# Inertial Reference Frames Practice Problems

Positrons (the mass of the positron is 9.11 × 10^{-31} kg and its charge +1.6 ×10^{-19} C ) emitted during a β^{+} decay emission of carbon 11 are sent toward a neutral copper (Z = 29) foil. During the head-on-collision, the positrons elastically scattered by copper. The distance from the center of copper at which a positron stops is 1.33 × 10^{-10} m. Calculate i) the potential energy in eV of the positron when it momentarily stops. Calculate ii) the initial kinetic energy and iii) the initial speed of the positron.

A nuclear scientist fired alpha particles with 6 MeV kinetic energy directly at a gold foil. During the head-on-collision, the charged alpha particles are elastically scattered by the gold atom that remains at rest. i) Determine the minimum distance (d) that can be reached between an alpha particle and the gold nucleus (Z = 79). ii) At this minimal distance, find the force (F) acting on an alpha particle.

What is the electrical potential difference required to accelerate a doubly ionized helium atom (m_{He}= 6.64 × 10^{-27} kg) so that it has the same (i) wavelength or (ii) energy as 0.01 nm X-ray radiation?

A 2.25-g projectile is released from an air gun with a speed of 820 m/s. i) What is its de Broglie wavelength? ii) Does the projectile behave like a wave?

A proton (m_{p}=1.67 × 10^{-27} kg) and a helium nucleus (m_{He}= 6.64 × 10^{-27} kg) have the same de Broglie wavelength. The speed of the proton is 2.75 × 10^{5} m/s. Calculate the speed of the helium nucleus.

Alpha particles striking beryllium atoms cause the release of neutrons. Calculate the de Broglie wavelength of a neutron (m = 1.674 × 10^{-27} kg) released with a kinetic energy of 4.0 MeV from the beryllium disintegration reaction.

During an experiment, a physicist used a proton gun to fire protons at a speed of v. If the proton de Broglie wavelength is 0.49 nm, calculate the proton's (i) momentum (p) and (ii) kinetic energy (K). For ii), give your answer in joules and electron volts.

What is the energy, in electron volts, of an X-ray photon used in dental radiography if it has a 0.250 nm wavelength?

The carbon-14 nucleus decays, releasing an electron and an antineutrino. Suppose the electron is emitted at a speed of 7.9 × 10^{7} m/s. i) Calculate the de Broglie wavelength of the emitted electron. ii) What is the de Broglie wavelength of a positively charged hydrogen atom moving at the same speed?

Consider a vertical screen placed at 1.50 m from a narrow slit through which a horizontal beam of light passes. The wavelength of light is 560 nm and the slit width is 0.0625 mm(i) For each photon in the beam determine the least uncertainty in the vertical component of momentum. (ii) Calculate the width of the diffraction pattern for which a maximum number of photons will pass through the slit.

Determine the ionization energy of a hypothetical element that exhibits the energy levels shown in the figure below.

Draw an energy level diagram for Li^{2+}. Label the energy of the level that marks ionization on the diagram.

The figure below shows the wave function of an electron in the first excited state, Ψ _{2}(x), inside an infinite quantum well. Plot the electron probability density.

Consider a particle confined along the y direction and characterized by the following wave function:

Ψ(y)=Ψ_{0}√(1 -y^{2}/4 mm ^{2}) if |y| ≤ 2.0 mm and Ψ(y)=0 if |y| ≥ 2.0 mm.

Determine the probability of locating the particle within a distance of 0.50 mm from y=0.0 mm.

The wave function of a certain particle is given by the function:

Ψ(x) = (√(0.3) cm ^{(-1/2)}) (x / 5 cm) if |x| ≤ 5.0 cm and Ψ(x) = 0 if |x| ≥ 5.0 cm.

Determine the probability of locating the particle between x = -2.0 cm and x = +2.0 cm, and illustrate your answer by marking the corresponding region on the probability density graph.

An 18-nm long box, divided into 3-nm and 15-nm sections by a removable partition, contains an electron at its third energy level (n=2) inside the smaller region. After temporary partition removal and reinsertion, the electron is in the larger section. What's its new quantum state?

An electron has the following energies: (i) 0.60 eV, (ii) 1.40 eV, and (iii) 1.80 eV. Determine the penetration distance for this electron in a potential well of depth U_{0}=4.00 eV.

A molecule with O-H bond absorbs infrared light at a wavelength of 2.9 micrometers (μm). Calculate the energy for the first three states of vibration for this O-H bond.

Calculate the quantum number for a glycerol droplet with a radius of 1.2 μm while it moves in a 22 μm long box at a speed of 1.2 μm/s. (Assume the density of glycerol is 1261 kg/m^{3}).

An electron is confined within a one-dimensional rigid box with a length of 0.159 nm, approximately thrice the Bohr radius. Determine the three lowest electron energy levels.

Identify the wavelengths in the atom's emission spectrum due to quantum shifts between its three lowest energy states. Considering the atom as an electron trapped in a one-dimensional box of 0.159 nm, about three times the Bohr radius. Label each wavelength as λ_{n→m} to indicate the specific transition.

In a rigid box with a length of L = 1.00 nm and for the quantum state with quantum number n=4, where is the particle most likely to be found along the box's length?

In a finite potential well with a width of 2 nm and a depth of 20 eV, determine the probability ratio between finding the particle within a small interval δx at a position x=L+0.5η and finding it within the same interval at position x=L.

