What is the kinetic energy of the external electron after the collision, and what is the frequency of the photon needed for the trapped electron to make the transition?
(a) The kinetic energy of the external electron after the collision is approximately -1.5 eV.
(b) The frequency of the photon needed for the trapped electron to make the transition is approximately 4.92 x 10^14 Hz.
Kinetic Energy Calculation:
Kinetic Energy Calculation: The kinetic energy of the external electron after the collision can be calculated using the conservation of energy principle. Initially, the external electron at rest has zero kinetic energy. The final kinetic energy is the sum of the initial kinetic energy and the work done by the electric field: E = qV, where q is the charge of the electron and V is the voltage. Given the charge of an electron (-1.6 x 10^-19 C) and the voltage (1.5 V), the kinetic energy is calculated as: KE = (-1.6 x 10^-19 C) * (1.5 V) = -2.4 x 10^-19 J. Converting this to electron volts (eV) by dividing by the elementary charge e, we get KE = -1.5 eV.
Photon Frequency Calculation:
Photon Frequency Calculation: The energy difference between the ground state and the first excited state in a 1D box is determined by the formula: ΔE = (n^2 * h^2) / (8 * m * L^2), where n is the quantum number, h is Planck's constant, m is the mass of the electron, and L is the length of the box. For the transition from the ground state to the first excited state (n = 2), the energy difference is calculated as: ΔE = (2^2 * (6.626 x 10^-34 J.s)^2) / (8 * (9.109 x 10^-31 kg) * (1 x 10^-9 m)^2) = 3.26 x 10^-19 J. Converting this energy to frequency by dividing by Planck's constant, we get: f = ΔE / h = (3.26 x 10^-19 J) / (6.626 x 10^-34 J.s) ≈ 4.92 x 10^14 Hz.
Therefore, the kinetic energy of the external electron after the collision is approximately -1.5 eV, and the frequency of the photon required for the trapped electron to undergo the same transition is approximately 4.92 x 10^14 Hz.