{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# pert - Time Independent Multi-Level Perturbation Theory...

This preview shows pages 1–4. Sign up to view the full content.

Time Independent, Multi-Level Perturbation Theory Given solutions H E n n n ( ) ( ) ( ) ( ) 0 0 0 0 ψ ψ = (1) Find the perturbed energy and wavefunctions. Noting that λ will be set to 1, the perturbed Hamiltonian and wavefunction are H H H H = + + + ( ) ( ) ( ) ... 0 1 2 2 λ λ (2) ψ ψ λψ λψ n n n n = + + + ( ) ( ) ( ) ... 0 1 2 2 (3) which has energies E E E E n n n n = + + + ( ) ( ) ( ) ... 0 1 2 2 λ λ (4)

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
the Schroedinger equation is H H H E E E n n n n n n n n n ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ... ... ... ... 0 1 2 2 0 1 2 2 0 1 2 2 0 1 2 2 + + + + + + = + + + + + + λ λ ψ λψ λψ λ λ ψ λψ λψ (5) multiplying-out, collecting terms in λ λ ψ ψ λ ψ ψ ψ ψ λ ψ ψ ψ ψ ψ ψ 0 0 0 0 0 1 0 1 1 0 1 1 0 2 0 2 1 1 2 0 0 2 1 1 2 0 0 H E H H E E H H H E E E n n n n n n n n n n n n n n n n n n ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ... - + + - - + + + - - - + = n s n s n s (6) since λ is arbitrary, each power of λ is zero separately, giving H E n n n ( ) ( ) ( ) 0 0 0 ψ ψ = (7) H H E E n n n n n n ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 1 1 0 0 1 1 0 ψ ψ ψ ψ + = + (8) H H H E E E n n n n n n n n n ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 2 1 1 2 0 0 2 1 1 2 0 ψ ψ ψ ψ ψ ψ + + = + + (9)
Write eq. (8) in bra-ket notation H n H n E n E n n n ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 1 1 0 0 1 1 0 + = + (10) and multiply by n ( ) 0 n H n n H n n E n n E n n n ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 0 1 0 1 0 0 0 1 0 1 0 + = + (11) n E n n H n n E n E n n n n n ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) | 0 0 1 0 1 0 0 0 1 1 0 0 + = + (12) n H n E n ( ) ( ) ( ) ( ) 0 1 0 1 = (13) i.e. first-order correction to energy

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 10

pert - Time Independent Multi-Level Perturbation Theory...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online