Question

A one-dimensional infinite square-well potential is given by:
\(v(x)=0\,\,for\,\,-\frac{a}{2}<x<+\frac{a}{2}\)= ∞ elsewhere
Let 𝐸_𝑒 (𝑥) and ψ_𝑒 (𝑥) be the ground state energy and the corresponding wave function, respectively, if an electron (e) is trapped in that well. Similarly, let 𝐸_µ(𝑥) and ψ_µ (𝑥) be the corresponding quantities, if a muon (µ) is trapped in the well. Choose the correct option:

• A.
• B.
• C.
• D.

Answer 1

The Correct Option is C

In the case of a particle in a one-dimensional infinite square well, the energy levels are quantized, and the energy depends on the mass of the particle. The energy levels are given by:

\[ E_n = \frac{n^2 \pi^2 \hbar^2}{2 m a^2} \] 

where:

• \( n \): Quantum number (e.g., \( n = 1 \) for the ground state)
• \( m \): Mass of the particle
• \( a \): Width of the square well

Comparison of Electron and Muon:

• The mass of the muon (\( m_\mu \)) is greater than the mass of the electron (\( m_e \)).
• Since the energy \( E_n \) is inversely proportional to the mass (\( m \)), the energy levels for the muon will be lower than those for the electron.

Conclusion:

Therefore, the ground state energy of the electron will be greater than the ground state energy of the muon.