Question

The muon has the same charge as an electron but a mass that is 207 times greater. The negatively charged muon can bind to a proton to form a new type of hydrogen atom. How does the binding energy \(E_{\mathrm{B} \mu}\) of the muon in the ground state of a muonic hydrogen atom compare with the binding energy \(E_{\mathrm{Be}}\) of an electron in the ground state of a conventional hydrogen atom? a) \(\left|E_{\mathrm{B} \mu}\right| \approx\left|E_{\mathrm{Be}}\right|\) d) \(\left|E_{\mathrm{B} \mu}\right| \approx 200 \mid E_{\mathrm{Be}}\) b) \(\left|E_{\mathrm{B} \mu}\right| \approx 100\left|E_{\mathrm{Be}}\right|\) e) \(\left|E_{\mathrm{B} \mu}\right| \approx\left|E_{\mathrm{Be}}\right| / 200\) c) \(\left|E_{\mathrm{B} \mu}\right| \approx\left|E_{\mathrm{Be}}\right| / 100\)

Short answer

a) |E(Bo)| ≈ 10 |E(Be)| b) |E(Bo)| ≈ 50 |E(Be)| c) |E(Bo)| ≈ 100 |E(Be)| d) |E(Bo)| ≈ 200 |E(Be)| Answer: d) |E(Bo)| ≈ 200 |E(Be)|

Step-by-step solution

Calculate the reduced mass for the muon-proton system

To start, we need to calculate the reduced mass (\(\mu\)) for the muon-proton system. This is given by the formula: $$\mu = \frac{m_\mu m_p}{m_\mu + m_p}$$ where \(m_\mu\) is the mass of the muon, and \(m_p\) is the mass of the proton. Given that the muon is 207 times heavier than an electron, we can rewrite this formula as: $$\mu = \frac{207m_e m_p}{207m_e + m_p}$$

Replace proton mass with electron mass in the formula

Since the mass of the proton is much larger than the mass of the electron (approximately 1836 times larger), we can simplify this expression to: $$\mu \approx \frac{207m_e m_p}{m_p}$$ $$\mu \approx 207m_e$$

Calculate the muonic hydrogen binding energy

Now we can calculate the binding energy for the ground state of the muonic hydrogen atom using the Rydberg Formula for reduced mass: $$E_{B\mu} = -\frac{13.6\,\text{eV} \times \mu}{m_e}$$ Therefore, we have: $$E_{B\mu} = -\frac{13.6\,\text{eV} \times 207m_e}{m_e}$$

Simplify the expression and compare binding energies

Now, we can simplify the expression as: $$E_{B\mu} = -13.6\,\text{eV} \times 207$$ We know that the binding energy of an electron in a conventional hydrogen atom is \(E_{Be} = -13.6\,\text{eV}\). Comparing these energies, we have: $$\left|E_{B\mu}\right| \approx 207\left|E_{Be}\right|$$ This answer is not provided in the options, since the factor should be within a range 200-210. Based on this, the closest answer is option (d): d) \(\left|E_{\mathrm{B} \mu}\right| \approx 200 \mid E_{\mathrm{Be}}\)

Key concepts

Binding Energy

In the context of atoms, binding energy refers to the energy required to disassemble a whole system into separate parts. For a hydrogen atom, it is the energy needed to remove an electron from the proton's attraction. In a conventional hydrogen atom, the binding energy of the electron is famously known as 13.6 eV. This value is derived from the electrostatic attraction between the proton and the electron.
In a muonic hydrogen atom, the scenario changes due to the presence of the muon instead of the electron. As the muon's mass is 207 times that of an electron, the force of attraction between the muon and the proton is considerably stronger. This leads to an increased binding energy in the muonic hydrogen atom. Following calculations using the reduced mass and Rydberg Formula, it's found that this binding energy is roughly 200 times that of the regular hydrogen atom. This indicates that the muonic hydrogen atom is held together much more tightly than its conventional counterpart.

Reduced Mass

The concept of reduced mass arises when studying systems of two interacting particles, like the electron-proton system in hydrogen or the muon-proton system in muonic hydrogen. It accounts for the fact that both particles are moving, not just the electron or muon. The formula for reduced mass ( \(\mu\) ) is: \[\mu = \frac{m_1 m_2}{m_1 + m_2}\] where \(m_1\) and \(m_2\) are the masses of the two particles.
In the muon-proton system, \(m_1\) is 207 times the mass of the electron, making the reduced mass roughly \(207m_e\). This change affects the energy levels and binding energy of the atom, as heavier particles (like muons) alter the balance between the two particles. Calculating binding energy requires knowing this reduced mass, since it changes due to the replacement of the electron with a much heavier muon.

Rydberg Formula

The Rydberg Formula is an essential tool in quantum mechanics for determining the binding energies of hydrogen-like atoms, such as hydrogen and muonic hydrogen. It is originally used to determine the spectral lines of the hydrogen atom, but it's also instrumental in calculating energy levels. The formula depends on the reduced mass of the system: \[E = - \frac{13.6 \,\text{eV} \times \mu}{m_e}\] Here, \(E\) represents the energy of a particular state, \(\mu\) is the reduced mass, and \(m_e\) is the electron mass.
In a muonic hydrogen atom, since \(\mu\) is significantly larger (207 times), the binding energy is also larger. The Rydberg Formula shows how this increased reduced mass leads to much stronger binding between the proton and the muon, resulting in a binding energy approximately 200 times that of standard hydrogen atoms.

Muon-Proton System

The muon-proton system forms the basis of a muonic hydrogen atom. In this system, a muon replaces the electron, forming a structure similar to a hydrogen atom. The muon shares qualities with the electron, such as charge, but its greater mass (207 times heavier) gives this system unique properties.
Because of this increased mass, the orbit of the muon around the proton is much closer compared to the electron's orbit. This proximity leads to vastly different characteristics, especially in terms of binding energy and energy levels, compared to regular hydrogen.

• It exhibits much stronger binding energy.
• The energy levels are also different, due to altered forces between the proton and the muon, a result of the muon's greater mass.

Understanding the muon-proton system helps in grasping the concept of muonic atoms, their unique position in atomic physics, and their applications in studying fundamental forces and tests of quantum electrodynamics.