Photon Box

An empty box of total mass M with perfectly reflecting walls is at rest in the lab frame. Then electromagnetic standing waves are introduced along the x direction, consisting of N photons, each of frequency v₀ (see Figure 1.1).

Figure 1.1

a) State what the rest mass of the system (box + photons) will be when the photons are present.

b) Show that this answer can be obtained by considering the momentum and/or energy of the box-plus-photon system in any inertial frame moving along the axis.

SOLUTION

a) Consider the initial state of the system. Write the 4-momentum of the box and photons as p^μ_box and p^μ_ph, respectively:

(1)

where we have used the fact that for a standing wave (which can be represented as the sum of traveling waves with opposite momenta) the total momentum is zero. Therefore, from (1), the 4-momentum of the system P^μ is given by

(2)

From (2), we can find the rest mass of the total system M (which is defined by M² c² = P^μ P_μ)

(3)

b) Transform the 4-momentum by going into a frame moving with velocity –v along the x axis. We have in this frame for energy ε' and momentum P'

where ε and P are the total energy and momentum in the rest frame, respectively. So

Therefore in the moving frame

and

We expect this to be true, of course, since mass is a relativistic invariant under a Lorentz transformation.

Another way to look at it is to consider a transformation of energy and momentum of the photons and the box separately. The frequencies of these photons will be Doppler shifted:

The energy of the photons

The energy of the box ε'_box = γM₀ c². The momentum of the photons

The momentum of the box

So the 4-momentum is the same as found above

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