Question

(a) Derive the conditions for the validity of the WKB approximation for the one-dimensional time-independent Schrodinger equation, and show that the approximation must fail in the immediate neighborhood of a classical turning point. 

(b) Explain, using perturbation theory, why the ground state energy of an atom always decreases when the atom is placed in an external electric field.

Answer 1

(a) The WKB method starts from the Schrodinger equation

(3) and (4) are the conditions for the validity of the WKB method. Integration of Eq. (5) gives

Near a turning point V(x) \(\sim\) E, p → 0 and (6) is not satisfied. Hence the WKB method cannot be used near classical turning points.

(b) Consider an atom in an external electric field \(\varepsilon\) in the z direction. The perturbation Hamiltonian is

H’ = -e\(\varepsilon\)z,

where z = \(\sum_iz_i\) is the sum of the z coordinates of the electrons of the atom, and the energy correction is

As z is an odd operator and the parity of the ground state is definite, H'₀₀ = 0. Futhermore E₀ - E_n < 0. Hence \(\Delta\)E₀ < 0. This means that the energy of the ground state decreases when the atom is placed in an electric field.