Question

To illustrate the variational method, use the ground state, normalized Harmonic Oscillator Wavefunction: ψ(x) = (α/π)^(1/4) * e^(-αx^2/2) a. Using this as the "guess" for the wavefunction, find the expectation value of the energy and find an expression for α which minimizes this energy. b. Show that it is in fact the same final energy expression as that given by solving the Schrödinger Equation. Just look up the equation, don't solve the Schrödinger Equation.

Name: to illustrate the variational method use the ground state normalized harmonic oscillator wavefunction x 14 e x22 a using this as the guess for the wavefunction find the expectation value of 57782
Uploaded: 2022-06-26T18:18:45-08:00
Duration: 2 min 53 s
Channel: Suman K
Description: to illustrate the variational method use the ground state normalized harmonic oscillator wavefunction x 14 e x22 a using this as the guess for the wavefunction find the expectation value of 57782

          To illustrate the variational method, use the ground state, normalized Harmonic Oscillator Wavefunction:

ψ(x) = (α/π)^(1/4) * e^(-αx^2/2)

a. Using this as the "guess" for the wavefunction, find the expectation value of the energy and find an expression for α which minimizes this energy.

b. Show that it is in fact the same final energy expression as that given by solving the Schrödinger Equation. Just look up the equation, don't solve the Schrödinger Equation.

Added by Elizabeth O.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Suman K

06/26/2022

Step 1

The expectation value of the energy is given by: E = ∫ψ*(x) * Ĥ * ψ(x) dx Where Ĥ is the Hamiltonian operator for the Harmonic Oscillator, which is given by: Ĥ = -ħ^2/(2m) * d^2/dx^2 + (1/2) * mω^2x^2 Substitute the given wavefunction into the expression for Show more…

Show all steps

Thanks for your feedback!

To illustrate the variational method, use the ground state, normalized Harmonic Oscillator Wavefunction: ψ(x) = (α/π)^(1/4) * e^(-αx^2/2) a. Using this as the "guess" for the wavefunction, find the expectation value of the energy and find an expression for α which minimizes this energy. b. Show that it is in fact the same final energy expression as that given by solving the Schrödinger Equation. Just look up the equation, don't solve the Schrödinger Equation.