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Consider a particle subjected to a harmonic oscillator potential of the form x)m. The allowed values...
3. The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an...
3. The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Consider an electron trapped by a one-dimensional harmonic potential V(x)=-5 mo?x” (where m is the electron mass, o is a constant angular frequency). In this case, the Schrödinger equation takes the following form, **...
4. (20 points). Consider a quantum harmonic oscillator with characteristic frequency w. The syste...
4. (20 points). Consider a quantum harmonic oscillator with characteristic frequency w. The system is in thermal equilibrium at temperature T. The oscillator is described by the following density matrix: A exp kaT where H is the usual harmonic oscillator Hamiltonian and kB is Boltzmann's constant. Working in the Fock (photon number) basis: a. Find the diagonal elements of ρ b. Determine the normalization constant A. c. Calculate the expectation value of energy (E
4. (20 points). Consider a quantum...
[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...
Quantum, 1D harmonic oscillator. Please answer in full. Thanks. Q3. The energy levels of the 1D harmonic oscillator are given by: En = (n +2)ha, n=0. 1, 2, 3, The CO molecule has a (reduced) mass of...
Quantum, 1D harmonic oscillator. Please answer in full.
Thanks.
Q3. The energy levels of the 1D harmonic oscillator are given by: En = (n +2)ha, n=0. 1, 2, 3, The CO molecule has a (reduced) mass of mco = 1.139 × 10-26 kg. Assuming a force constant of kco 1860 N/m, what is: a) The angular frequency, w, of the ground state CO bond vibration? b) The energy separation between the ground and first excited vibrational states? 7 marks] The...
19. Suppose that an electron in a one-dimensional harmonic-oscillator potential muo2 is subjected to an oscillating...
19. Suppose that an electron in a one-dimensional harmonic-oscillator potential muo2 is subjected to an oscillating electric field o) cos wt in the x direction (a) If the electron is initially in the ground state, what is the proba- bility that the electron will be in the nth excited state at time t? (b) I , perturbation theory will fail at some time t. What is the critical time?
[4] Consider a harmonic oscillator of mass m and angular frequency ω. At time t-0, the state of this oscillator is given by y(о) со фо) + с ф.) where the states I 0) .) represent the ground state...
[4] Consider a harmonic oscillator of mass m and angular frequency ω. At time t-0, the state of this oscillator is given by y(о) со фо) + с ф.) where the states I 0) .) represent the ground state and first excited state respectively. (a) Write the normalization condition for lv(o) and determine the mean value (H) of the energy in terms of co and ci. (b) With the additional requirement (H)-ho. calculate eoand o,p.
[4] Consider a harmonic oscillator...
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...
Please do this problem about quantum mechanic harmonic oscillator and show all your steps thank you. Q1. Consider a particle of mass m moving in a one-dimensional harmonic oscillator potential. 1....
Please do this problem about quantum mechanic harmonic
oscillator and show all your steps thank you.
Q1. Consider a particle of mass m moving in a one-dimensional harmonic oscillator potential. 1. Calculate the product of uncertainties in position and momentum for the particle in 2. Compare the result of (a) with the uncertainty product when the particle is in its the fifth excited state, ie. (OxơP)5. lowest energy state.
Q1. Consider a particle of mass m moving in a one-dimensional...
2. Now consider a particle in the ground state of the harmonic oscillator. ok gives the...
2. Now consider a particle in the ground state of the harmonic oscillator. ok gives the wave function for the ground state, but not the value of the constant A. Determine what it has to be if the ground state is normalized. (b) Suppose a classical particle has an energy equal to the ground state energy E. This particle will, of course, oscillate back and forth as though it were attached to a spring. What would its turning points be?...
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.
3. The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Consider an electron trapped by a one-dimensional harmonic potential V(x)=-5 mo?x” (where m is the electron mass, o is a constant angular frequency). In this case, the Schrödinger equation takes the following form, **...
4. (20 points). Consider a quantum harmonic oscillator with characteristic frequency w. The system is in thermal equilibrium at temperature T. The oscillator is described by the following density matrix: A exp kaT where H is the usual harmonic oscillator Hamiltonian and kB is Boltzmann's constant. Working in the Fock (photon number) basis: a. Find the diagonal elements of ρ b. Determine the normalization constant A. c. Calculate the expectation value of energy (E
4. (20 points). Consider a quantum...
[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...
Quantum, 1D harmonic oscillator. Please answer in full.
Thanks.
Q3. The energy levels of the 1D harmonic oscillator are given by: En = (n +2)ha, n=0. 1, 2, 3, The CO molecule has a (reduced) mass of mco = 1.139 × 10-26 kg. Assuming a force constant of kco 1860 N/m, what is: a) The angular frequency, w, of the ground state CO bond vibration? b) The energy separation between the ground and first excited vibrational states? 7 marks] The...
19. Suppose that an electron in a one-dimensional harmonic-oscillator potential muo2 is subjected to an oscillating electric field o) cos wt in the x direction (a) If the electron is initially in the ground state, what is the proba- bility that the electron will be in the nth excited state at time t? (b) I , perturbation theory will fail at some time t. What is the critical time?
[4] Consider a harmonic oscillator of mass m and angular frequency ω. At time t-0, the state of this oscillator is given by y(о) со фо) + с ф.) where the states I 0) .) represent the ground state and first excited state respectively. (a) Write the normalization condition for lv(o) and determine the mean value (H) of the energy in terms of co and ci. (b) With the additional requirement (H)-ho. calculate eoand o,p.
[4] Consider a harmonic oscillator...
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...
Please do this problem about quantum mechanic harmonic
oscillator and show all your steps thank you.
Q1. Consider a particle of mass m moving in a one-dimensional harmonic oscillator potential. 1. Calculate the product of uncertainties in position and momentum for the particle in 2. Compare the result of (a) with the uncertainty product when the particle is in its the fifth excited state, ie. (OxơP)5. lowest energy state.
Q1. Consider a particle of mass m moving in a one-dimensional...
2. Now consider a particle in the ground state of the harmonic oscillator. ok gives the wave function for the ground state, but not the value of the constant A. Determine what it has to be if the ground state is normalized. (b) Suppose a classical particle has an energy equal to the ground state energy E. This particle will, of course, oscillate back and forth as though it were attached to a spring. What would its turning points be?...
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.
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