Consider a one-dimensional crystal lattice of copper ions with an equilibrium spacing of 0.25 nm. The copper ions have mass m, and charge e, and are arranged in a regular pattern. Determine the energies of these ions' three lowest vibrational states within the lattice. Consider the net electric force on the middle charge is given approximately by F = - e^{2}•x/[d^{3}•π•ε_{0}]**i** where d is the distance between the adjacent charges.

Within a 1D box of length 8 nm, a quantum particle exhibits energy levels: E_{n} = 17.3 eV and E_{n+1} = 23.8 eV. Create an energy-level diagram, indicating all levels from 1 to n+1, while labeling each level with its respective energy value.

Within a 1D container measuring 12 fm, a quantum particle exhibits energy levels: E_{n} = 20.8 MeV and E_{n+1} = 28.6 MeV. Create a visual representation of the wave function at the (n+1)^{th} energy level (E_{n+1}).

Consider the decay of two unstable isotopes, ^{14}C and ^{13}N. Determine the decay mode and identify the daughter nucleus for each isotope.

Polonium-218 decays by alpha emission and has a half-life of 3.1 minutes. A nuclear research laboratory with a 5.0 g sphere of ^{218}Po wishes to use the sphere for heating 2005 liters of water to generate electricity. Determine the time taken to heat the water from 20°C to its boiling point at 100°C. Useful data: m_{Po-218} = 218.008966 u, m_{Pb-214} = 213.999804 u, m_{He} = 4.002603, the specific heat of water is 4190 J/kg•°C.

A bag is filled with two radioactive isotopes: N_{A} = 40% and N_{B} = 60%. The half-life of isotope A is 6.00 × 10^{8} yr, and the half-life of isotope B is 3.20 × 10^{9} yr. After a time t, percentages of the nuclides are N_{A} = 2.70% and N_{B} = 97.3%. What is the value of t?

A particle localized in the region 0 ≤ y ≤ X obeys the wave function shown below. The value of the wave function outside these boundaries is zero. What is the probability of locating the particle in the limits 0 ≤ y ≤ X.

Particle in the 1D space −2 dm ≤ x ≤ 2 dm obeys the function ψ(y)= a√(3 - 0.5y ^{2}). If there are 162 particles in the range −2 dm ≤ x ≤2 dm, how many particles are in the range -1 dm ≤ x ≤ 1 dm?

Consider the potential-energy function U(x) of a particle, as illustrated in Figure. By solving the Schrödinger equation, it is determined that two quantum states have energies of E_{4} = 1.2 eV (for n = 4) and E _{7} = 2.8 eV (for n = 7). Redraw the Figure to include the graph's energy lines representing these n=4 and n=7 states.

A scanning tunneling microscope (STM) is a powerful tool for imaging the surface of a material down to individual atoms with high resolution. Determine by what factor the tunneling current increase if the nickel atom has a height of 0.032 nm and its work function is Φ_{w}= 5.15 eV. The barrier's width is 0.41 nm.

In an STM, the tunneling probability is very sensitive to the distance between the probe and the surface. Consider a sodium atom with a work function of Φ_{w}= 2.8 eV. The gap from the sodium surface to the STM probe is 0.47 nm. If we can accurately detect an 8.0% current change, determine the smallest height change that this STM can detect.

Carbon (^{14}C), an isotope of carbon, decays into Nitrogen ( ^{14}N), a stable isotope of nitrogen, in the reaction below via beta decay.

^{14} C → ^{14} N + e⁻ + ν_{e}

Determine the total kinetic energy (ignoring antineutrino) in the appropriate unit (MeV, GeV, TeV) of the Nitrogen (^{14}N) and the electron.

Iodine (^{131}I), which is initially at rest, decays into xenon (^{131}Xe) through the emission of a beta particle, in this case, an electron. The decay is :

¹³¹I → ¹³¹Xe + e⁻

Formulate an equation for the conservation of relativistic energy in this decay in terms of masses of iodine (mᵢ), xenon (mₓ), and the electron (mₑ), along with their relativistic factors (xenon(γₓ) and the electron (γₑ)).

The simplified (ignoring the antineutrino) equation for the decay of cobalt-60 (initially at rest) into nickel-60 is:

^{60}Co → ^{60}Ni + e⁻

Formulate an equation that represents the relativistic momentum conservation for this decay. Your equation should be in terms of speed (v).

An oscillating mass (m) is described by the wave function ψ(y) = Dyexp (−y²/d²). The wave function holds for energy levels with E = 0. Plot the wave function.

Determine the probability that a 2p electron in the hydrogen atom will be found at a distance r > a_{B} from the nucleus. The 2P electron's wave function is .

The probability density of a proton passing through a rectangular slit and hitting a detector at position y is shown below. Determine the probability that a proton hits the detector between y = 0.25 mm and y = 0.50 mm.

The wave function of the hydrogen ground state is given by: Ψ(r) = (1/√(πa₀ ^{3)}) e^{(-r/a₀) }where Ψ(r) represents the wave function, a₀ is the Bohr radius, and r is the distance from the proton. Plot the wave function and the probability density as functions of r/a₀.

The motion of a particle follows ψ(y)= a√(8 - 2y^{2}). The wave function exists in the range -2 nm≤ y ≤ 2 nm and decays to zero for other values of y. The unit of y is nm. Plot |ψ(x)|^{2} for the interval −2.5 nm≤x≤2.5 nm, giving appropriate numerical labels on the axes.

A particle restricted in the region 0 ≤ y ≤ 1.00 nm follows the wave function plotted below. The value of the wave function outside these limits is zero. Sketch a well-labeled graph of the probability density.