SUMMARY
The discussion centers on calculating the frequency of a free electron using two different approaches. The first method employs the momentum-wavelength relationship \( p = \frac{h}{\lambda} \) and results in the equation \( f = \frac{mv^2}{h} \). The second method utilizes kinetic energy \( E = \frac{1}{2} mv^2 \) and Planck's relation \( E = hf \), leading to \( f = \frac{mv^2}{2h} \). The discrepancy arises from the distinction between phase velocity and group velocity, as highlighted in Griffiths' "Introduction to Quantum Mechanics".
PREREQUISITES
- Understanding of quantum mechanics principles, specifically wave-particle duality.
- Familiarity with the concepts of phase velocity and group velocity.
- Knowledge of momentum and energy equations in physics.
- Ability to interpret equations from quantum mechanics literature, such as Griffiths' textbook.
NEXT STEPS
- Study Griffiths' "Introduction to Quantum Mechanics", specifically Section 2.4, Equation 2.90.
- Research the differences between phase velocity and group velocity in quantum mechanics.
- Explore the implications of momentum and energy relationships for free electrons.
- Learn about the applications of Planck's relation in various quantum systems.
USEFUL FOR
Students and professionals in physics, particularly those studying quantum mechanics, as well as educators seeking to clarify concepts related to electron behavior and wave-particle duality.