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Chapter 4: Q48E (page 137)
The p0 is a subatomic particle of fleeting existence. Data tables don't usually quote its lifetime. Rather, they quote a "width," meaning energy uncertainty, of about 150MeV. Roughly what is its lifetime?
The lifetime is .
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Classically and nonrelativistically, we say that the energyof a massive free particle is just its kinetic energy. (a) With this assumption, show that the classical particle velocityis. (b) Show that this velocity and that of the matter wave differ by a factor of 2. (c) In reality, a massive object also has internal energy, no matter how slowly it moves, and its total energyis, where.Show thatisand thatisIs there anything wrong with it? (The issue is discussed further in Chapter 6.)
In Exercise 45, the case is made that the position uncertainty for a typical macroscopic object is generally so much smaller than its actual physical dimensions that applying the uncertainty principle would be absurd. Here we gain same idea of how small an object would have♦ to be before quantum mechanics might rear its head. The density of aluminum is , is typical of solids and liquids around us. Suppose we could narrow down the velocity of an aluminum sphere to within an uncertainty ofper decade. How small would it have to be for its position uncertainty to be at least as large asof its radius?
According to the energy-time uncertainty principle, the lifetimeof a state and the uncertaintyin its energy are invertible proportional. Hydrogen's red spectral line is the result of an electron making a transition "downward" frum a quantum state whose lifetime is about.
(a) What inherent uncertainty in the energy of the emitted person does this imply? (Note: Unfortunately. we might use the symbol for the energy difference-i.e., the energy of the photon-but here it means the uncertainly in that energy difference.)
(b) To what range in wavelength s does this correspond? (As noted in Exercise 2.57. the uncertainty principle is one contributor to the broadening of spectral lines.)
(c) Obtain a general formula relating .
What is the range of frequencies in a 1 ns pulse of
(a) 1060nmInfrared laser light and
(b) 100MHzRadio waves?
(c) For which is the "uncertainty" in frequency, relative to its approximate value, larger?
Generally speaking, why is the wave nature of matter so counterintuitive?
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