Question

A muon is a particle very similar to an electron. It has the same charge but its mass is \(1.88 \cdot 10^{-28} \mathrm{~kg}\). a) Calculate the reduced mass for a hydrogen-like muonic atom consisting of a single proton and a muon. b) Calculate the ionization energy for such an atom, assuming that the muon is initially in its ground state.

Short answer

Question: Calculate the reduced mass and ionization energy for a hydrogen-like muonic atom consisting of a single proton and a muon. Answer: The reduced mass of the muonic atom is approximately \(1.74 \cdot 10^{-28}\) kg, and the ionization energy for this atom is approximately -207.9 eV.

Step-by-step solution

Calculate the reduced mass of the muonic atom

To calculate the reduced mass, we will use the following formula: \[ \mu = \frac{m_1 \times m_2}{m_1 + m_2} \] where \(m_1\) and \(m_2\) are the masses of the two interacting particles (muon and proton). Given that the mass of the muon is \(1.88 \cdot 10^{-28}\) kg, and the mass of the proton is \(1.67 \cdot 10^{-27}\) kg, we can now calculate the reduced mass: \[\mu = \frac{(1.88 \cdot 10^{-28} \mathrm{~kg})(1.67 \cdot 10^{-27} \mathrm{~kg})}{(1.88 \cdot 10^{-28} \mathrm{~kg}) + (1.67 \cdot 10^{-27} \mathrm{~kg})}\]

Compute the reduced mass of the muonic atom

Now compute the reduced mass: \[\mu \approx 1.74 \cdot 10^{-28} \mathrm{~kg}\]

Compute the ionization energy for the muonic atom

To compute the ionization energy for a hydrogen-like muonic atom, we will use the following formula: \[ E_n = -\frac{Z^2 \times R \times \mu}{m_\mathrm{e} \times n^2} \] where \(E_n\) is the energy of the nth level, \(Z\) is the atomic number, \(R\) is the Rydberg constant, \(\mu\) is the reduced mass, \(m_\mathrm{e}\) is the electron mass, and \(n\) is the principal quantum number (which is 1 for the ground state). For our muonic atom, we have \(Z = 1\), \(\mu = 1.74 \cdot 10^{-28}\ \mathrm{kg}\), \(m_\mathrm{e} = 9.11 \cdot 10^{-31}\ \mathrm{kg}\), and the Rydberg constant, \(R = 13.6\ \mathrm{eV}\). Now, compute the ionization energy for the ground state (n = 1): \[E_1 = -\frac{(1)^2 \times (13.6\ \mathrm{eV}) \times (1.74 \cdot 10^{-28}\ \mathrm{kg})}{(9.11 \cdot 10^{-31}\ \mathrm{kg}) \times (1)^2}\]

Calculate the ionization energy

Now calculate the ionization energy for the muonic atom: \[E_1 \approx -207.9\ \mathrm{eV}\] The ionization energy for this hydrogen-like muonic atom is approximately -207.9 eV. The negative sign indicates that it takes 207.9 eV to remove a muon from the ground state to ionize the atom.

Key concepts

Reduced Mass Calculation

Understanding the concept of reduced mass is crucial when analyzing systems with two bodies interacting under mutual forces, such as in the case of a muonic atom. The reduced mass, denoted as \(\mu\), is a measure that simplifies two-body problems by reducing them to equivalent one-body problems. This is especially helpful in quantum mechanics and celestial mechanics.

The calculation for reduced mass involves the masses of the two particles in question. If we have two particles with masses \(m_1\) and \(m_2\), the reduced mass \(\mu\) is given by the equation:\[ \mu = \frac{m_1 \times m_2}{m_1 + m_2} \]In practical terms, this means that in a hydrogen-like muonic atom, where a muon orbits around a proton similar to how an electron would in a regular hydrogen atom, we use the masses of the proton and the muon for \(m_1\) and \(m_2\) to calculate the reduced mass. This calculation gives us a single mass with which to compute properties like energy levels more conveniently, as shown in the textbook solution for muonic atoms.

Hydrogen-like Atoms

Hydrogen-like atoms are a special set of atoms that consist of a nucleus with a single electron orbiting it. In essence, these are single-electron systems and can include ions where other electrons have been removed, leaving a single electron behind. A great example of such an atom that deviates from the norm is the muonic atom, which, instead of an electron, has a heavier particle - a muon - in orbit around the nucleus.

Since muons share the same charge as electrons but have a much higher mass, hydrogen-like muonic atoms display unique properties. Their behavior is governed by quantum mechanics, and despite their increased mass, the muon in a muonic atom follows similar laws and principles that apply to an electron in a regular hydrogen atom, albeit with some deviations in calculated values such as energy levels due to the difference in mass.

Rydberg Constant

The Rydberg constant is a fundamental quantity in spectroscopy and quantum mechanics, symbolized by \(R\). It represents the limit of the highest wavenumber (inverse wavelength) of any photon that can be emitted from the hydrogen atom or hydrogen-like elements. The significance of the Rydberg constant lies in its role in calculating the wavelengths of spectral lines of hydrogen.

Its value is given by:\[ R = 13.6\ \mathrm{eV} \]When referring to a hydrogen-like atom, the Rydberg formula needs to be adjusted to incorporate the reduced mass of the system. This modification results in a more accurate prediction of the energy levels in these systems. As such, in any system with hydrogen-like properties, whether it's dealing with a traditional electron or a muon as in muonic atoms, the Rydberg constant remains a key component for calculating energy-related properties of the atoms.

Quantum Energy Levels

Quantum energy levels are discrete energy values that an electron (or in the case of muonic atoms, a muon) can occupy in an atom. According to the Bohr model and the more advanced quantum mechanical models, these particles cannot possess arbitrary energy but are limited to specific levels.

For hydrogen-like atoms, these energy levels are dictated by the principal quantum number \(n\) and can be calculated using the formula:\[ E_n = -\frac{Z^2 \times R \times \mu}{m_\mathrm{e} \times n^2} \]In this equation, \(E_n\) represents the energy of the state corresponding to the principal quantum number \(n\), \(Z\) is the atomic number, \(R\) is the Rydberg constant, \(\mu\) is the reduced mass, \(m_\mathrm{e}\) is the electron mass (or the mass of the orbiting particle, such as a muon). These levels are critical in determining the spectral lines and also the ionization energy required to remove the electron or muon from its ground state, as in our exercise with muonic atoms.