1. Perturbation Theory
Only a few QM systems that can be solved exactly: Hydrogen Atom(simplified), harmonic oscillator, infinite+finite well
solve using perturbation theory which starts from a known solution and makes successive approx- imations
start with time independent. V’(x)=V(x)+v(x)
V(x) has solutions to the S.E. and so known eigenvalues and eigenfunctions
let perturbation v(x) be small compared to V(x)
As yl form complete set of states (linear algebra)
Sometimes Einstein convention used. Implied sum if 2 of same index
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2. Plug into Schrod. Eq. know solutions for V use orthogonality
multiply each side by wave function* and integrate
matrix element of potential v is defined:
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3. One solution: assume perturbed wave function very close to unperturbed (matrix is unitary as “size” of wavefunction doesn’t change)
assume last term small. Take m=n. Energy difference is expectation value of perturbing potential
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4. Redo compact notation eigenvalues/functions for a “base” Hamiltonian
want to solve (for l small)(l keeps track or order)
define matrix element for Hamiltonian H
finite or infinite dimensional matrix. If finite (say 3x3) can use diagonalization techniques. If infinite can use perturbation theory
write wavefunction in terms of eigenfunctions but assume just small change
P460 - perturbation

5. compact notation- energy
look at first few terms (book does more)
which simplifies to (first order in l)
rearranging
take the scalar product of both sides with
first approximation of the energy shift is the expectation value of the perturbing potential
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6. compact notation- wavefunction
look at wavefunction
and repeat equation for energy
take the scalar product of both sides with
gives for first order in l
note depends on overlap of wavefunctions and energy difference
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7. Time independent example
know eigenfunctions/values of infinite well.
Assume mostly in ground state n=1
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8. Time independent example
Get first order correction to wavefunction
only even Parity terms remain (rest identically 0) as
gives
Even Parity
P460 - perturbation