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A harmonic oscillator is in a state such that a measurement of the energy would yield...
1. Quantum harmonic oscillator (a) Derive formula for standard deviation of position measurement ...
1. Quantum harmonic oscillator (a) Derive formula for standard deviation of position measurement on a particle prepared in the ground state of harmonic oscillator. The formula will depend on h, m andw (b) Estimate order of magnitude of the standard deviation in (a) for the LIGO mirror of mass 10 kg and w 1 Hz. (c) A coherent state lo) is defined to be the eigenstate of the lowering operator with eigenvalue a, i.e. à lo)a) Write la) as where...
A particle with mass m is in a one dimensional simple harmonic oscillator potential. At timet0 it is described by the superposition state where Vo, 1 and Vz are normalised energy eigenfunctions of th...
A particle with mass m is in a one dimensional simple harmonic oscillator potential. At timet0 it is described by the superposition state where Vo, 1 and Vz are normalised energy eigenfunctions of the harmonic oscillator potential corresponding to energies Eo, E1 and E2 (a) Show that the wavefunction is normalised (b) If an observation of energy is made, what is the most likely value of energy and with what probability would it be obtained? (c) If the experiment is...
5. A particle in the harmonic oscillator potential has the initial wave function Psi(x, 0) = A[\psi_{0}(x) + \psi_{1}(x)] for some constant A. Here to and ₁ are the normalized ground state and the first excited state wavefunctions of the harmonic oscill
5. A particle in the harmonic oscillator potential has the initial wave function Psi(x, 0) = A[\psi_{0}(x) + \psi_{1}(x)] for some constant A. Here to and ₁ are the normalized ground state and the first excited state wavefunctions of the harmonic oscillator, respectively. (a) Normalize (r, 0). (b) Find the wavefunction (r, t) at a later time t and hence evaluate (x, t) 2. Leave your answers involving expressions in to and ₁. c) sing the following normalized expression of...
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...
Estimate the ground-state energy of a one-dimensional simple harmonic oscillator using (50) = e-a-l as a...
Estimate the ground-state energy of a one-dimensional simple harmonic oscillator using (50) = e-a-l as a trial function with a to be varied. For a simple harmonic oscillator we have H + jmwºr? Recall that, for the variational method, the trial function (HO) gives an expectation value of H such that (016) > Eo, where Eo is the ground state energy. You may use: n! dH() ||= TH(c) – z[1 – H(r)], 8(2), dx S." arcade an+1 where (x) and...
The lowest energy wavefunction of the quantum harmonic oscillator has the form (c) Determine σ an...
The lowest energy wavefunction of the quantum harmonic oscillator has the form (c) Determine σ and Eo (the energy of this lowest-energy wavefunction) by using the time-independent Schrödinger equation (H/Ho(x)- E/Ho(x) In Lecture 3, we found that the solution for a classical harmonic oscillator displaced from equilibrium by an amount o and released at rest was x(t)cos(wt) (d) Classically, what is the momentum of this harmonic oscillator as a function of time? (e) Show that 〈z) (expectation value of x)...
3- A one-dimensional harmonic oscillator wave function is ψ(x) = Axe-bx2 a) Find the total energy...
3- A one-dimensional harmonic oscillator wave function is ψ(x) = Axe-bx2 a) Find the total energy E b) Find the constant b c) Find the normalization constant A. d) Find the expectation value of x, e) Find the uncertainty in x, Ох. f) Find the expectation value of p g) Find the uncertainty in p, Op For the Hamiltonian matrix shown below:
3- A one-dimensional harmonic oscillator wave function is ψ(x) = Axe-bx2 a) Find the total energy E b)...
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...
Question A2: Coherent states of the harmonic oscillator Consider a one-dimensional harmonic oscillator with the Hamiltonian...
Question A2: Coherent states of the harmonic oscillator Consider a one-dimensional harmonic oscillator with the Hamiltonian 12 12 m2 H = -2m d. 2+ 2 Here m and w are the mass and frequency, respectively. Consider a time-dependent wave function of the form <(x,t) = C'exp (-a(x – 9(t)+ ik(t)z +io(t)), where a and C are positive constants, and g(t), k(t), and o(t) are real functions of time t. 1. Express C in terms of a. [2 marks] 2. By...
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. Quantum harmonic oscillator (a) Derive formula for standard deviation of position measurement on a particle prepared in the ground state of harmonic oscillator. The formula will depend on h, m andw (b) Estimate order of magnitude of the standard deviation in (a) for the LIGO mirror of mass 10 kg and w 1 Hz. (c) A coherent state lo) is defined to be the eigenstate of the lowering operator with eigenvalue a, i.e. à lo)a) Write la) as where...
A particle with mass m is in a one dimensional simple harmonic oscillator potential. At timet0 it is described by the superposition state where Vo, 1 and Vz are normalised energy eigenfunctions of the harmonic oscillator potential corresponding to energies Eo, E1 and E2 (a) Show that the wavefunction is normalised (b) If an observation of energy is made, what is the most likely value of energy and with what probability would it be obtained? (c) If the experiment 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...
Estimate the ground-state energy of a one-dimensional simple harmonic oscillator using (50) = e-a-l as a trial function with a to be varied. For a simple harmonic oscillator we have H + jmwºr? Recall that, for the variational method, the trial function (HO) gives an expectation value of H such that (016) > Eo, where Eo is the ground state energy. You may use: n! dH() ||= TH(c) – z[1 – H(r)], 8(2), dx S." arcade an+1 where (x) and...
The lowest energy wavefunction of the quantum harmonic oscillator has the form (c) Determine σ and Eo (the energy of this lowest-energy wavefunction) by using the time-independent Schrödinger equation (H/Ho(x)- E/Ho(x) In Lecture 3, we found that the solution for a classical harmonic oscillator displaced from equilibrium by an amount o and released at rest was x(t)cos(wt) (d) Classically, what is the momentum of this harmonic oscillator as a function of time? (e) Show that 〈z) (expectation value of x)...
3- A one-dimensional harmonic oscillator wave function is ψ(x) = Axe-bx2 a) Find the total energy E b) Find the constant b c) Find the normalization constant A. d) Find the expectation value of x, e) Find the uncertainty in x, Ох. f) Find the expectation value of p g) Find the uncertainty in p, Op For the Hamiltonian matrix shown below:
3- A one-dimensional harmonic oscillator wave function is ψ(x) = Axe-bx2 a) Find the total energy E b)...
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...
Question A2: Coherent states of the harmonic oscillator Consider a one-dimensional harmonic oscillator with the Hamiltonian 12 12 m2 H = -2m d. 2+ 2 Here m and w are the mass and frequency, respectively. Consider a time-dependent wave function of the form <(x,t) = C'exp (-a(x – 9(t)+ ik(t)z +io(t)), where a and C are positive constants, and g(t), k(t), and o(t) are real functions of time t. 1. Express C in terms of a. [2 marks] 2. By...
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).
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