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Q7. Consider a harmonic oscillator system that has an object which is initially in the state:...
At time t = 0 a particle in a Harmonic Oscillator potential is in the state...
At time t = 0 a particle in a Harmonic Oscillator potential is in the state plcx.e = 0) = va (23*43+(iv]+213) por mayorale * a. Find the expectation value of the momentum (p). b. What is the probability of measuring the state to have energy E = 9ħw/2? E = 3ħw/2? c. Find y(x, t).
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...
[20 points] A particle in the simple harmonic oscillator potential with angular frequency a is initially...
[20 points] A particle in the simple harmonic oscillator potential with angular frequency a is initially in the ground state: c,y, (x) =Yo(x Att = 0 , the angular frequency of the oscillator suddenly doubles: a} → a½-2.4 The initial wave function can be written in terms of the modified potential (denoted with a tilde:~: Recall that the general form of the first few stationary states for the harmonic oscillator are given on page 56 of your text. a. What...
4. (20 points) Harmonic Oscillator The ground state wave function of a simple harmonic oscillator is...
4. (20 points) Harmonic Oscillator The ground state wave function of a simple harmonic oscillator is (a) = Ae-42", where a = (a) Using the normalization condition, obtain the constant A. (b) Find (c), (), and Az, using the result of A obtained in (a). Again, A.= V(32) - (2) (c) Find (p) and Ap. For the latter, you need to evaluate (p). Hint: For a harmonic oscillator, the time-averaged kinetic energy is equal to the time-averaged potential energy, and...
Consider the dimensionless harmonic oscillator Hamiltonian, (where m = h̄ = 1). Consider the orthogonal wave...
Consider the dimensionless harmonic oscillator Hamiltonian,
(where m = h̄ = 1).
Consider the orthogonal wave functions
and
, which are eigenfunctions of H with eigenvalues 1/2 and 5/2,
respectively.
with p=_ïda 2 2 We were unable to transcribe this imageY;(r) = (1-2x2)e-r2/2 (a) Let фо(x-AgVo(x) and φ2(x) = A2V2(x) and suppose that φ。(x) and φ2(x) are normalized. Find the constants Ao and A2. (b) Suppose that, at timet0, the state of the oscillator is given by Find the constant...
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...
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-...
1. Consider a charged particle bound in the harmonic oscillator potential V(x) = mw x2. A...
1. Consider a charged particle bound in the harmonic oscillator potential V(x) = mw x2. A weak electric field is applied to the system such that the potential energy, U(X), now has an extra term: V(x) = -qEx. We write the full Hamiltonian as H = Ho +V(x) where Ho = Px +mw x2 V(x) = –qEx. (a) Write down the unperturbed energies, EO. (b) Find the first-order correction to E . (c) Calculate the second-order correction to E ....
A one-particle two-dimensional harmonic oscillator has the potential energy function V=V(x,y)=k/2(x2+y2). write the t...
A one-particle two-dimensional harmonic oscillator has the potential energy function V=V(x,y)=k/2(x2+y2). write the time-independent SchrÖdinger equation for the system and the energy eigenvalues. Define clearly the symbols you used.
A particle in the harmonic oscillator potential starts out at t = 0 in the state...
A particle in the harmonic oscillator potential starts out at t = 0 in the state Ψ(x, 0) = A (5ψ0(x) + 12ψ1(x)) (a) [2 points] Find A. (b) [5 points] Find <x> and <p> as a function of time. (c) [3 points] Check Ehrenfest’s theorem that d<p>/dt = − <dV/dx>
At time t = 0 a particle in a Harmonic Oscillator potential is in the state plcx.e = 0) = va (23*43+(iv]+213) por mayorale * a. Find the expectation value of the momentum (p). b. What is the probability of measuring the state to have energy E = 9ħw/2? E = 3ħw/2? c. Find y(x, t).
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...
[20 points] A particle in the simple harmonic oscillator potential with angular frequency a is initially in the ground state: c,y, (x) =Yo(x Att = 0 , the angular frequency of the oscillator suddenly doubles: a} → a½-2.4 The initial wave function can be written in terms of the modified potential (denoted with a tilde:~: Recall that the general form of the first few stationary states for the harmonic oscillator are given on page 56 of your text. a. What...
4. (20 points) Harmonic Oscillator The ground state wave function of a simple harmonic oscillator is (a) = Ae-42", where a = (a) Using the normalization condition, obtain the constant A. (b) Find (c), (), and Az, using the result of A obtained in (a). Again, A.= V(32) - (2) (c) Find (p) and Ap. For the latter, you need to evaluate (p). Hint: For a harmonic oscillator, the time-averaged kinetic energy is equal to the time-averaged potential energy, and...
Consider the dimensionless harmonic oscillator Hamiltonian,
(where m = h̄ = 1).
Consider the orthogonal wave functions
and
, which are eigenfunctions of H with eigenvalues 1/2 and 5/2,
respectively.
with p=_ïda 2 2 We were unable to transcribe this imageY;(r) = (1-2x2)e-r2/2 (a) Let фо(x-AgVo(x) and φ2(x) = A2V2(x) and suppose that φ。(x) and φ2(x) are normalized. Find the constants Ao and A2. (b) Suppose that, at timet0, the state of the oscillator is given by Find the constant...
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...
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-...
1. Consider a charged particle bound in the harmonic oscillator potential V(x) = mw x2. A weak electric field is applied to the system such that the potential energy, U(X), now has an extra term: V(x) = -qEx. We write the full Hamiltonian as H = Ho +V(x) where Ho = Px +mw x2 V(x) = –qEx. (a) Write down the unperturbed energies, EO. (b) Find the first-order correction to E . (c) Calculate the second-order correction to E ....
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