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

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

View Full Document Right Arrow Icon
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
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.
Ask a homework question - tutors are online