Math Review Practice Problems

A quantum particle has energy levels in a 1D box of length 0.87 nm: E _n = 12.5 eV and E _n+1 = 18.9 eV. Find the particle's mass and also identify the particle.

Use the energy level diagram for Li²⁺ to show transitions corresponding to emissions that can be observed for the energy level, n = 4.

A proton has a de Broglie wavelength equal to the radius of a ²⁰⁰Hg nucleus. Determine the energy of the proton in MeV.

Mass is converted to energy when a low-energy electron and its anti-particle (positron) annihilate each other and produce at least two photons. If the particle's initial kinetic energy is 0.231 MeV, what should the value for the energy, frequency, and wavelength of each photon be, if the collision is head-on? 

Evenly distributed free protons occupy the range 0 ≤ y ≤ 5 m. No protons are detectable outside this range. Calculate the probability density at y = 2.5 m.

An alpha particle (He-nucleus) is propelled toward a potassium nucleus (Z = 19). The diameter and energy levels of the nucleus are depicted in the figure below. Initially, the potassium nucleus is in its ground state. However, after the collision, it emits a photon with a wavelength of approximately 64.91 fm. Determine the minimum initial speed of the alpha particle. Hint: Consider the principle of energy conservation while analyzing the alpha particle-nucleus interaction.

A bioluminescent fish emits a light pulse with a peak intensity of 480 nm. Each light pulse lasts for a duration of 60 ms and has a power output of 2.7 mW. Assuming that all light is emitted at this specific peak wavelength, determine what the number of photons emitted during one pulse will be.

In a photoelectric effect experiment where tungsten is used as the cathode, a photoelectric current is detected when the incident light frequency is above a frequency threshold, f₀. Determine f₀. The work function of tungsten is 4.66 eV.

Consider a hydrogen-like atom with three energy levels. The energies of these levels are measured to be 1.50 eV, 2.75 eV, and 3.25 eV. Visualize the energy-level diagram for this atom and provide appropriate labels indicating the energy values and their corresponding quantum numbers. The ground state energy is defined as zero.

In an experiment, a physicist adjusts the electron's speed to achieve specific de Broglie wavelengths: (i) 2.0 μm, (ii) 2.0 nm, and (iii) 2.0 fm. Determine the required speed for each wavelength.

Consider a lithium atom with an energy of 2.0 eV in a finite potential well below the U₀. Determine the distance at which an atom enters the classically forbidden region.

Consider an unstable atomic nucleus that experiences alpha decay and releases 6.05 MeV energy. When the mass of the parent nucleus and the resulting daughter nucleus are added together, it amounts to 420 u. Can you determine the original parent nucleus mass before it decayed?

An isotope of ²⁴Na₁₁ undergoes beta-minus decay. What is the total energy released (in MeV)?

Calculate the radiation dose in Grays if a 120g tumor in the brain requires 0.30 J of X-ray radiation during a radiotherapy session. 

During a geological survey, an 80 kg field researcher is accidentally exposed to 40 µJ of naturally occurring radon radiation. Calculate the dose equivalent in mrem.

A nuclear research institute wishes to have a collision between a 32 MeV (at impact) alpha particle and a stationary polonium-209. Determine the kinetic energy of the alpha particle at the instant it is fired toward the polonium nucleus. Take the polonium nuclei to be rest at all times, and do not treat the alpha particle like a point particle.

Suppose two neon atoms (²⁰Ne) combine to form a calcium atom (⁴⁰Ca); how much energy is emitted? The binding energy per nucleon in ²⁰Ne and ⁴⁰Ca is 8.0 MeV and 8.6 MeV respectively. 

Determine the age of a lignin fossil that has a ¹⁴C/¹²C ratio of 2.52 × 10 ^-13.

A radioactive material has two types of radioactive atoms, W and Y. At t = 0, the number of atoms is related by N_W = 15 N_Y. After a duration of 3 hours, the numbers are N_W = N_Y. If t_1/2 for W is 0.65 hours, what is t_1/2 for Y?

An atom of Polonium-210 is placed within a magnetic field of strength 0.85 T. During its radioactive decay process, it releases an alpha particle. This alpha particle moves in a direction that is perpendicular to the magnetic field B, creating a circular path with a diameter of 0.80 m. Determine the energy, in MeV, that the alpha particle possessed at the time it was emitted.

The nucleus of a gold atom ¹⁹⁷Au (atomic number = 79) has a diameter of 16.0 fm. Determine the speed required for a proton to be fired towards the gold nucleus such that it has a turning point 3.0 fm from the surface. Assume the nucleus remains stationary.

Determine the nuclear radius of a theoretical element ²⁵⁰Y

Atomic nuclei have diameters commonly measured in fm (10^-15). Which stable nuclei have a 7.851 fm diameter?

Determine the binding energy per nucleon (in MeV) for isotopes ⁶⁰Ni and ⁶⁰Co. Consider that the proton and neutron masses are 1.00783 amu and 1.00866 amu, respectively.

Consider a pair of nucleons separated by 1.5 fm with a nuclear potential energy of -40 MeV. Use the potential energy graph below to estimate the ratio of the gravitational potential energy to nuclear potential energy.

Demonstrate the energy-level diagrams for the A=10 nuclei of Beryllium and Carbon. Include all the occupied neutron and proton levels in your diagram.

A nuclear engineer studies a spherical sample of Plutonium-239 (²³⁹Pu) with a 19,816 kg/m³ density. The sample exhibits an activity of 1.3 Ci. Given that the half-life of Plutonium-239 is 24,000 years, what is the radius of the Plutonium-239 sphere?

