Consider an electron undergoing cyclotron motion in a magnetic field. According to Bohr, the electron’s angular momentum must be quantized in units of ℏ. Find an expression for the allowed energy levels E_n in terms of ℏ and the cyclotron frequency f_cyc.

Verified step by step guidance

1. Start by recalling the quantization condition for angular momentum according to Bohr: \( L = n\hbar \), where \( L \) is the angular momentum, \( n \) is a positive integer (quantum number), and \( \hbar \) is the reduced Planck's constant.
2. The angular momentum of a particle in circular motion is given by \( L = m_e r v \), where \( m_e \) is the mass of the electron, \( r \) is the radius of the circular motion, and \( v \) is the tangential velocity of the electron.
3. The centripetal force required for circular motion is provided by the magnetic Lorentz force: \( \frac{m_e v^2}{r} = q_e v B \), where \( q_e \) is the charge of the electron and \( B \) is the magnetic field strength. Solve for \( v \) in terms of \( r \), \( q_e \), \( B \), and \( m_e \).
4. The cyclotron frequency is defined as \( f_{\text{cyc}} = \frac{q_e B}{2\pi m_e} \). Use this to express the velocity \( v \) in terms of \( f_{\text{cyc}} \) and \( r \). Substitute this expression for \( v \) into the quantization condition \( L = n\hbar \) to find \( r \) in terms of \( n \), \( \hbar \), \( f_{\text{cyc}} \), and \( m_e \).
5. The total energy of the electron in cyclotron motion is purely kinetic: \( E_n = \frac{1}{2} m_e v^2 \). Use the expressions for \( v \) and \( r \) derived earlier to express \( E_n \) in terms of \( n \), \( \hbar \), and \( f_{\text{cyc}} \). This will give the quantized energy levels \( E_n \).

Key Concepts

Cyclotron Motion

Cyclotron motion refers to the circular motion of a charged particle, such as an electron, in a magnetic field. The magnetic force acts as a centripetal force, causing the particle to move in a circular path. The frequency of this motion, known as the cyclotron frequency, is directly proportional to the strength of the magnetic field and the charge-to-mass ratio of the particle.

Quantization of Angular Momentum

In quantum mechanics, the angular momentum of a particle is quantized, meaning it can only take on discrete values. According to Bohr's model, the angular momentum of an electron in an atom is quantized in units of ℏ (h-bar), where ℏ = h/(2π) and h is Planck's constant. This principle is crucial for determining the allowed energy levels of the electron in a magnetic field.

Energy Levels and Cyclotron Frequency

The energy levels of an electron in cyclotron motion can be expressed in terms of the cyclotron frequency (fcyc), which is the frequency at which the electron orbits due to the magnetic field. The allowed energy levels are quantized and can be derived from the relationship between angular momentum and cyclotron frequency, leading to the expression En = nℏfcyc, where n is a quantum number representing the energy level.