35. Special Relativity

What is the difference in the wavelengths emitted in a 199→2 transition and a 200→2 transition?

Find the radius of the electron’s orbit, the electron’s speed, and the energy of the atom for the first three stationary states of He+.

A hydrogen atom in its fourth excited state emits a photon with a wavelength of 1282 nm. What is the atom's maximum possible orbital angular momentum (as a multiple of ℏ ) after the emission?

Consider the electron wave function where x is in cm. Determine the normalization constant c.

A muon—a subatomic particle with charge −e and a mass 207 times that of an electron—is confined in a 15-pm-long, one-dimensional box. ( 1pm=1picometer=10⁻¹² m.) What is the wavelength, in nm, of the photon emitted in a quantum jump from n = 2 to n = 1?

Heavy nuclei often undergo alpha decay in which they emit an alpha particle (i.e., a helium nucleus). Alpha particles are so tightly bound together that it’s reasonable to think of an alpha particle as a single unit within the nucleus from which it is emitted. The probability that a nucleus will undergo alpha decay is proportional to the frequency with which the alpha particle reflects from the walls of the nucleus. What is that frequency (reflections/s) for a maximum-speed alpha particle within a ²³⁸U nucleus?

An electron in a one­-dimensional box has ground ­state energy $2.00$ eV. What is the wavelength of the photon absorbed when the electron makes a transition to the second excited state?

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.' Repeat your calculations of part a if the rocket is replaced with a laser beam.

(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?

An ultrashort pulse has a duration of $9.00$ fs and produces light at a wavelength of $556$ nm. What are the momentum and momentum uncertainty of a single photon in the pulse?

Suppose you put five electrons into a 0.50-nm-wide one-dimensional rigid box (i.e., an infinite potential well). What is the ground-state energy—that is, the total energy of all five electrons in the ground-state configuration?

A laser emits 1.0 × 10¹⁹ photons per second from an excited state with energy E₂ = 1.17 eV. The lower energy level is E₁ = 0 eV. What is the wavelength of this laser?

CALC Determine the normalization constant A1 for the n = 1 ground-state wave function of the quantum harmonic oscillator. Your answer will be in terms of b.

How many electrons, protons, and neutrons are contained in the following atoms or ions: (a) ¹⁰B, (b) ¹³N⁺, and (c) ¹⁷O⁺⁺⁺?

Consider a wave function given by $\psi \left(x\right)=Asinkx$, where $k=2\pi /\lambda$ and $A$ is a real constant.

(a) For what values of $x$ is there the highest probability of finding the particle described by this wave function? Explain.

(b) For which values of $x$ is the probability zero? Explain.