Hydrogen in Capacitor

A hydrogen atom in its ground state is placed between the parallel plates of a capacitor. For times t < 0, no voltage is applied. Starting at t = 0,an electric field E(t) = E₀e^-t/τ is applied, where τ is a constant. Derive the formula for the probability that the electron ends up in state j due to this perturbation. Evaluate the result for j:

a) a 2s state

b) a 2p state

SOLUTION

For time-dependent perturbations a general wave function is

(1)

where the ѱ_j satisfy

(2)

For the time-dependent perturbation V(t),

(3)

From Schrodinger’s equation we can derive an equation for the time development of the amplitudes a_j (t):

(4)

(5)

If the system is initially in the ground state, we have a_1S (0) = 1 and the other values of a_j (0) are zero. For small perturbations it is sufficient to solve the equation for j ≠ 1S:

(6)

(7)

The general probability P_j that a transition is made to state j is given by

(8)

This probability is dimensionless. It should be less than unity for this theory to be valid.

a) For the state j = 2S the probability is zero. It vanishes because the matrix element of z is zero: ⟨2S|z|1S⟩ = 0 because of parity. Both S-states have even parity, and z has odd parity.

b) For the state j = 2P the transition is allowed to the L = 1, M = 0 orbital state, which is called 2P_z. The matrix element is similar to the earlier problem for the Stark effect. The 2P eigenstate for L = 1, S = 0 and that for the 1S state is exp  The integral is

(9)

where a₀ is the Bohr radius of the hydrogen atom.