Question

Which of the following statements is false? (1) de Broglic wavelength associated with matter particle is inversely proportional to momentum. (2) de Broglic equation is a relationship between a moving particle and its momentum. (3) de Broglie equation suggests that an electrons has particle wave nature. (4) de Broglie equation is applicable to electrons only.

Short answer

Statement 4 is false.

Step-by-step solution

Identify de Broglie Equation

The de Broglie equation is given by \ \[ \lambda = \frac{h}{p} \ \], where \ \lambda \ \ is the wavelength, \ \h \ \ is Planck's constant, and \ \p \ \ is the momentum of the particle.

Analyze Statement 1

Statement 1 states that the de Broglie wavelength is inversely proportional to momentum. From the de Broglie equation, \ \[ \lambda = \frac{h}{p} \ \], it is clear that \ \lambda \ \ is indeed inversely proportional to \ \p \ \. Thus, Statement 1 is true.

Analyze Statement 2

Statement 2 claims that the de Broglie equation describes the relationship between a moving particle and its momentum. The de Broglie equation \ \[ \lambda = \frac{h}{p} \ \] directly relates the wavelength of a moving particle to its momentum. Hence, Statement 2 is true.

Analyze Statement 3

Statement 3 suggests that the de Broglie equation implies that an electron has both particle and wave nature. The equation \ \[ \lambda = \frac{h}{p} \ \] indeed indicates the wave-particle duality of particles such as electrons. So, Statement 3 is true.

Analyze Statement 4

Statement 4 asserts that the de Broglie equation is only applicable to electrons. This is false because the de Broglie equation can be applied to any moving matter particle, not just electrons.

Key concepts

Wave-Particle Duality

Wave-particle duality is a fundamental concept in quantum mechanics. It suggests that every particle or quantum entity can exhibit both wave-like and particle-like properties. This means that particles such as electrons, which we usually think of as solid objects, can also behave like waves under certain conditions. This concept was revolutionary because it challenged the classical notion that particles and waves are entirely separate phenomena.

Momentum

Momentum is a measure of the quantity of motion an object has and is calculated as the product of an object's mass and velocity. In the de Broglie equation, momentum ( p ) plays a crucial role. The equation is , demonstrating that the wavelength ( ) of a moving particle is inversely proportional to its momentum. This means that as the momentum of a particle increases, its wavelength decreases, and vice versa.

Planck's Constant

Planck's Constant ( h ) is a fundamental constant in physics that describes the size of the quantums. It is a critical element in the de Broglie equation. The value of Planck's Constant is approximately J·s. The use of this constant in the de Broglie equation helps to bridge the gap between the wave and particle natures of matter, reinforcing the concept of wave-particle duality by connecting the wave properties of a particle (its wavelength) to its motion (momentum).

Matter Waves

Matter waves, also known as de Broglie waves, refer to the wave-like behavior of particles. According to the de Broglie hypothesis, every moving particle has an associated wavelength. This wave-like behavior of particles can be observed in experiments such as electron diffraction. The de Broglie equation, , fully encapsulates this idea by expressing the wavelength of a particle as a function of Planck's constant and its momentum. This principle is not limited to electrons; it applies to all matter.