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4. Write the electronic Hamiltonian operator for H, under the Born-Oppenheimer approximation. Show that, once the...
52. Write out the full Hamiltonian for a Li atom. You may assume the Born-Oppenheimer approximation holds. Identify eac...
52. Write out the full Hamiltonian for a Li atom. You may assume the Born-Oppenheimer approximation holds. Identify each of the terms (e.g. kinetic energy of electron 1, potential energy for nuclear-electron 1 attraction).
The key idea in the Born-Oppenheimer Approximation (BOA) is that the electronic energy depends on...
The key idea in the Born-Oppenheimer Approximation (BOA) is that the electronic energy depends on the geometry. Explain this, and include a drawing of how the electronic energy (energy of ) depends on a geometry coordinate, R. Example, how does the electronic energy of O2 depend on the O-O bond length? e(r) e(r)
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...
Part c and h please Help (a) Describe the essence of the orbital approximation. 3 pts...
Part c and h please
Help (a) Describe the essence of the orbital approximation. 3 pts (b) Suggest antisymmetrized wave functions of the Helium atom in the singlet (1s)2 ground state, and the (c) Normalize the (1s) wave function of (b), provided that individual space orbitals and spin functions are (d) Explain the energy ordering and degeneracy of the lowest three singlet and triplet states of the Helium singlet (1s) (2s) and triplet (1s)(2s) excited-state configurations in the orbital approximation....
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. 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...
Consider an H2+ ion. In the figure below, HẠ, HB, and e represent the two nuclei...
Consider an H2+ ion. In the figure below, HẠ, HB, and e represent the two nuclei and the electron, respectively, and all the relevant distances are defined. You must use atomic units for this problem. (a) (4 points) Write down the Born-Oppenheimer Hamiltonian for the H2+ ion. R HA H (b) (6 points) The anti-bonding molecular orbital is expressed as V = N_(ls (r) – 1s (r)), where ls, (r) and 1s; (r) are hydrogen Is orbitals centered at nuclei...
is A system with an unper tubed Hamiltonian H Subjected to a perturbation HC Mon ,...
is A system with an unper tubed Hamiltonian H Subjected to a perturbation HC Mon , 3 . • Il (1 1 . i eil 2 0 ; H = da o vz; 0 0 -2 0 II 131 l-ai vei o 2, where a is a real Constant with The dimesions of an energy anda is a dimension less real Constant. as show that the following vectors are the eigen vectors of M" and determine their associated eigenvalues. 14>*3)...
In this optional assignment you will find the eigenfunctions and eigenenergies of the hydrogen atom using an operator method which involves using Supersymmetric Quantum Mechanics (SUSY QM). In t...
In this optional assignment you will find the eigenfunctions and eigenenergies of the hydrogen atom using an operator method which involves using Supersymmetric Quantum Mechanics (SUSY QM). In the SUSY QM formalism, any smooth potential Vx) (or equivalently Vr)) can be rewritten in terms of a superpotential Wix)l (Based upon lecture notes for 8.05 Quantum Krishna Rajagopal at MIT Physics II as taught by Prof Recall that the Schroedinger radial equation for the radial wavefunction u(r)-r Rfr) can be rewritten...
52. Write out the full Hamiltonian for a Li atom. You may assume the Born-Oppenheimer approximation holds. Identify each of the terms (e.g. kinetic energy of electron 1, potential energy for nuclear-electron 1 attraction).
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...
Part c and h please
Help (a) Describe the essence of the orbital approximation. 3 pts (b) Suggest antisymmetrized wave functions of the Helium atom in the singlet (1s)2 ground state, and the (c) Normalize the (1s) wave function of (b), provided that individual space orbitals and spin functions are (d) Explain the energy ordering and degeneracy of the lowest three singlet and triplet states of the Helium singlet (1s) (2s) and triplet (1s)(2s) excited-state configurations in the orbital approximation....
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. 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...
Consider an H2+ ion. In the figure below, HẠ, HB, and e represent the two nuclei and the electron, respectively, and all the relevant distances are defined. You must use atomic units for this problem. (a) (4 points) Write down the Born-Oppenheimer Hamiltonian for the H2+ ion. R HA H (b) (6 points) The anti-bonding molecular orbital is expressed as V = N_(ls (r) – 1s (r)), where ls, (r) and 1s; (r) are hydrogen Is orbitals centered at nuclei...
is A system with an unper tubed Hamiltonian H Subjected to a perturbation HC Mon , 3 . • Il (1 1 . i eil 2 0 ; H = da o vz; 0 0 -2 0 II 131 l-ai vei o 2, where a is a real Constant with The dimesions of an energy anda is a dimension less real Constant. as show that the following vectors are the eigen vectors of M" and determine their associated eigenvalues. 14>*3)...
In this optional assignment you will find the eigenfunctions and eigenenergies of the hydrogen atom using an operator method which involves using Supersymmetric Quantum Mechanics (SUSY QM). In the SUSY QM formalism, any smooth potential Vx) (or equivalently Vr)) can be rewritten in terms of a superpotential Wix)l (Based upon lecture notes for 8.05 Quantum Krishna Rajagopal at MIT Physics II as taught by Prof Recall that the Schroedinger radial equation for the radial wavefunction u(r)-r Rfr) can be rewritten...
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