Understanding Wavelength and Kinetic Energy
In quantum mechanics, the wavelength of a particle is inversely related to its momentum, which is affected by its kinetic energy. The relationship between kinetic energy (K) and wavelength (λ) can be understood through the de Broglie wavelength formula.
De Broglie Wavelength Formula
- The de Broglie wavelength (λ) is given by the formula:
λ = h / p
where h is Planck's constant and p is the momentum of the particle.
- Momentum (p) for a particle is defined as:
p = √(2mK)
where m is the mass of the particle and K is its kinetic energy.
Effects of Increased Kinetic Energy
- When the kinetic energy is increased to 4K, the new momentum becomes:
p' = √(2m(4K)) = √(8mK) = 2√(2mK) = 2p
- Since the new momentum is doubled, we can substitute this into the de Broglie wavelength formula:
λ' = h / p' = h / (2p) = λ / 2
Conclusion
- Therefore, when the kinetic energy of the particle is increased to 4K, the new wavelength λ' becomes:
λ' = λ / 2
- This shows that the wavelength is halved when the kinetic energy is quadrupled, illustrating the profound relationship between energy and wave properties in quantum mechanics.