An unknown isotope Y decay equation is Y → ⁵⁰ Ti + e⁺ + v. Identify the unknown isotope Y in the following decay.

A quantum particle moves freely in the positive x-direction following the equation ψ(x,t) = A[e^i(k₁^{x - ω}₁^t) + e^i(k₂^{x - ω}₂^t)]. Suppose k₂ = 4k₁ = 4k and a maximum of the probability distribution function |ψ(x,t)|² occurs at t = 0 and x = 0. i) Given that ω = ħk²/2m, determine the least positive value of x that gives a maximum in the probability distribution function at t = 4π/ω. ii) Use the result in part i) to determine the average speed in the negative x-direction of the probability distribution function. Also, determine the average speed using v_av = (ω₂ - ω₁)/(k₂ - k₁).

The probability of detecting a particle with a normalized wave function ψ(y) within the limits of y and y + dy is given by (|ψ|²)dy. The normalized wave function for a particle in a box with boundaries at y = 0 and y = L is ψ_n(y)=√(2/L)•sin⁡[(nπy/L)]. The possible values of n are n = 1,2,3... Assuming the particle to be in the ground state, i) At which values of y in the range 0 ≤ y ≤ L the particle is impossible to be found? ii) At which values of y within the boundary the particle is most likely to be found? iii) Do your values agree or differ with these curves?

During a nuclear plant accident, a 25 kg dog becomes accidentally contaminated with 0.13 Ci of the released radioactive strontium . The ingested strontium substitutes the calcium in the bones since the two elements are chemically identical and can be stored for a long period of time. Consider that the half-life is 29.1 years, the energy absorbed by the bones from each decay equals 0.546 MeV, and the electrons' relative biological effectiveness (RBE) factor is 1.25. i) Over the course of two weeks, what is the absorbed dose in rad and its equivalent in rem? ii) The energy released during the decay is higher than 0.546 MeV. Explain why some of the energy of the decay is not absorbed by the bones.

Consider a photon found in the light emitted by a laser device that has a momentum of 9.56 × 10^-28 kg•m/s. Calculate the energy of the photon in Joules (J) and electron-volts (eV).

The wave function ψ(x,t) = A[e^{i(kx - ωt)} - e^{i(3kx - 9ωt)}] is a one-dimensional function of a free particle in space. The constants k and ω are positive and real. i) Determine the least positive value of x that gives a maximum in the probability function |ψ(x,t)|² at t = 0. ii) Determine the least positive value of x that gives a maximum in the probability function |ψ(x,t)|² at t = 4π/ω.

The wave function ψ(x,t) = A[e^{i(2kx - 4ωt)} - e^{i(4kx - 16ωt)}] describes a particle oscillating in a one-dimensional path. Constants k and ω are real and positive. Determine v_av using the approach where v_av is the ratio of the distance moved by the maxima divided by the change in time. Hint: Use values of t in the form nπ/ω.

Suppose has solutions ψ_a and ψ_b, and their respective energies are E_a and E_b. However, E_a ≠ E_b. For the non-zero constants C and D, determine if ψ = Cψ_a + Dψ_b satisfies to be a solution of the equation, giving a reason for your choice.

Assuming the Bohr model of the atom, an electron in Li²⁺ emits a photon of wavelength 13.5 nm for the transition n = 2 to n = 1. i) Treat the atom like an electron in a one-dimensional box. Find the width of a box that would allow the emission of a photon of wavelength 13.5 nm. ii) Determine the ground state energy of a box with the width you calculated in part i). iii) Comparing spacing between adjacent energy levels, state whether a one-dimension box is a suitable model for the Bohr atomic model. 

When an electron in a hydrogen atom transitions from the excited state (n = 5) to the ground state (n = 1), the wavelength of the light it gives off is 432 nm. Active Galactic Nuclei (AGN) is a small area at the center of some galaxies that give off massive amounts of energy and is extremely bright. Assume that the 432 nm line in this distant galaxy has redshifted to 620.3 nm. Find out how many light years this galaxy is from Earth.

The H_γ radiation coming from a moving galaxy is measured at a wavelength of 451 nm. This wavelength is redshifted 17 nm from the H_γ wavelength coming from a source on earth. Determine the distance of the galaxy from the earth.

In a certain game, there are five potential outcomes. The options are noted as A, B, C, D, and E. The occurrence probabilities for options A, B, and C are all identical and equal to 15%. However, the probability of option D is four times the probability of option E. Determine the probabilities of options D and E.

Consider a bead of radius 0.60 mm that reciprocates between two barriers due to perfectly elastic collisions. The two barriers are 80 mm apart. What is the probability of finding the center of the bead at a random moment in time, precisely at the midpoint between the barriers?

A cube of length 2.5 cm is confined in a one-dimensional space given by y = 0 to y = 80 cm. The cube bounces back and forth in the boundaries of this space. The magnitude of the velocity remains constant before and after bouncing. At an arbitrary instant, determine the probability that the cube's center lies in the range 38 cm ≤ y ≤ 42 cm.

Free protons with even distribution are confined in the region of space ranging from 0 ≤ y ≤ 4 cm. No protons are detected outside this region. Suppose 1.0 × 10 ⁵ protons are in the entire range; how many protons are located in the range 0.65 cm to 0.72 cm?

Imagine an experiment being performed in a laboratory to study the photoelectric effect. Some researchers observe the emission of photoelectrons when they illuminate a metallic surface with light wavelengths below 350 nm. Determine what the work function will be for this metallic surface used in this particular experiment.