“I know this world is ruled by infinite intelligence. Everything that surrounds us – everything that exists – proves that there are infinite laws behind it. There can be no denying this fact. It is mathematical in its precision.”

– Thomas A. Edison

A free response practice question involving on atomic physics and quantum effects was given to you in the post dated 23^{rd} January 2011. As promised, I give below a model answer along with the question:

An electron and its *antiparticle* positron, moving with the same speed, undergo a head-on collision and get annihilated to produce two photons of equal wave length. This process is called *pair-annihilation*. Positron has the same mass as the electron, but its charge is positive.

(a) Determine the rest energy of a positron in electron volt.

(b) Determine the *maximum* wave length of the photons generated in the electron-positron pair annihilation.

(c) Which one of the following statements [(i), (ii) and (iii)] is correct regarding the direction of motion of the two photons? Put a tick (√) mark against the correct choice.

(i) The photons move along the same direction……..

(ii) The photons move along opposite directions……..

(iii) The photons move along perpendicular directions……..

Justify your answer to part (c) above.

(d) Explain why *two* photons (and not a single one) have to be produced in pair annihilation.

(e) If the particles undergoing pair annihilation were proton and its antiparticle *antiproton* (instead of electron and positron), what can you say about the maximum wave length (*λ*_{1})* *of the photons generated, in comparison with the value obtained in the case of electron-positron pair annihilation? Put a tick (√) mark against the correct answer choosing from the options (i), (ii) and (iii) given below:

(i) *λ*_{1} is increased…….

(ii) *λ*_{1} is the same…….

(iii) *λ*_{1} is decreased…….

Justify your answer.

(a) If the rest mass of the positron is *m*_{0} and the speed of light in free space is *c*, the rest energy *E* of the positron, as given by Einstein’s mass-energy relation, is

*E = m*_{0}*c*^{2} = (9.11×10^{– 31} kg)(3×10^{8} ms^{–1})^{2} = 8.20×10^{– 14 }J

[The mass of the positron is the same as that of the electron]

Since one electron volt is equivalent to 1.6×10^{– 19 }J, the energy in eV is given by

*E =* (8.20×10^{– 14 }J)/(1.6×10^{– 19 }J) = 5.12×10^{5} eV.

(b) The wave length of the photon generated in pair annihilation will be *maximum* if the electron and positron have *minimum *energy which is equal to the rest energy. Therefore, the photon of maximum wave length has energy equal to 8.20×10^{– 14 }J.

Since *E = hc/λ* where *h* is Planck’s constant and *λ* is the wave length of the photon, we have

*λ = hc/E*

Therefore, maximum wave length, *λ*_{max}* =* (6.63×10^{– 34} Js×3×10^{8} ms^{–1})/(8.20×10^{– 14 }J)

Or, *λ*_{max}* =* 2.43×10^{–12 }m

(c) Statement (ii) is correct.

The photons have to move along *opposite *directions since the total momentum of the two photon system must be zero. This follows from the law of conservation of momentum. The total momentum of the electron-positron system was zero since they moved with the *same* speed in *opposite *directions

[Note that the magnitudes of the momenta of the photons generated are the same since they have the same wave length *λ* (Remember de Broglie relation *p = h/**λ* where *p* is the momentum].

(d) A photon can never be at rest and hence it has momentum. The condition of zero total momentum for the system can be satisfied only if there are two photons (at least) so that the momentum of one can be nullified by that of the other.

(e) The correct choice is (iii).

The rest mass of the proton (or anti-proton) is much greater than that of the electron (or positron) and hence the rest energy is greater. The energy of the generated photon is greater and so the wave length is smaller.

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