EP301 Quantum Mechanics II • Variational principle is also called Rayleigh-Ritz apporoximation • Have discussed the essential features of this approximation • Helps to get a good estimate(upper bound) on ground state energy for systems not solvable exactly
Properties of the ground state wavefunction • Ground state is non degenerate- how to prove ? • Take a rotational invariant system • Take complex wavefunction • Expectation value of V(r) is independent of phase of the wavefunc • Evaluate kinetic energy- choose phase factor such that it is minimum
Proof f (r ) >=0 Ground state is nodeless and non-degenerate
Corollaries • If ground state is degenerate, then we must find orthogonal function g( r ) >=0 which is not possible • If the potential obeys V(r) = V(-r) then the ground state function f (r) >=0 implies only symmetric wavefunction
• Did you check expectation value of Ze 2 / r in hydrogen like atom is 2Z 2 E 0 • Expectation value of e 2 /|r 1 - r 2 | in for Helium ground state in first order perturbation theory
Helium Using the dimensionless var
where
Double harmonic oscillator Symmetric potential
Variational estimate
Work out for n=0
Particle with V(x)= g|x| • Solution is Airy function Ai[(2m g/h 2 ) 1/3 (x-E 0 /g)] with ground state energy • E 0 = 0.809 (g 2 h 2 / 2m) 1/3 • Take a normalised Gaussian trial function • Show that <H> = 0.813 (g 2 h 2 / 2m) 1/3
Recommend
More recommend
