2. Variational method. We can approximate the true ground-state wavefunction of the harmonic -프sxs and w(x) =D0 cos(cx...
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...
1. Variational method In this problem, you will approximate the ground state wave function of a quantum system using the variational theory. Use the trial wave function below 2 cos/T) , 1x1 trial a/2 to approximate the ground state of a harmonic oscillator given by 2.2 2 using a as an adjustable parameter. (a) Calculate the expectation value for the kinetic energy, (?) trial 4 points (b) Calculate the expectation value for the potential energy, Virial. Sketch ??tria, (V)trial, 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...
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...
Questions 3-5 3. The predecessor to Hartree-Fock was the Hartree method, where the main difference is that the Hartree-Fock method includes an trial wavefunction by writing it as a Slater Determinant, while the Hartree method uses a simple product wavefunction that does not capture anti- symmetry. In particular, for a minimal-basis model of, the Hartree method's trial wavefunction is given in the while the Hartree-Fock trial wav is given by where and are molecular orbitals, and and coordinates of electron...
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?...
4.2 The potential energy in a MOFSET device near the metal oxide interface is approximately V(x) - qEx forx > 0 where q is the electron charge, and E is the electric field strength. Use the variational technique to estimate the ground state energy of an electron in this configuration. (Hints: a) use the un-normalized trial function ф(x)-x exp(-ax2)). b) Find the normalized trial wave-function c) Compute the energy functional (i.e. the expectation value of the Hamiltonian for the state...
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...
For the ground state of a quantum harmonic oscillator, given by ?_0 = (?^(-1/2)*?^(-1/4))*?^(-(x^2)/(2?^2)) , show that the expectation values for potential and kinetic energy are equal.
If you were to use a trial function of the form φ(x):: (1 + cax 2)r"r/2, where α (kμ/h2)1/2 and c is a variational parameter, to calculate the ground-state energy of a harmonic oscillator, what do you think the value ofc will come out to be? Why? 8-6.