35. Special Relativity

Our Milky Way galaxy is 100,000 ly in diameter. A spaceship crossing the galaxy measures the galaxy’s diameter to be a mere 1.0 ly. How long is the crossing time as measured in the galaxy’s reference frame?

Through what potential difference must electrons be accelerated if they are to have:

(a) the same wavelength as an x ray of wavelength $0.220$ nm; and

(b) the same energy as the x ray in part (a)?

A cosmic ray travels 60 km through the earth's atmosphere in 400 μs, as measured by experimenters on the ground. How long does the journey take according to the cosmic ray?

All the very heavy atoms found in the earth were created long ago by nuclear fusion reactions in a supernova, an exploding star. The debris spewed out by the supernova later coalesced into the gases from which the sun and the planets of our solar system were formed. Nuclear physics suggests that the uranium isotopes ²³⁵U and ²³⁸U should have been created in roughly equal numbers. Today, 99.28% of uranium is ²³⁸U and only 0.72% is ²³⁵U. How long ago did the supernova occur?

An electron has a de Broglie wavelength of $2.80\times {10}^{-10}$ m. Determine (a) the magnitude of its momentum and (b) its kinetic energy (in joules and in electron volts).

It might seem strange that in beta decay the positive proton, which is repelled by the positive nucleus, remains in the nucleus while the negative electron, which is attracted to the nucleus, is ejected. To understand beta decay, let's analyze the decay of a free neutron that is at rest in the laboratory. We'll ignore the antineutrino and consider the decay n → p⁺ + e⁻. The analysis requires the use of relativistic energy and momentum, from Chapter 36. Write the equation that expresses the conservation of relativistic energy for this decay. Your equation will be in terms of the three masses m_n, m_p and m_e and the relativistic factors y_p and y_e.

A rocket is fired from the earth to the moon at a speed of 0.990c. Let two events be 'rocket leaves earth' and 'rocket hits moon.' In the earth's reference frame, calculate ∆x, ∆t, and the spacetime interval s for these events.

What is the probability of finding a 1s hydrogen electron at distance r > a_B from the proton?

A particle of mass m travels at a speed v = 0.32 c. At what speed will its momentum be doubled?

What is the momentum of a proton traveling at v = 0.72c?

The graph in FIGURE EX40.15 shows the potential-energy function U(x) of a particle. Solution of the Schrödinger equation finds that the n = 3 level has E₃ = 0.5 eV and that the n = 6 level has E₆ = 2.0 eV. Redraw this figure and add to it the energy lines for the n = 3 and n = 6 states.

INT Model an atom as an electron in a rigid box of length 0.100 nm, roughly twice the Bohr radius. Calculate all the wavelengths that would be seen in the emission spectrum of this atom due to quantum jumps between these four energy levels. Give each wavelength a label λ_n→m to indicate the transition.

There is evidence that low-energy x rays have an RBE slightly greater than 1. Suppose that 10 keV photons with an RBE of 1.2 are used to make a chest x ray. A 60 kg person receives a 0.30 mSv dose from a chest x ray that exposes 25% of the patient's body. How many x ray photons are absorbed in the patient's body?

Which of the following quantities definitely changes when the reference frame changes in classical mechanics?