The correct option is **B. λ = h/P**. ### Detailed Explanation: 1. **Understanding the Equation**: - The equation \( \lambda = \frac{h}{P} \) relates the wavelength (\( \lambda \)) of a matter wave to its momentum (\( P \)). Here, \( h \) is Planck's constant, which is a fundamental constant in quantum mechanics, approximately equal to \( 6.626 \times 10^{-34} \, \text{Js} \). - This equation is derived from the de Broglie hypothesis, which states that all matter exhibits wave-like properties. According to de Broglie, the wavelength of a particle is inversely proportional to its momentum. 2. **Momentum and Wavelength**: - Momentum (\( P \)) for a particle is defined as \( P = mv \), where \( m \) is the mass and \( v \) is the velocity of the particle. Thus, as the momentum increases, the wavelength decreases, indicating a wave-particle duality. 3. **Why Option B is Correct**: - The equation \( \lambda = \frac{h}{P} \) directly illustrates the wave nature of matter, showing how particles can exhibit wave-like behavior through their wavelength. This is a cornerstone of quantum mechanics and is essential for understanding phenomena such as electron diffraction and the behavior of particles at the quantum level. ### Analysis of Other Options: - **Option A: \( \lambda = \frac{hc}{E} \)**: - This equation relates the wavelength of a photon to its energy (\( E \)). While it is correct for photons (light particles), it does not apply to matter waves in the same way that de Broglie's equation does. Therefore, it does not demonstrate the particle-wave nature of matter itself. - **Option C: \( \lambda = \frac{c}{f} \)**: - This equation describes the relationship between the speed of light (\( c \)), frequency (\( f \)), and wavelength (\( \lambda \)) of electromagnetic waves. While it is valid for light, it does not pertain to the wave nature of matter, which is what the question is asking about. - **Option D: \( \lambda = 2d \sin \theta \)**: - This equation is related to the interference pattern of waves, particularly in the context of diffraction and interference in a double-slit experiment. It does not directly relate to the wave nature of matter in the context of de Broglie's hypothesis. ### Summary of Key Points: - The correct equation demonstrating the particle and wave nature of matter is \( \lambda = \frac{h}{P} \). - This equation shows that as the momentum of a particle increases, its wavelength decreases, illustrating wave-particle duality. - Other options either pertain to light (photons) or describe wave behavior in different contexts, not specifically related to matter waves. - Understanding these relationships is crucial for grasping fundamental concepts in quantum mechanics and the behavior of particles at microscopic scales. ### Revision Summary: - **Correct Answer**: B. \( \lambda = \frac{h}{P} \) (de Broglie wavelength). - **Wave-Particle Duality**: Matter exhibits both particle and wave characteristics. - **Planck's Constant**: Fundamental in relating wavelength to momentum. - **Other Options**: Relate to light or specific wave phenomena, not matter waves.