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1. Variational method In this problem, you will approximate the ground state wave function of a...
2. Variational method. We can approximate the true ground-state wavefunction of the harmonic -프sxs and w(x) =D0 cos(cx...
2. Variational method. We can approximate the true ground-state wavefunction of the harmonic -프sxs and w(x) =D0 cos(cx) in the range oscillator by the trial wavefunction p(x) = X 2c 2c outside this range. (This wavefunction is already normalized). (A) Compute the energy expectation value of b as a function of c. (B) Determine the value of c that gives the minimal energy. (C) Compare the minimal energy to the energy of the true ground-state wavefunction
2. Variational method. We...
2. Variational Principle. The energy of a system with wave function ψ is given by where...
2. Variational Principle. The energy of a system with wave function ψ is given by where H is the energy operator. The variational principle is a method by which we guess a trial form for the wave function φ, with adjustable parameters, and minimize the resulting energy with respect to the adjustable parameters. This essentially chooses a "best fit" wave function based on our guess. Since the energy of the system with the correct wave function will always be minimum...
(a) Use the variational method to estimate the ground state energy of a particle of |mass...
(a) Use the variational method to estimate the ground state energy of a particle of |mass m in a potential Vx)kx, k > 0. (b) Calculate the energy shift in the ground state and in the degenerate 1t excited state of a 2-dimensional harmonic oscillator H(P2P,2/2m m(x +y)due to the perturbation V 2Axy. (20 pts)
(a) Use the variational method to estimate the ground state energy of a particle of |mass m in a potential Vx)kx, k > 0. (b)...
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 variational method can be used to solve for the ground state wavefunction and energy of a harmonic oscillator. Using a trail wavefunction of , where the function is defined between . The Hamilto...
The variational method can be used to solve for the ground state wavefunction and energy of a harmonic oscillator. Using a trail wavefunction of , where the function is defined between . The Hamiltonian operator for a 1D harmonic oscillator is Solving for the wavefunction gives Find that gives the lowest energy and compare from the trial function to the exact value, where coS We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to...
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 exact ground state energy of He is -79.0 eV. Using the variational method, you calculate an a...
The exact ground state energy of He is -79.0 eV. Using the variational method, you calculate an approximate energy to be -83.0 eV. You must have made an error because the variational method energies must A. Equal the exact ground state energy B. Be positive C. (Equal or) lie above the ground state D. (Equal or) lie below the ground state
1) Wave function for the ground state of an harmonic oscillator is given by. (x) =...
1) Wave function for the ground state of an harmonic oscillator is given by. (x) = A1/2 (a/T)1/4 e-ax /2 Evaluate the expectation value <x<> for this wave state (ove (Hint: Joo.co u² e-a u du = 2;. ue-au du = (1/2a) (Tc/a)2) pace)
Two students have a very pressing homework deadline concerning the application of the variational principle to...
Two students have a very pressing homework deadline concerning the application of the variational principle to estimate the ground state energy of the harmonic oscillator. The Hamiltonian operator of such system is î H -12d = 24 d.22 + 2 .2. in which u is the reduced mass of the oscillator and w = (force constant/u)/2 its natural frequency. The correct energies for this system are well known Eo = (v +) , v= 0,1,2, ... As the trial function...
Quantum Chemistry. Thx in Advance! 1. For a harmonic oscillator with unit mass and unit frequency,...
Quantum Chemistry. Thx in Advance!
1. For a harmonic oscillator with unit mass and unit frequency, the Schrödinger equation for its eigenfunction is given by where n 0, 1, 2, . . .. Answer the following questions. Given a trial wave function, ?(x)-?000CnUn(x), where expression for the expectation value is is assumed to be real, the Prove that Eo2 h/2 2. Assume that the trial wave function for the ground state eigenfunction in Eq. (1) is ?(x) = cos Xx,...
2. Variational method. We can approximate the true ground-state wavefunction of the harmonic -프sxs and w(x) =D0 cos(cx) in the range oscillator by the trial wavefunction p(x) = X 2c 2c outside this range. (This wavefunction is already normalized). (A) Compute the energy expectation value of b as a function of c. (B) Determine the value of c that gives the minimal energy. (C) Compare the minimal energy to the energy of the true ground-state wavefunction
2. Variational method. We...
2. Variational Principle. The energy of a system with wave function ψ is given by where H is the energy operator. The variational principle is a method by which we guess a trial form for the wave function φ, with adjustable parameters, and minimize the resulting energy with respect to the adjustable parameters. This essentially chooses a "best fit" wave function based on our guess. Since the energy of the system with the correct wave function will always be minimum...
(a) Use the variational method to estimate the ground state energy of a particle of |mass m in a potential Vx)kx, k > 0. (b) Calculate the energy shift in the ground state and in the degenerate 1t excited state of a 2-dimensional harmonic oscillator H(P2P,2/2m m(x +y)due to the perturbation V 2Axy. (20 pts)
(a) Use the variational method to estimate the ground state energy of a particle of |mass m in a potential Vx)kx, k > 0. (b)...
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...
1) Wave function for the ground state of an harmonic oscillator is given by. (x) = A1/2 (a/T)1/4 e-ax /2 Evaluate the expectation value <x<> for this wave state (ove (Hint: Joo.co u² e-a u du = 2;. ue-au du = (1/2a) (Tc/a)2) pace)
Two students have a very pressing homework deadline concerning the application of the variational principle to estimate the ground state energy of the harmonic oscillator. The Hamiltonian operator of such system is î H -12d = 24 d.22 + 2 .2. in which u is the reduced mass of the oscillator and w = (force constant/u)/2 its natural frequency. The correct energies for this system are well known Eo = (v +) , v= 0,1,2, ... As the trial function...
Quantum Chemistry. Thx in Advance!
1. For a harmonic oscillator with unit mass and unit frequency, the Schrödinger equation for its eigenfunction is given by where n 0, 1, 2, . . .. Answer the following questions. Given a trial wave function, ?(x)-?000CnUn(x), where expression for the expectation value is is assumed to be real, the Prove that Eo2 h/2 2. Assume that the trial wave function for the ground state eigenfunction in Eq. (1) is ?(x) = cos Xx,...
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