Nondegenerate TimeIndependent Perturbation Theory
Joseph C. Brossett
Historical Background
• In 1900, Max Planck introduces quantum
theory to explain thermal radiation.
• In 1926, Erwin Schrodinger introduces
wave (quantum) mechanics and his
famous equation.
• Also in 1926, Max Born develops a
probabilistic interpretation of Schrodinger’s
wave functions.
Fundamentals of Quantum Theory
• The Schrodinger Equation: (-(h/2pi)2/2m * d2/dx2
+ V) * Psin = En * Psin
• The Statistical Interpretation: integrating the
magnitude of Psi(x,t)2 (complex conjugation)
from x to x+dx gives the probability of finding the
particle in this domain at time t
• Probability: the magnitude of Psi(x,t)2 can be
though of as a probability density between x and
x+dx
Fundamentals of Quantum Theory
• The expectation value of any observable is
the value of the probability (the integral of
|Psi(x,t)|2) with that observable in the
integral for the probability.
• In Dirac notation, this is <Q> =
<Psi|Q|Psi> with Q some observable.
Elementary Unperturbed Systems
• The infinite square well where V(x) =
infinity for all x other than 0 <= x <= a
(where V(x) = 0) with a the length of the
well
• The harmonic oscillator where V(x) =
(½)mx2ω2
• The hydrogen atom where V(r) = -(1/4piE0)
* 1/r
Perturbation Theory
• A procedure for obtaining approximate
solutions to a perturbed system by altering
(perturbing) the exact solutions to the
unperturbed case of the system.
• E1n = <Psi0n|H-|Psi0n> where E1n is the firstorder correction to the nth energy state
• Psi1n = the sum on m not equal to n of
(<Psi0m|H- |Psi0n> / (E0n – E0m)) * Psi0m
• Corresponding higher-order corrections
An Example: The Infinite Square
Well
• For the unperturbed infinite square well, the
energies are given by E00 = (n2 pi2 (h/2pi)2 ) /
(2ma2 ) with a again the width of the well.
• The Psins are given by Psin = (2/a) sin (kx).
• Perturbing the well by “raising the floor” to some
potential V0, a constant, we have the correction
to the ground state energy given by E10 =
<Psi00|V0|Psi00> = V0 .