Evaluate the average kinetic and potential energies for the second excited state (n = 2) of...
Evaluate the average kinetic and potential energies for the second excited state (n = 2) of the harmonic oscillator. (Hint: this problem requires some integration)
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Evaluate the average kinetic and potential energies for the
second excited state (n = 2) of...
The wave function for a harmonic oscillator in its first excited state is Consider the harmonic o...
The wave function for a harmonic oscillator in its first excited state is Consider the harmonic oscillator with Hamiltonian and let H --hk and r-cr d. Evaluate E0) for the first excited state using perturbation theory 2m dx 2
The wave function for a harmonic oscillator in its first excited state is Consider the harmonic oscillator with Hamiltonian and let H --hk and r-cr d. Evaluate E0) for the first excited state using perturbation theory 2m dx 2
1000 diatomic molecules with vibrational state described by N molecules in the ground vibrational...
1000 diatomic molecules with vibrational state described by N molecules in the ground vibrational state O molecules in the lowest potential Total Energy Potential Energy state M molecules in an excited vibrational states P molecules in an excited potential energy states Schematic of Energy Eigenfunctions and the 1. Consider a sample of 1000 identically prepared diatomic molecules, each of which can be Potential Energy function of the Harmonic Oscillator described by the ground state of the Harmonic oscillator: Ψ-ψ。. If...
Solve the following problem: 6. a) Find an expectation value<p> in the n" state of the...
Solve the following problem: 6. a) Find an expectation value<p> in the n" state of the harmonic oscillator. Hint: it may be useful to recall the following equations: a.y-n+1 1 (Fip+max) /2 hma n+1 and а. b) Compute an expectation value of the kinetic energy in the n" state of the harmonic oscillator. Show that it is exactly half of the total energy in that state
By evaluating expectation values of the kinetic and potential energies directly, show that the vi...
By evaluating expectation values of the kinetic and potential
energies directly,
show that the virial theorem V 2 T holds in the bound state of a
delta well
potential (x)
. By evaluating expectation values of the kinetic and potential energies directly, show that the virial theorem )--2 T) holds in the bound state of a delta well potential -a5(
. By evaluating expectation values of the kinetic and potential energies directly, show that the virial theorem )--2...
Classical Mechanics Let us consider the following kinetic (T) and potential (U) energies of a two-dimensional...
Classical Mechanics Let us consider the following kinetic (T) and potential (U) energies of a two-dimensional oscillator : ?(?,̇ ?̇)= ?/2 (?̇²+ ?̇²) ?(?,?)= ?/2 (?²+?² )+??? where x and y denote, respectively, the cartesian displacements of the oscillator; ?̇= ??/?? and ?̇= ??/?? the time derivatives of the displacements; m the mass of the oscillator; K the stiffness constant of the oscillator; A is the coupling constant. 1) Using the following coordinate transformations, ?= 1/√2 (?+?) ?= 1/√2 (?−?)...
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...
Consider a particle subjected to a harmonic oscillator potential of the form x)m. The allowed values...
Consider a particle subjected to a harmonic oscillator potential of the form x)m. The allowed values of energy for the simple harmonic oscillator is (a) What is the energy corresponding to the ground state (3 points)? (b) What is the energy separation between the ground state and the first excited state (3 points)? (c) The selection rule allows only those transitions for which the quantum number changes by 1. What is the energy of photon necessary to make the transition...
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...
Consider a classical particle confined on a segment with length 2L. If a harmonic potential is...
Consider a classical particle confined on a segment with length 2L. If a harmonic potential is introduced on this segment, then the Hamiltinian becomes and application of the equipartition theorem predicts the average values of the kinetic and potential energies, e.g., <V>-kT/2. On the hand, making L sufficiently small, one can ensure that V(x)=ma,2x2/2ckT/2 for all x between-L and L. We conclude that the average value is larger than any allowed value of V! Please explain this paradox (assume that...
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...
The wave function for a harmonic oscillator in its first excited state is Consider the harmonic oscillator with Hamiltonian and let H --hk and r-cr d. Evaluate E0) for the first excited state using perturbation theory 2m dx 2
The wave function for a harmonic oscillator in its first excited state is Consider the harmonic oscillator with Hamiltonian and let H --hk and r-cr d. Evaluate E0) for the first excited state using perturbation theory 2m dx 2
1000 diatomic molecules with vibrational state described by N molecules in the ground vibrational state O molecules in the lowest potential Total Energy Potential Energy state M molecules in an excited vibrational states P molecules in an excited potential energy states Schematic of Energy Eigenfunctions and the 1. Consider a sample of 1000 identically prepared diatomic molecules, each of which can be Potential Energy function of the Harmonic Oscillator described by the ground state of the Harmonic oscillator: Ψ-ψ。. If...
Solve the following problem: 6. a) Find an expectation value<p> in the n" state of the harmonic oscillator. Hint: it may be useful to recall the following equations: a.y-n+1 1 (Fip+max) /2 hma n+1 and а. b) Compute an expectation value of the kinetic energy in the n" state of the harmonic oscillator. Show that it is exactly half of the total energy in that state
By evaluating expectation values of the kinetic and potential
energies directly,
show that the virial theorem V 2 T holds in the bound state of a
delta well
potential (x)
. By evaluating expectation values of the kinetic and potential energies directly, show that the virial theorem )--2 T) holds in the bound state of a delta well potential -a5(
. By evaluating expectation values of the kinetic and potential energies directly, show that the virial theorem )--2...
Consider a particle subjected to a harmonic oscillator potential of the form x)m. The allowed values of energy for the simple harmonic oscillator is (a) What is the energy corresponding to the ground state (3 points)? (b) What is the energy separation between the ground state and the first excited state (3 points)? (c) The selection rule allows only those transitions for which the quantum number changes by 1. What is the energy of photon necessary to make the transition...
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...
Consider a classical particle confined on a segment with length 2L. If a harmonic potential is introduced on this segment, then the Hamiltinian becomes and application of the equipartition theorem predicts the average values of the kinetic and potential energies, e.g., <V>-kT/2. On the hand, making L sufficiently small, one can ensure that V(x)=ma,2x2/2ckT/2 for all x between-L and L. We conclude that the average value is larger than any allowed value of V! Please explain this paradox (assume that...
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...
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