2. Using the trial wave function calculate the ground state energy of the Hamiltonian (34 pts.)
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...
2. The unational method is an incredibly simple but surprisingly powerful method for understanding the low- energy behavior of quantu systems. It is used constantly in marny-body physics and in quantum chemistry. The main idea is thst for any physical Hamiltonian, there is a lowest energy state, i.e. the ground state Ipo). All other states (ignoring degeneracy) have higher energy that this one. Therefore we have Therefore, to get an upper bound on the energy of Eo, it suffices to...
Consider the Hamiltonian P2 V8 (tk)? exp (–2(k:X)2) H = 2 m (6.63) where k > 0. We consider the following trial wave function y = A exp (a) Is this trial wave function justified? Give reasons. (b) Find the expectation value for the kinetic energy and the potential energy. Express your answers in hk2 multiples of 2m (c) Compute the upper bound for the ground state energy.
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...
Problem 8 (30 pts). The ground state wave function for the hydrogen atom is: W... (7,0,) - (a, 15pts) Find (-2) for an electron in this state. Find <x> and <x>
Problem 8 (30 pts). The ground state wave function for the hydrogen atom is: W... (1,0,0) - (a, 15pts) Find (-) for an electron in this state. Find <x> and <p>
( 25 marks) The wave function for a hydrogen atom in the ground state is given by \(\psi(r)=A e^{-r / a_{s}}\), where \(A\) is a constant and \(a_{B}\) is the Bohr radius. (a) Find the constant \(A\). (b) Determine the expectation value of the potential energy for the ground state of hydrogen.
8-4. Use a trial function of the form φ(x)-1/(1 + β?) to calculate the ground-state energy of a harmonic oscillator. The necessary integrals are (2n-3)(2n-5)(2n-7) . . . (1) π -w (1 + β?)" (2n-2)(2n-4)(2n-6) . . . (2) β1/2 and oo x2dx (2n-5)(2n-7) (1) π n2 3 -oo (1 + f3x2)" (2n-2)(2n-4) . . . (2) β3/2 8-4. Use a trial function of the form φ(x)-1/(1 + β?) to calculate the ground-state energy of a harmonic oscillator. The necessary...
Need solution to part 2, Hamiltonian keeps breaking down to zero 3(a) The normalised ground state of a one dimensional system in a simple harmonic potential V(x) = aur2 is where α- /mw/ћ and the normalisation constant is given as A-(o2/7) i. Compute the expectation value of the potential energy 〈V〉 in the ground state by explicit integration using the standard integral oO expadz 1x3x5.2n 2na(2n+1)' ii. Show that po is an eigenfunction of the Hamiltonian operator and the correspond-...