Consider the Hamiltonian, , where and a) Determine the energies and eigenfunctions of the undisturbed Hamiltonian....
, where 
and 
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Consider the Hamiltonian, , where and
a) Determine the energies and eigenfunctions of the
undisturbed Hamiltonian....
(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...
Please solve the problem as soon as possible. Problem 1: Consider a two level system with Hamiltonian: Using the first order time-dependent perturbation theory, obtain the probability coefficients cn...
Please solve the problem as soon as possible.
Problem 1: Consider a two level system with Hamiltonian: Using the first order time-dependent perturbation theory, obtain the probability coefficients cn (t) if the perturbation is applied at t >0 and the system is originally in the ground state. Hint: When solving the problem, first you may need to find the energies and wave functions of the unperturbed Hamiltonian A0.
Problem 1: Consider a two level system with Hamiltonian: Using the first...
1. (25 points) The Hamiltonian operator H, for a particular molecule has a complete set of...
1. (25 points) The Hamiltonian operator H, for a particular molecule has a complete set of orthonormal eigenfunctions on (where n = 1,2,3,...) with corresponding eigenvalues (n-1)h. The molecule in state n is subject to measurement of the dipole moment, for which the mathematical operator is represented by M. After the measurement of the dipole moment, the wave function of the particle is: Y = 0.5 0.1 +0.7071 Q. + 0.5 Pn+1 a) Show that is normalized. Now consider a...
Quantum mechanics. A Hamiltonian of the form , is equivalent to the Hamiltonian of a harmonic...
Quantum mechanics. A Hamiltonian of the form , is equivalent to the Hamiltonian of a harmonic oscillator with its equilibrium point displaced where and C are constant, find them. With the previous result, find the exact spectrum of H. Calculate the same spectrum using the theory of disturbances to second order with . Compare your results. Calculate the wave functions up to first order using as a perturbation. P2 22 P2 Tm We were unable to transcribe this imageWe were...
Exercise 4: Fine structure of hydrogenic atoms a) Consider a Hamiltonian H-Ho + λΗ. with Mr a small perturbation. Show...
Exercise 4: Fine structure of hydrogenic atoms a) Consider a Hamiltonian H-Ho + λΗ. with Mr a small perturbation. Show that in (non-degenerate) perturbation theory the first order correction to the unperturbed, discrete energy level E(Holis given by and the second order by b) Apply this to evaluate the first order corrections to the energy levels (the so-called fine structure) of a hydrogenic atom, that arise due to relativistic corrections. Confirm that the answer for the total first order correction...
Consider a quantum mechanical system with 4 states and an unperturbed Hamiltonian given by 1 0 0 0 Ho E0 0 2 0 a small perturbation is added to this Hamiltonian 0 0 1 0 where e is much smaller than E...
Consider a quantum mechanical system with 4 states and an unperturbed Hamiltonian given by 1 0 0 0 Ho E0 0 2 0 a small perturbation is added to this Hamiltonian 0 0 1 0 where e is much smaller than E a) [10pts] What are the energy eigenvalues of the unperturbed system of the following states? 1 o 2o 0 and which energy levels are degenerate? b) [10pts Find a good basis for degenerate perturbation theory instead of c)...
4. 3D Infinite Square Well Perturbation (20 pts) Consider the three-dimensional infinite cubic well: otherwise The stationary states are where n, ny, and n, are integers. The corresponding allowed en...
4. 3D Infinite Square Well Perturbation (20 pts) Consider the three-dimensional infinite cubic well: otherwise The stationary states are where n, ny, and n, are integers. The corresponding allowed energies are Now let us introduce the perturbation otherwise a) Find the first-order correction to the ground state energy b) Find the first-order correction to the first excited state
4. 3D Infinite Square Well Perturbation (20 pts) Consider the three-dimensional infinite cubic well: otherwise The stationary states are where n, ny,...
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...
