Quantum Mechanics II Variational principle is also called - - PowerPoint PPT Presentation

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

Ze2 / r in hydrogen like atom is 2Z2 E0

• Expectation value of e2 /|r1 - r2 | in

for Helium ground state in first

• rder perturbation theory

Helium

Using the dimensionless var

where

Double harmonic

• scillator

Symmetric potential

Variational estimate

Work out for n=0

Particle with V(x)= g|x|

• Solution is Airy function

Ai[(2m g/h2 )1/3(x-E0/g)] with ground state energy

• E0 = 0.809 (g2 h2 / 2m)1/3
• Take a normalised Gaussian trial

function

• Show that <H> = 0.813 (g2 h2 / 2m)1/3