1) Show that H' is Hermitian. 2) Express H' by the step-operators a+ and a 3) Calculate the...
1. (50 points) Consider the particle in a one-dimensional box (0 s x S L). Assume a term is added to the Hamiltonian of the form: πχ V(x)g sin Sketch the potential and the expected eigenfunction (small g). In the limit of small g, find the second order correction to the ground state energy 2. (50 points) For a diatomic molecule rotating in free space, the Hamiltonian may be written: 12 21 Where L is the total angular momentum operator,...
Consider the finite difference matrix operator for the 1D model problem u(/d2- f(x) on domain [0, 1] with boundary conditions u(0) = 0 and u(1) = 0, given by [-2 1 1-2 1 E RnXn h2 1 -2 1 This matrix can be considered a discrete version of the continuous operator d/da2 that acts upon a function(r). (a) Show that the n eigenvectors of A are given by the vectors ) (p-1,... , n) with components and with eigenvalues h2...
Question #2: 6 pts] Find the eigenvalues and the normalized eigenvectors of the matrix 21 2 -1 2 Question #3: 10 pts] The electron in a hydrogen atom is a linear combination of eigenstates. Let us assume a limited linear combination to provide some sample calculations $(r, θ, φ) 2 ,1,0,0 + '2,1,0 (a) Normalize the above equation. (b) What are the possible results of individual measurements of energy, angular momentum, and the z-component of angular momentum? (c) What are...
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