Hektas Ticaret TAS, Kemeralti cad 28, Karakoy, 8030 Istanbul atillag@hektas.com

For a time independent potential V(xyz) the most general solution of the Schrodinger equation with y _i the energy eigenfunctions is:

Y (t) = S a_k y _i e^{-i w k t}

a_k are constant and accordingly d/dt <E> = 0. However the energy of an accelerated charged wave packet must dissipate by radiation so that <E> must decrease in time . To achieve this one must allow the coefficients a_k to change in time.

We will consider first a simple case of a particle with the elementary charge e in a one dimensional potential box of length s ( 0<x<s ) with infinitely high potential walls . Lets consider the superposition of the ground state y ₁ and the first excited state y ₂ that belong to the energy eigenvalues E₁ and E₂ respectively

Y (x,t) = a₁ y ₁(x) e^{-iw 1t} + a₂ y ₂(x) e^{-iw 2t} (2)

The charge density e|Y | ² can be calculated for real y _k and real a_k

e|Y |² = a₁² e|y ₁|² + a₂² e|y ₂|² + 2a₁a₂ ey _{2y 1} cos(w ₂ - w ₁)t (3)

Snapshots of Y for a₁ = a₂ = 2^-1/2 are shown in Figure 1.

It looks, as if the "charge cloud" makes an oscillation between the potential walls. We will use the classical oscillating dipole as an approximation in our model. The radiation power P_r of a classical radiating dipole is [1]

P_r = (p² w ⁴ ) / (12p e ₀ c³ ) (4)

Where p is the dipole moment, w is the oscillation frequency, e ₀ is the dielectric constant of vacuum and c is light velocity. The length of the dipole is approximately the distance between two peaks which is approximately s/2. According to (3) The oscillating portion of the electric charge q is proportional to a₁a₂ so that we write q = ka₁a₂ with 0<k<2. The dipolemoment is then approximately given by (5)

p = (s/2) ka₁a₂ e (5)

Using (5) in (4) we obtain for the emitted energy per time

P_r = K a₁² a₂² (6)

where K = (k²s²e² w ⁴ ) / (48p e ₀ c³) _.

The energy that is transferred to the electromagnetic field during radiation can be supplied only by the decrease of the energy of the particle.

- d/dt <E> = P_r (7)

Putting (6) in (7) using <E> = a₁² E₁ + a₂² E₂ and eliminating a₁² using a₁² + a₂² = 1 , the equation (7) becomes a differential equation for a₂²(t) with ^D E = E₂ - E₁

- D E d/dt (a₂² ) = K [a₂² (1 - a₂²) ] (8)

We can solve (8) by separating the variables and obtain

a₂² (t) = 1/(1 + Ce^{Kt/D E} ) (9)

Only the solutions with C >= 0 satisfy the condition |a₂| <= 1 and are physically meaningful . a₂²(t) and a₁²(t) = 1 - a₂²(t) for C = 1 are plotted in figure 2.

If there are many states with lower energy, a quasiindeterminacy emerges as we will show below. Let's generalize (6) and denote P_ik the Energy dissipation rate by radiation for a transition from an energy eigensstate |i> to energy eistate |k> in a superposition as in (1) . We assume that P_ik is a function of |a_i| and |a_k| .

P_ik = f_ik(|a_i|, |a_k| ) (10)

By f_ik we mean that the functional form can be different for each ik pair. What is common for all transitions is that f_ik(|a_i|, |a_k| ) has such a form that it is 0 for a_i = 0 or for a_k = 0. The rate of change of |a_i| is determined by an increase caused by the inflow from higher energy states and a competing decrease caused by the outflow to lower energy states. For n states in consideration we have n equations each with n-1 terms . The equation system should satisfy following two conditions :

1. the normalization should be preserved.

d/dt S |a_i|² = 0 (11)

2.Energy conservation condition (7) must be satisfied

- d/dt S E_i |a_i|² = S f_ik (|a_i|, |a_k| )

This can be achieved by generalizing (8)

d|a_i|²/dt = S f_ik (|a_i|, |a_k| )/E_k-E_i (13)

                    k, k¹ i

the terms with E_k > E_i are positive giving the inflow to state |i> and negative for E_k < E_i giving the outflow from the state |i> .

To verify that (13) satisfies (11) we have to add left and right sides of all equations seperately . To obtain (12) we have to multiply each equation with E_i before adding left and right sides seperately . We have to take into account that for a given pair of states |r> |s> we have two terms containing f_ik(|a_r|, |a_s|) in the equation system ; the term f_ik (|a_r|, |a_s|) / E_s-E_r in the equation for d/dt|a_r|² and the term f_ik (|a_r|, |a_s|) / E_r-E_s in the equation for d/dt|a_s|².

For 3 states the time development of |a_i| have been calculated numerically with a spreadsheet program using the approximation (6) f_ik(|a_i|, |a_k| ) = |a_i|² |a_k|² (with K = 1 to see the qualitative behavior) for energies E₁=2, E₂= 6, E₃ = 8. Starting with initial values for |a_i|² , D |a_i|² for a unit time step D t are calculated by (14) and next |a_i|^{2 '}s are obtained by adding D |a_i|² to the previous |a_i|² 's and so on .

D |a_i|² = S |a_i|² |a_k|² / E_k-E_i (14)

                 k, k¹ i

The results, obtained for two slightly differing initial states in close neighborhood of a₃ = 1 are plotted in Figure 3. Figure 3a represents the direct transition from |3> to |1> while figure 3b represents a transition first from |3> to |2> and then from |2> to |1>

[1] David J.Griffiths, Introduction to Electrodynamics, (Prentice-Hall 1989) 2^th ed., pp 405

Figure 1

The time evolution of |Y |² given as snapshots at different instants. Y is here the superposition of the ground state (w ₁) and the first excited state (w ₂) in a box potential, each of equal weight in superposition . T = 2p / (w ₂ - w ₁) is the period of the oscillation .

Figure 2

The time evolution of the coefficients a₁ and a₂ as a consequence of radiation for a particle of charge e in a potential box of length s. a₁ and a₂ are the contribution of ground state and first excited state to the superposition respectively . K = (k²s²e² w ⁴ ) / (48p e ₀ c³) and D E = E₂ - E₁ , k is a constant close to 2, w = D E/h.

Figure 3

The time evolution of the coefficients |a₁| |a₂| and |a₃| as a consequence of radiation for a superposition of 3 different energy eigenstates with a₁ the contribution of the lowest one, a₂ the contribution of the next higher one and a₃ the contribution of the highest one of the considered energy eigenstates . The calculations are made for two slightly differing initial compositions both in close neighborhood of a₃ = 1 . The unit of time axis is given as one calculational step of (14).

Figure 3 a for |a₁|²(t₀) = ,0009999999 |a₂|²(t₀) = 0,0000000001 and |a₃|²(t₀) = 0,999 and

Figure 3b for |a₁|²(t₀) = ,00000001 |a₂|²(t₀) = 0,00000999 and |a₃|²(t₀) = 0,99999