# homework2 - 4 Use the variational method with wave...

This preview shows page 1. Sign up to view the full content.

4. Use the variational method with wave functions of the form φ α ( r )= r a π 2 1 r 2 + a 2 and φ α ( r e r/a π a 3 to estimate the ground state energy of a particle subject to a Coulomb potential V ( r e 2 /r. Note that the second wave function above reduces, for the correct value of a, to the exact ground state wave function for the Coulomb problem. 5. Evaluate to f rst order the energy shifts in the spectrum of the harmonic oscillator Hamiltonian H 0 = 1 2 ~ ω q 2 p 2 ) due to a perturbation ˆ V = λ ˆ q 4 .(H e r q and ˆ p are dimensionless position and momentum variables, such that [ˆ q, ˆ p ]= i . See Chapter 3 of the class notes for details, and for useful techniques for calculating expectation values of powers of ˆ q =( a + a + ) / 2 . ) 6. A system with Hamiltonian H (0) is subject to a perturbation H (1) , which in a certain ONB can be repre- sented by the following matrices h H (0) i = ε 0 000 0 04 ε 0 00 0 6 ε 0 0 8 ε 0 0 0 1 0 ε 0 h H (1) i = γ 0
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 12/19/2010 for the course PHYSICS ph 463 taught by Professor Paule. during the Fall '10 term at Missouri S&T.

Ask a homework question - tutors are online