Problems are listed in approximate order of difficulty. A single dot (•) indicates straigh...

Problems are listed in approximate order of difficulty. A single dot (•) indicates straightforward problems involving just one main concept and sometimes requiring no more than substitution of numbers in the appropriate formula. Two dots (••) identify problems that are slightly more challenging and usually involve more than one concept. Three dots (•••) indicate problems that are distinctly more challenging, either because they are intrinsically difficult or involve lengthy calculations. Needless to say, these distinctions are hard to draw and are only approximate.

••• An atom (or a nucleus) is ordinarily found in its ground state, or state of lowest energy. However, by supplying some energy, one can lift it to an excited state. An excited state of an atom X is sometimes denoted X* and, because of the additional energy ΔE, has a mass m* slightly greater than that of the ground state (m):

 m* = m + Δm

where Δm = ΔE/c2. If left in isolation, the excited state X* usually drops back to the ground state emitting a single photon.

 X* → X + γ

If the atom were immovable, the photon would carry off all the additional energy.

 Eγ = Δmc2

In reality, conservation of momentum requires the atom to recoil, so that a little of the energy goes to kinetic energy of the recoiling atom. (a) Using conservation of momentum and energy and assuming that the excited atom X* was at rest, show that

(b) The energy needed to lift a hydrogen atom from its ground state to the lowest excited state is 10.2 eV. Evaluate the fraction Δm/m* for this stale. (Does it make a significant difference whether you use m or m* in the denominator?) What percentage of the available energy Δmc2 goes to the photon in the decay of this excited state? (Your result illustrates what a good approximation it usually is to assume that all the energy goes to the photon in an atomic transition.)