35. Special Relativity

CALC A particle of mass m has the wave function ψ(x) = Ax exp (−x²/a²) when it is in an allowed energy level with E = 0. Draw a graph of ψ(x) versus x.

CALC A particle of mass m has the wave function ψ(x) = Ax exp (−x²/a²) when it is in an allowed energy level with E = 0. At what value or values of x is the particle most likely to be found?

Figure 40.17 showed that a typical nuclear radius is 4.0 fm. As you’ll learn in Chapter 42, a typical energy of a neutron bound inside the nuclear potential well is E_n = −20 MeV. To find out how “fuzzy” the edge of the nucleus is, what is the neutron’s penetration distance into the classically forbidden region as a fraction of the nuclear radius?

A proton’s energy is 1.0 MeV below the top of a 10-fm-wide energy barrier. What is the probability that the proton will tunnel through the barrier?

What is the probability that an electron will tunnel through a 0.50 nm air gap from a metal to a STM probe if the work function is 4.0 eV?

The probe passes over an atom that is 0.050 nm “tall.” By what factor does the tunneling current increase?

If a 10% current change is reliably detectable, what is the smallest height change the STM can detect?

(II) Suppose that three main-sequence stars could undergo the three changes represented by the three arrows, A, B, and C, in the H–R diagram of Fig. 44–35. For each case, describe the changes in temperature, intrinsic luminosity, and size.

How much energy would be required to break a helium nucleus into its constituents, two protons and two neutrons? The rest masses of a proton (including an electron), a neutron, and neutral helium are, respectively, 1.00783 u, 1.00867 u, and 4.00260 u. (This energy difference is called the total binding energy of the ${}_2^4\text{He}$ nucleus.)

Two protons, each having a speed of 0.945c in the laboratory, are moving toward each other. Determine (a) the momentum of each proton in the laboratory, (b) the total momentum of the two protons in the laboratory, and (c) the momentum of one proton as seen by the other proton.

The mean lifetimes listed in Table 43–2 are in terms of proper time, measured in a reference frame where the particle is at rest. If a tau lepton is created with a kinetic energy of 920 MeV, how long would its track be as measured in the lab, on average, ignoring any collisions?

(I) Calculate the rest energy of an electron (m = 9.11 x 10⁻³¹ kg) in joules and in MeV (1 MeV = 1.60 x 10⁻¹³ J) .

(I) The total annual energy consumption in the United States is about 1 x 10²⁰ J How much mass would have to be converted to energy to fuel this need?

(II) What is the Schwarzschild radius for a typical galaxy (like ours)?

(II) What is the maximum sum-of-the-angles for a triangle on a sphere?