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1. Suppose I have a harmonic oscillator with a small quartic perturbation: 2 рґ 2m 2...
Consider a one-dimensional (1D) harmonic oscillator problem, where the perturbation V causes a modification of the...
Consider a one-dimensional (1D) harmonic oscillator problem, where the perturbation V causes a modification of the oscillator frequency: 2 K22 H = H. +V, (1) 2 K2 V = - K +K > 0. Of course, this problem (1), (2) is trivially solved exactly yielding the oscillator solutions with a new frequency. Show that corrections to the nth energy level as calculated within the perturbation theory indeed reproduce the exact result, restricting yourselves to terms up to the second order...
H2 d2 1 2m dx22 m ω2 + γχ4, use perturbation theory to estimat 1. For the HamiltonianH - the grou...
h2 d2 1 2m dx22 m ω2 + γχ4, use perturbation theory to estimat 1. For the HamiltonianH - the ground state energy (A) What would be a good choice for the reference, or unperturbed, Hamiltonian? (B) The ground state wavefunction for harmonic oscillator is ψ(x) e 2 wit mc /h. W energy rite down the expression the first order perturbation contribution to the (C) Evaluate the integral from part (B). The relevant integral should be in the Useful Integral...
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...
Suppose a particle is in a one-dimensional harmonic oscillator potential. Suppose that a perturbation is added...
Suppose a particle is in a one-dimensional harmonic oscillator
potential. Suppose that
a perturbation is added at time t = 0 of the form . Assume that at time t = 0 the
particle
is in the ground state. Use first order perturbation theory to find
the probability that at some
time t1 > 0 the particle is in the first excited state of the
harmonic oscillator.
H' = ext.
4. Let us revisit the shifted harmonic oscillator from problem set 5, but this time through...
4. Let us revisit the shifted harmonic oscillator from problem set 5, but this time through the lens of perturbation theory. The Hamiltonian of the oscillator is given by * 2m + mw?f? + cî, and, as solved for previously, it has eigenenergies of En = hwan + mwra and eigenstates of (0) = N,,,a1 + role of (rc)*/2, where Do = 42 and a=(mw/h) (a) By treating the term cî as a perturbation, show that the first-order correction to...
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...
3. (a) Consider a 1-dim harmonic oscillator in its ground state (0) of the unperturbed Hamiltonian at t--0o. Let a pert...
3. (a) Consider a 1-dim harmonic oscillator in its ground state (0) of the unperturbed Hamiltonian at t--0o. Let a perturbation Hi(t)--eEXe t2 (e, E and rare constants) be applied between - and too. What is the probability that the oscillator will be in the state n) (of the unperturbed oscillator) as t-> oo?(15%) (b) The bottom of an infinite well is changed to have the shape V(x)-ε sin® for 0Sxa. Calculate the energy shifts for all the excited states...
4 A nonlinear oscillator Consider a perturbed harmonic oscillator. Using x p2 H + ke? 2...
4 A nonlinear oscillator Consider a perturbed harmonic oscillator. Using x p2 H + ke? 2 + ex4 2m 1. write this Hamiltonian in terms of â and at 2. At what frequency or frequencies could this system absorb radiation if € = 0, i.e. the oscillator is unperturbed 1 3. Qualitatively, what do the states look like for the perturbed Hamiltonian? Write the new states as a sum of the unperturbed states, without worrying too much about the amplitude...
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)...
As a result of a sudden perturbation of the harmonic oscillator originally in the ground state, t...
As a result of a sudden perturbation of the harmonic oscillator originally in the ground state, the restoring force coefficient k in its potential energy U(a) (1/2)k2 changes to k' ak, a>0. Find the proba- bility to find the new oscillator in an excited state.
As a result of a sudden perturbation of the harmonic oscillator originally in the ground state, the restoring force coefficient k in its potential energy U(a) (1/2)k2 changes to k' ak, a>0. Find the proba-...
Consider a one-dimensional (1D) harmonic oscillator problem, where the perturbation V causes a modification of the oscillator frequency: 2 K22 H = H. +V, (1) 2 K2 V = - K +K > 0. Of course, this problem (1), (2) is trivially solved exactly yielding the oscillator solutions with a new frequency. Show that corrections to the nth energy level as calculated within the perturbation theory indeed reproduce the exact result, restricting yourselves to terms up to the second order...
h2 d2 1 2m dx22 m ω2 + γχ4, use perturbation theory to estimat 1. For the HamiltonianH - the ground state energy (A) What would be a good choice for the reference, or unperturbed, Hamiltonian? (B) The ground state wavefunction for harmonic oscillator is ψ(x) e 2 wit mc /h. W energy rite down the expression the first order perturbation contribution to the (C) Evaluate the integral from part (B). The relevant integral should be in the Useful Integral...
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...
Suppose a particle is in a one-dimensional harmonic oscillator
potential. Suppose that
a perturbation is added at time t = 0 of the form . Assume that at time t = 0 the
particle
is in the ground state. Use first order perturbation theory to find
the probability that at some
time t1 > 0 the particle is in the first excited state of the
harmonic oscillator.
H' = ext.
4. Let us revisit the shifted harmonic oscillator from problem set 5, but this time through the lens of perturbation theory. The Hamiltonian of the oscillator is given by * 2m + mw?f? + cî, and, as solved for previously, it has eigenenergies of En = hwan + mwra and eigenstates of (0) = N,,,a1 + role of (rc)*/2, where Do = 42 and a=(mw/h) (a) By treating the term cî as a perturbation, show that the first-order correction to...
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...
3. (a) Consider a 1-dim harmonic oscillator in its ground state (0) of the unperturbed Hamiltonian at t--0o. Let a perturbation Hi(t)--eEXe t2 (e, E and rare constants) be applied between - and too. What is the probability that the oscillator will be in the state n) (of the unperturbed oscillator) as t-> oo?(15%) (b) The bottom of an infinite well is changed to have the shape V(x)-ε sin® for 0Sxa. Calculate the energy shifts for all the excited states...
4 A nonlinear oscillator Consider a perturbed harmonic oscillator. Using x p2 H + ke? 2 + ex4 2m 1. write this Hamiltonian in terms of â and at 2. At what frequency or frequencies could this system absorb radiation if € = 0, i.e. the oscillator is unperturbed 1 3. Qualitatively, what do the states look like for the perturbed Hamiltonian? Write the new states as a sum of the unperturbed states, without worrying too much about the amplitude...
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)...
As a result of a sudden perturbation of the harmonic oscillator originally in the ground state, the restoring force coefficient k in its potential energy U(a) (1/2)k2 changes to k' ak, a>0. Find the proba- bility to find the new oscillator in an excited state.
As a result of a sudden perturbation of the harmonic oscillator originally in the ground state, the restoring force coefficient k in its potential energy U(a) (1/2)k2 changes to k' ak, a>0. Find the proba-...
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