1. The two dimensional Harmonic Oscillator has the Hamiltonian n, n'>denotes the state In> of the...
Please solve with the
explanations of notations
1. The two dimensional Harmonic Oscillator has the Hamiltonian n, n'>denotes the state In> of the x-oscillator and In'> of the y-oscillator. This system is perturbed with the potential energy: Hi-Kix y. The perturbation removes the The perturbation removes the degeneracy of the states | 1,0> and |0,1> a) In first order perturbation theory find the two nondegenerate eigenstates of the full b) Find the corresponding energy eigenvalues. На Hamiltonian as normalized linear...
Let Ho be the Hamiltonian of the non-relativistic hydrogen atom neglecting spin. Consider H1 = e|E\r cos 0 with e|E|af...
Let Ho be the Hamiltonian of the non-relativistic hydrogen atom neglecting spin. Consider H1 = e|E\r cos 0 with e|E|af < 1. This Hamiltonian describes a weak constant electric field in the z-direction interacting with the atomic dipole. We want to understand the effect such a field has on the first excited energy level, E2, of hydrogen. Remember that this energy level is degenerate with corresponding eigenstates |2lm) Use first-order perturbation theory to find the aproximate energies of Ho+ H1...
(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...
Please solve the problem as soon as possible.
Problem 1: Consider a two level system with Hamiltonian: Using the first order time-dependent perturbation theory, obtain the probability coefficients cn (t) if the perturbation is applied at t >0 and the system is originally in the ground state. Hint: When solving the problem, first you may need to find the energies and wave functions of the unperturbed Hamiltonian A0.
Problem 1: Consider a two level system with Hamiltonian: Using the first...
1. (25 points) The Hamiltonian operator H, for a particular molecule has a complete set of orthonormal eigenfunctions on (where n = 1,2,3,...) with corresponding eigenvalues (n-1)h. The molecule in state n is subject to measurement of the dipole moment, for which the mathematical operator is represented by M. After the measurement of the dipole moment, the wave function of the particle is: Y = 0.5 0.1 +0.7071 Q. + 0.5 Pn+1 a) Show that is normalized. Now consider a...
Exercise 4: Fine structure of hydrogenic atoms a) Consider a Hamiltonian H-Ho + λΗ. with Mr a small perturbation. Show that in (non-degenerate) perturbation theory the first order correction to the unperturbed, discrete energy level E(Holis given by and the second order by b) Apply this to evaluate the first order corrections to the energy levels (the so-called fine structure) of a hydrogenic atom, that arise due to relativistic corrections. Confirm that the answer for the total first order correction...
Consider a quantum mechanical system with 4 states and an unperturbed Hamiltonian given by 1 0 0 0 Ho E0 0 2 0 a small perturbation is added to this Hamiltonian 0 0 1 0 where e is much smaller than E a) [10pts] What are the energy eigenvalues of the unperturbed system of the following states? 1 o 2o 0 and which energy levels are degenerate? b) [10pts Find a good basis for degenerate perturbation theory instead of c)...
4. 3D Infinite Square Well Perturbation (20 pts) Consider the three-dimensional infinite cubic well: otherwise The stationary states are where n, ny, and n, are integers. The corresponding allowed energies are Now let us introduce the perturbation otherwise a) Find the first-order correction to the ground state energy b) Find the first-order correction to the first excited state
4. 3D Infinite Square Well Perturbation (20 pts) Consider the three-dimensional infinite cubic well: otherwise The stationary states are where n, ny,...
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...
Please solve with the
explanations of notations
1. The two dimensional Harmonic Oscillator has the Hamiltonian n, n'>denotes the state In> of the x-oscillator and In'> of the y-oscillator. This system is perturbed with the potential energy: Hi-Kix y. The perturbation removes the The perturbation removes the degeneracy of the states | 1,0> and |0,1> a) In first order perturbation theory find the two nondegenerate eigenstates of the full b) Find the corresponding energy eigenvalues. На Hamiltonian as normalized linear...
Let Ho be the Hamiltonian of the non-relativistic hydrogen atom neglecting spin. Consider H1 = e|E\r cos 0 with e|E|af < 1. This Hamiltonian describes a weak constant electric field in the z-direction interacting with the atomic dipole. We want to understand the effect such a field has on the first excited energy level, E2, of hydrogen. Remember that this energy level is degenerate with corresponding eigenstates |2lm) Use first-order perturbation theory to find the aproximate energies of Ho+ H1...
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