According to the results from part 1 of section V (i.e Harmonic oscillator), what are the...
According to the results from part 1 of section V (i.e Harmonic oscillator), what are the relations among E, T, and V for harmonic oscillator?
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According to the results from part 1 of section V (i.e Harmonic
oscillator), what are the...
The most general wave function of a particle in the simple harmonic oscillator potential is: V(x,...
The most general wave function of a particle in the simple harmonic oscillator potential is: V(x, t) = (x)e-1st/ where and E, are the harmonic oscillator's stationary states and their corresponding energies. (a) Show that the expectation value of position is (hint: use the results of Problem 4): (v) = A cos (wt - ) where the real constants A and o are given by: 1 2 Ae-id-1 " Entichtin Interpret this result, comparing it with the motion of a...
7 Harmonic oscillator in "energy space" Consider the harmonic oscillator in "energy space", i.e., in terms...
7 Harmonic oscillator in "energy space" Consider the harmonic oscillator in "energy space", i.e., in terms of the basis of eigenvectors n) of the harmonic oscillator Hamiltonian, with Hn) -hwn1/2)]n). We computed these in terms of wavefunctions in position space, ie. pn(x)-(zln), but we can also work purely in terms of the abstract energy eigenvectors in Dirac notation. PS9.pdf 1. You computed the matrix elements 〈nleln) on an earlier problem set. Now find (nn) for general n,n' 2. Find the...
6 The Fermionic Oscillator Suppose that we constructed a harmonic oscillator Hamiltonian H in terms of...
6 The Fermionic Oscillator Suppose that we constructed a harmonic oscillator Hamiltonian H in terms of raising and lowering operators a+,a in the usual way, such that but now whereaa obey the anticommutation relationn (Be careful! The a+,a are operators, rather than numbers.) (a) Suppose I give you a wavefunction that solves the time-independent Schrödinger equation, i.e. such that HUn-EUn-hw (n + ) ψη. Is a+Un also a solution to the time-independent Schrödinger equation If so, what is its energy...
question no 4.22, statistical physics by Reif Volume 5 4.92 Mean energy of a harmonic oscillator A harmonic oscillator...
question no 4.22, statistical physics by Reif Volume 5
4.92 Mean energy of a harmonic oscillator A harmonic oscillator has a mass and spring constant which are such that its classical angular frequency of oscllation is equal to w. In a quantum- mechanical description, such an oscillator is characterized by a set of discrete states having energies En given by The quantum number n which labels these states can here assume all the integral values A particular instance of a...
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...
1. An ideal (frictionless) simple harmonic oscillator is set into motion by releasing it from rest...
1. An ideal (frictionless) simple harmonic oscillator is set into motion by releasing it from rest at X +0.750 m. The oscillator is set into motion once again from x=+0.750 m, except the oscillator now experiences a retarding force that is linear with respect to velocity. As a result, the oscillator does not return to its original starting position, but instead reaches = +0.700 m after one period. a. During the first full oscillation of motion, determine the fraction of...
In the lecture notes, we only solved the TISE for the quantum harmonic oscillator 1 Now,...
In the lecture notes, we only solved the TISE for the quantum harmonic oscillator 1 Now, write down the actual solution of the wavefunction of the quantum harmonic oscillator, i.e. the solution that solves TDSE not TISE. 2. We consider the Quantum Harmonic Oscillator In Heisenberg Picture: (a) Hamiltonian to use is the quantum harmonic oscillator Hamiltonian Solve the Heisenberg equations of motion for the operators X (t) and P(t) where the Calculate the commutator [X(t), X (0)] and show...
A) The velocity of a simple harmonic oscillator is given by v = -7.35sin(26.8t) (mks units)...
A) The velocity of a simple harmonic oscillator is given by v = -7.35sin(26.8t) (mks units) What is its angular frequency? B) What is the amplitude of the motion in Problem 1 in meters to two decimal places? C) To the nearest hundredth of a meter where is the mass in Problem 1 at the time t = 44.03 seconds? D) If the mass in Problem C is 0.75 kg, what is the spring's potential energy to the nearest tenth...
-The velocity of a simple harmonic oscillator is given by v = -7.93sin(29.4t) (mks units) What...
-The velocity of a simple harmonic oscillator is given by v = -7.93sin(29.4t) (mks units) What is its angular frequency? -What is the amplitude of the motion in Problem 1 in meters to two decimal places? -To the nearest hundredth of a meter where is the mass in Problem 1 at the time t = 22.64 seconds? -If the mass in Problem 3 is 0.34 kg, what is the spring's potential energy to the nearest tenth of a joule? -What...
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 ....
The most general wave function of a particle in the simple harmonic oscillator potential is: V(x, t) = (x)e-1st/ where and E, are the harmonic oscillator's stationary states and their corresponding energies. (a) Show that the expectation value of position is (hint: use the results of Problem 4): (v) = A cos (wt - ) where the real constants A and o are given by: 1 2 Ae-id-1 " Entichtin Interpret this result, comparing it with the motion of a...
7 Harmonic oscillator in "energy space" Consider the harmonic oscillator in "energy space", i.e., in terms of the basis of eigenvectors n) of the harmonic oscillator Hamiltonian, with Hn) -hwn1/2)]n). We computed these in terms of wavefunctions in position space, ie. pn(x)-(zln), but we can also work purely in terms of the abstract energy eigenvectors in Dirac notation. PS9.pdf 1. You computed the matrix elements 〈nleln) on an earlier problem set. Now find (nn) for general n,n' 2. Find the...
6 The Fermionic Oscillator Suppose that we constructed a harmonic oscillator Hamiltonian H in terms of raising and lowering operators a+,a in the usual way, such that but now whereaa obey the anticommutation relationn (Be careful! The a+,a are operators, rather than numbers.) (a) Suppose I give you a wavefunction that solves the time-independent Schrödinger equation, i.e. such that HUn-EUn-hw (n + ) ψη. Is a+Un also a solution to the time-independent Schrödinger equation If so, what is its energy...
question no 4.22, statistical physics by Reif Volume 5
4.92 Mean energy of a harmonic oscillator A harmonic oscillator has a mass and spring constant which are such that its classical angular frequency of oscllation is equal to w. In a quantum- mechanical description, such an oscillator is characterized by a set of discrete states having energies En given by The quantum number n which labels these states can here assume all the integral values A particular instance of a...
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...
1. An ideal (frictionless) simple harmonic oscillator is set into motion by releasing it from rest at X +0.750 m. The oscillator is set into motion once again from x=+0.750 m, except the oscillator now experiences a retarding force that is linear with respect to velocity. As a result, the oscillator does not return to its original starting position, but instead reaches = +0.700 m after one period. a. During the first full oscillation of motion, determine the fraction of...
In the lecture notes, we only solved the TISE for the quantum harmonic oscillator 1 Now, write down the actual solution of the wavefunction of the quantum harmonic oscillator, i.e. the solution that solves TDSE not TISE. 2. We consider the Quantum Harmonic Oscillator In Heisenberg Picture: (a) Hamiltonian to use is the quantum harmonic oscillator Hamiltonian Solve the Heisenberg equations of motion for the operators X (t) and P(t) where the Calculate the commutator [X(t), X (0)] and show...
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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