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a) Calculate (x2) for the n=0 state of the harmonic oscilator. The result can be written...
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-...
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...
4&5 only thnkyouu :) 3. The force constant for 119F molecule is 966 N/m. a) Calculate...
4&5 only
thnkyouu :)
3. The force constant for 119F molecule is 966 N/m. a) Calculate the zero-point vibrational energy using a harmonic oscillator potential. b) Calculate the frequency of light needed to excite this molecule from the ground state to the first excited state. 4. Is 41(x) = *xe 2 an eigenfunction for the kinetic energy operator? Is it an eigenfunction for potential energy operator? 5. HCI molecule can be described by the Morse potential with De = 7.41...
6. a) Calculate the expectation value of x as a function of time for an electron...
6. a) Calculate the expectation value of x as a function of time for an electron in a state that is a (normalized) equal mixture of the ground state and 1st excited state of a 1D HO b) Graph x vs time for the case k = 1 eV/nm2. What is its value at t=0? What is the period of the oscillation in femtoseconds? For the one-dimensional (1D) harmonic oscillator (HO) the potential energy function has the form V(a) k2/2,...
A) Calculate the mean values <x^2> and <p^2x> for stationery state n. B) Calculate the rms...
A) Calculate the mean values <x^2> and <p^2x> for
stationery state n.
B) Calculate the rms deviations sigma(x) and sigma(px) for
stationery state n. Check that the Heisenberg uncertainty principle
is satisfied.
1. Consider a particle of mass m in an infinite square well with potential energy 0 for 0 saSa otherwise. V (z) = For simplicity, we may take the 'universe' here to be the region of 0 sas a, which is where the wave function is nontrivial. Consequently,...
One can assume a quantum mechanical harmonic oscillator model for the N-H stretching vibrations of the...
One can assume a quantum mechanical harmonic oscillator model for the N-H stretching vibrations of the peptide bonds. For the harmonic oscillator the energy levels are given by: E, = (V+})ħw where: W= /k/ u In the above express k is the force constant and u is the reduced mass. (a) Write the Schrödinger equation in terms of the reduced mass u, being sure to define all symbols. (b) Calculate the frequency of the infrared radiation absorbed by the N-H...
Consider the quantum mechanical vibration of H2 in the n = 1 state. Calculate the expectation...
Consider the quantum mechanical vibration of H2 in the n = 1 state. Calculate the expectation value of the potential energy, (Epotential), of this vibrational state. The wavefunction for the n = 1 state is: a = 6 where a = kr for Hz is 575 N/m and u = 8.368 x 10-20 kg The potential energy operator for the quantum harmonic oscillator is: Epotential = ***
Problem #1 The explicit wavefunction for a particle in the n-1 state of the quantum harmonic...
Problem #1 The explicit wavefunction for a particle in the n-1 state of the quantum harmonic oscillator is p1(x)- Axe-bx2 where mo 2h and ?1/4 (Note: In last week's homework there was an "h" where there should have been ?. This has been corrected in this week's assignment.) (a) By applying the lowering operator to ),obtain an explicit form for o(x) (i.e. the n-0 wavefunction) (b) By applying the raising operator to x), obtain an explicit form for p2(x) (i.e....
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...
[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...
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-...
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...
4&5 only
thnkyouu :)
3. The force constant for 119F molecule is 966 N/m. a) Calculate the zero-point vibrational energy using a harmonic oscillator potential. b) Calculate the frequency of light needed to excite this molecule from the ground state to the first excited state. 4. Is 41(x) = *xe 2 an eigenfunction for the kinetic energy operator? Is it an eigenfunction for potential energy operator? 5. HCI molecule can be described by the Morse potential with De = 7.41...
6. a) Calculate the expectation value of x as a function of time for an electron in a state that is a (normalized) equal mixture of the ground state and 1st excited state of a 1D HO b) Graph x vs time for the case k = 1 eV/nm2. What is its value at t=0? What is the period of the oscillation in femtoseconds? For the one-dimensional (1D) harmonic oscillator (HO) the potential energy function has the form V(a) k2/2,...
A) Calculate the mean values <x^2> and <p^2x> for
stationery state n.
B) Calculate the rms deviations sigma(x) and sigma(px) for
stationery state n. Check that the Heisenberg uncertainty principle
is satisfied.
1. Consider a particle of mass m in an infinite square well with potential energy 0 for 0 saSa otherwise. V (z) = For simplicity, we may take the 'universe' here to be the region of 0 sas a, which is where the wave function is nontrivial. Consequently,...
One can assume a quantum mechanical harmonic oscillator model for the N-H stretching vibrations of the peptide bonds. For the harmonic oscillator the energy levels are given by: E, = (V+})ħw where: W= /k/ u In the above express k is the force constant and u is the reduced mass. (a) Write the Schrödinger equation in terms of the reduced mass u, being sure to define all symbols. (b) Calculate the frequency of the infrared radiation absorbed by the N-H...
Consider the quantum mechanical vibration of H2 in the n = 1 state. Calculate the expectation value of the potential energy, (Epotential), of this vibrational state. The wavefunction for the n = 1 state is: a = 6 where a = kr for Hz is 575 N/m and u = 8.368 x 10-20 kg The potential energy operator for the quantum harmonic oscillator is: Epotential = ***
Problem #1 The explicit wavefunction for a particle in the n-1 state of the quantum harmonic oscillator is p1(x)- Axe-bx2 where mo 2h and ?1/4 (Note: In last week's homework there was an "h" where there should have been ?. This has been corrected in this week's assignment.) (a) By applying the lowering operator to ),obtain an explicit form for o(x) (i.e. the n-0 wavefunction) (b) By applying the raising operator to x), obtain an explicit form for p2(x) (i.e....
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...
[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...
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