35. Special Relativity

A particle is described by the wave function mm where L = 2.0 mm. Determine the normalization constant c.

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. A ²³⁸U nucleus, which decays by alpha emission, is 15 fm in diameter. Model an alpha particle within a ²³⁸U nucleus as being in a one-dimensional box. What is the maximum speed an alpha particle is likely to have?

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?

The electrons in a rigid box emit photons of wavelength 1484 nm during the 3→2 transition. What kind of photons are they—infrared, visible, or ultraviolet?

The electrons in a rigid box emit photons of wavelength 1484 nm during the 3→2 transition. How long is the box in which the electrons are confined?

A 16-nm-long box has a thin partition that divides the box into a 4-nm-long section and a 12-nm-long section. An electron confined in the shorter section is in the n = 2 state. The partition is briefly withdrawn, then reinserted, leaving the electron in the longer section of the box. What is the electron’s quantum state after the partition is back in place?

A finite potential well has depth U₀ = 2.00 eV. What is the penetration distance for an electron with energy (a) 0.50 eV, (b) 1.00 eV, and (c) 1.50 eV?

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 An electron is confined in a harmonic potential well that has a spring constant of 2.0 N/m. What wavelength photon is emitted if the electron undergoes a 3→1 quantum jump?

INT An electron is confined in a harmonic potential well that has a spring constant of 12.0 N/m. What is the longest wavelength of light that the electron can absorb?

Use the data from Figure 40.24 to calculate the first three vibrational energy levels of a C=O carbon-oxygen double bond.

An electron approaches a 1.0-nm-wide potential-energy barrier of height 5.0 eV. What energy electron has a tunneling probability of (a) 10%, (b) 1.0%, and (c) 0.10%?

CALC Suppose that ψ₁(x) and ψ₂(x) are both solutions to the Schrödinger equation for the same potential energy U(x). Prove that the superposition ψ(x)=Aψ₁(x) + Bψ₂(x) is also a solution to the Schrödinger equation.

A 2.0-μm-diameter water droplet is moving with a speed of 1.0 μm/s in a 20-μm-long box. Estimate the particle’s quantum number.

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.