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.

مواضيع ذات صلة

مصطلحات الفيزياء النووية

إزاحة حمراء REDSHIFT

المركبات النانوية (Nanocomposites)

مولد فان دي جراف الالكتروستاتي

التطويرات التقنية الحديثة في المعجلات الالكتروستاتية

معجلات الالكترونات الالكتروستاتية

معجلات الفان دي جراف الترادفية

الحد من زيادة الطاقة في معجل فان دي جراف

أنبوبة التعجيل قي معجلات الجسيمات من نوع فان دي جراف

ارتكاز معجل فان دي جراف

مبدأ تشغيل معجل الفان دي جراف

القضايا الأخلاقية المتعلقة بتكنولوجيا النانو