# Wave Nature of Matter and De Broglie’s Equation

One of physics’ most perplexing ideas is the wave nature of matter. A particle is constrained to a certain location, but a wave is dispersed over space. It has been demonstrated that light can have a particle or wave nature. As with a billiard ball, electrons and photons display particle characteristics in the photoelectric effect. However, you’ll recall the Diffraction experiment and the Interference Rings. Similar to the way two waves on a pond’s surface interact as they come together. In many instances, the waveform of light is evident. It’s a fascinating puzzle to solve. It even affects our sense of sight! The eye-lens light collecting and focussing mechanisms are in keeping with light’s wave nature. However, its absorption by the retina’s rods and cones corresponds to light’s particle nature! While we were still trying to figure out the riddle, Louis de Broglie threw a wrench in the works with his de Broglie Relationship.

### Wave Nature of Matter

Radiation is viewed as a wave in classical mechanics, while particles are viewed as hard billiard balls. Radiation was shown to be capable of behaving as both waves and particles. Radiation and moving particles may both supply energy and momentum to various things. De Broglie proposed in 1924 that matter should have a dual nature because of nature’s inherent symmetry. Particles do not have a specific location in the space where they reside. Quantum theory was built on the idea that radiation and matter have a dual nature.

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**De Broglie’s Equation**

Light and radiation are both particles and waves, according to De Broglie’s hypothesis, thus matter must also have a particle and wave character. Wave theory was born as a result of the de Broglie connection.

De Broglie’s equation is given as:

λ = h ⁄ p = h ⁄ (mv)where,

- λ is the wavelength of particle,
- p is the momentum of a particle,
- h is the Planck’s constant,
- m is the mass of partcile, &
- v is the velocity of the particle.

Since this connection shows that matter may act like a wave, it’s important to understand its importance. A moving particle, no matter how little or large it may be, has a unique wavelength according to De Broglie’s Equation. If we look closely at macroscopic objects, we can see the wave aspect of the matter. As an item grows in size, its wavelength shrinks until it is undetectable, which explains why macroscopic things in the actual world lack wave-like characteristics. Even the cricket ball you throw has a wavelength that you cannot see. The Plank’s constant links the wavelength and momentum in the equation.

**De Broglie’s Hypothesis**

The momentum of a photon having energy E is given as:

**p = E ⁄ c**

The speed of light in a vacuum is represented by the letter c.≥

Planck’s idea states that the energy of a photon is determined by its frequency and wavelength.

**E = h ****v**** = h c ⁄ λ**

The energy should be equal, implying:

h c ⁄ λ = p c

λ = h ⁄ p

De Broglie concluded that the aforementioned relationship should apply to particles as well. **p=mv** is the momentum of a particle with mass m moving at a speed of v. As a result, it must have a wavelength of

**λ = h ⁄ p = h ⁄ (mv)**

### Heisenberg’s Uncertainty Principle

By diffracting electrons through a crystal, the Davisson-Germer experiment demonstrated without a shadow of a doubt the nature of matter as a wave. De Broglie won the Nobel Prize in Physics in 1929 for his theory of matter waves, which opened up a whole new area of study known as Quantum Physics. Heisenberg’s Uncertainty Principle neatly incorporates the matter-wave hypothesis. According to the Uncertainty Principle, it is impossible to know an electron’s momentum and position at the same time for any other particle. Uncertainty exists in both the position ‘Δx’ and momentum ‘Δp’.

**Heisenberg’s Uncertainty Equation:**

The Uncertainty Principle says that a particle’s momentum and location cannot be determined with precision at the same time. To put it another way, there is always some degree of ambiguity about where something is located, as well as its velocity. The unknowns are linked by,

Δx Δp ≥ h ⁄ 4πwhere

- Δx is the uncertainty in position and
- Δp is the uncertainty in the momentum of the particle.

If a particle’s momentum is precisely measured (i.e. Δp=0), the uncertainty Δx in its location becomes infinite. According to de Broglie’s equation, a particle with a known momentum should also have a known wavelength. A certain wavelength can be found throughout all of space, all the way to infinity. According to Born’s Probability Interpretation, this implies that the particle is not localised in space and that the uncertainty of its location is thus limitless. The wavelengths in real life, on the other hand, have a defined boundary and aren’t limitless, thus uncertainty in terms of both location and momentum is limited. Localized waves (wave packets), which include wavelengths of varying lengths, should be used to represent any particle.

### Sample Problems

**Question 1: What do you understand by the** **Matter-wave packet?**

**Answer:**

In contrast to a progressive wave, a wave packet is a superposition of sinusoidal waves of various wavelengths that is confined in space. The location and momentum of a particle may be accurately represented using a wave packet. The particle’s velocity is calculated using the packet’s group velocity. The De Broglie hypothesis and the uncertainty principle are both used to describe a wave packet.

**Question 2: Can the De Broglie Equation Be Used to Calculate Photon Energy?**

**Answer:**

Radiation is made up of photons, which are massless particles. Even though a photon’s rest mass is zero, relativity says its energy equates to a momentum. A photon’s energy is related to its frequency and wavelength, according to Max Planck’s theory. The relationship between wavelength and photon momentum resembles that of the de Broglie equation for matter.

**Question 3: If a baseball weighs 0.1 kg and travels at 60 m ⁄ s, what is its de Broglie wavelength?**

**Answer:**

Given:

Mass of a baseball, m = 0.1 kg

Speed of a baseball, v = 60 m ⁄ s

Planck’s constant, h = 6.626 × 10

^{−34}J sThe de Broglie wavelength of an object is given as:

λ = h ⁄ (m v)

= 6.626 × 10

^{−34}⁄ ( 0.1 × 60) m= 1.104 × 10

^{−34}mHence, the de Broglie wavelength is

1.104 × 10.^{−34}m

**Question 4: Which has a longer de Broglie wavelength if electron and proton are the same speed?**

**Answer:**

Due to its far greater mass, the de Broglie wavelength of a proton is 1800 times smaller than that of an electron, and as a result, its momentum at the same speed is 1800 times more than that of an electron. The electron has a greater radiance because of its longer wavelength.

**Question 5: A molecule’s electron moves at a 20 m ⁄ s clip. The electron’s p-momentum uncertainty is 2p×10 ^{−6} that of the electron’s initial momentum. Calculate the uncertainty in position x for an electron weighing 9.1×10^{−31} kg.**

**Answer:**

Given:

Mass of the electron, m = 9.1×10

^{−31}kgSpeed of the electron, v = 20 m ⁄ s

Momentum of the electron, p = mv

= 9.1×10

^{−31 }× 20 kg m ⁄ s= 182×10

^{−31}kg m/sUncertainty in momentum, Δp = 2p×10

^{−6}= 364×10

^{−37}kg m/sPlanck’s constant, h = 6.626 × 10

^{−34}J sHeisenberg Uncertainty Formula is given as:

Δx Δp ≥ h ⁄ 4π

Δx ≥ h ⁄ (4π Δp)

Δx ≥ (6.626 × 10

^{−34}J s) ⁄ (4π × 364×10^{−37}kg m/s) = 1.44 mHence, the uncertainty in position for an electron is

1.44 m.