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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)

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perturbation theory
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

P460 - perturbation

plug into schrod eq
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:

P460 - perturbation

slide3
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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P460 - perturbation

redo compact notation
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

compact notation energy
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

P460 - perturbation

compact notation wavefunction
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

P460 - perturbation

time independent example
Time independent example
  • know eigenfunctions/values of infinite well.
  • Assume mostly in ground state n=1

P460 - perturbation

time independent example8
Time independent example
  • Get first order correction to wavefunction
  • only even Parity terms remain (rest identically 0) as
  • gives

Even Parity

P460 - perturbation