What You Need To Know About Mathematical Principles In Wave-Particle Duality?

The historical journey of wave-particle duality began in the early 20th century, with notably the famous double-slit experiment conducted by Thomas Young and later refined by scientists like Albert Einstein and Niels Bohr. These experiments demonstrated that even elementary particles, when observed under certain conditions, exhibit wave-like interference patterns, suggesting their wave-like nature. Simultaneously, these particles also behave as discrete entities, exhibiting particle-like behaviour when their position or momentum is measured.

To comprehend the profound nature of wave-particle duality, scientists have turned to mathematics as a powerful tool. Mathematical principles play a pivotal role in describing and quantifying the behaviour of quantum systems. Mathematical concepts such as wave equations, wavefunctions, probability amplitudes, and operators, allow us to explore the dynamic interplay between waves and particles in the quantum realm.

The concepts related to wave-particle duality are primarily discussed in the Class 12 modern physics chapters such as “Dual Nature Of Radiation And Matter” and “Atoms” and in Chemistry Class 11 chapter 2 “Structure Of Atoms”. Therefore, more comprehensive understanding and preparation are required for competitive exams as it is important for both subjects.

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Here, in this article, we will discuss Mathematical Principles In Wave-Particle Duality including the mathematics involved and some solved problems based on the concepts which gives an idea for competitive exams.

Mathematical Principles Behind The Wave-Particle Duality

At the heart of mathematical formalism lies the Schrödinger equation, a fundamental equation in quantum mechanics. This equation, developed by Erwin Schrödinger in 1925, describes the time evolution of a quantum system, providing insights into its wave-like behaviour. Through the Schrödinger equation, we can derive wavefunctions that represent the probability distributions of a particle's position, momentum, and other observable properties.

Probability amplitudes, known as wavefunctions, embody the probabilistic nature of quantum mechanics. By applying mathematical operations to these wavefunctions, we can extract valuable information about the particle's behaviour. For instance, the squared magnitude of the wavefunction yields the probability density, enabling us to determine the likelihood of finding a particle at a particular location.

The Schrödinger equation for a particle in one dimension can be written as:

ĤΨ(x, t) = iħ∂Ψ(x, t)/dt

Here, Ĥ represents the Hamiltonian operator, which is an expression involving the particle's kinetic and potential energies. Ψ(x, t) is the wave function of the particle, which depends on both position (x) and time (t). The symbol i represents the imaginary unit √(-1), and ħ is the reduced Planck's constant. Solving the Schrödinger equation yields the wave function, which provides information about the probability distribution of the particle's position. The square of the wave function, |Ψ(x, t)|², gives the probability density of finding the particle at a specific position x at time t. The normalisation condition ensures that the total probability of finding the particle in all possible positions is equal to 1.

Mathematically, this condition can be expressed as:

|Ψ(x, t)|² dx = 1

The wave function can also be used to calculate the expectation value of observable quantities, such as position and momentum. The expectation value of an observable A is given by:

〈A〉 = ∫Ψ*(x, t) ÂΨ(x, t) dx

Here, Â represents the operator associated with the observable A.

The operators in quantum mechanics act on the wave function to extract specific information about the particle.

Let's denote the momentum of an electron as p and its wavelength as λ. The de Broglie relation can be expressed as:

λ = h/p

where h is Planck's constant.

Now, let's consider the momentum of an electron, which is given by the product of its mass (m) and velocity (v):

p = m * v

Combining the above equations, we can rewrite the de Broglie relation as:

λ = h/mv

To proceed with the mathematical proof, we will use the concept of momentum operators and wavefunctions in quantum mechanics.

In quantum mechanics, a wave function Ψ describes the behaviour of a particle. For a one-dimensional case, we can write the wavefunction as Ψ(x, t), where x represents the position of the electron at a given time t.

The momentum operator in quantum mechanics is represented by the operator ħ * (d/dx), where ħ is the reduced Planck's constant and d/dx is the derivative with respect to the position x.

According to the principles of quantum mechanics, the momentum operator acting on the wavefunction gives the momentum of the particle. Mathematically, we can express this as:

p = ħ *(d/dx)* Ψ(x, t)

Now, let's consider the time-independent Schrödinger equation, which describes the behaviour of quantum systems. In one dimension, it is given by:

(-ħ² / 2m) * (d²/dx²) * Ψ(x) + V(x) * Ψ(x) = E * Ψ(x)

Where V(x) represents the potential energy and E is the energy of the system.

If we assume that the potential energy (V(x)) is zero, we have:

(-ħ² / 2m) * (d²/dx²) * Ψ(x) = E * Ψ(x)

Rearranging the equation, we get:

(d²/dx²) * Ψ(x) = -(2mE / ħ²) * Ψ(x)

Comparing this equation with the general form of the wave equation:

(d²/dx²) * Ψ(x) = (1/v²) * Ψ(x)

where v is the velocity of the wave, we can equate the two expressions:

-(2mE / ħ²) = 1/v²

Simplifying the equation, we find:

v = ħ / √(2mE)

Now, let's recall the relation between velocity, wavelength, and frequency for a wave:

v = λ * f

where λ is the wavelength and f is the frequency.

Substituting the expression for velocity derived above, we have:

ħ / √(2mE) = λ * f

Rearranging the equation, we obtain:

λ = (ħ * f) / √(2mE)

Comparing this equation with the de Broglie relation we derived earlier:

λ = h / (m * v)

we can see that they are equivalent, where h = ħ * f.

Hence, mathematically, we have shown that the moving electron behaves like a wave by establishing a relationship between its momentum, wavelength, and frequency.

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Solved Problems

Problem-1: (JEE Main 2023)

The de Broglie wavelength of an electron having kinetic energy E is λ. If the kinetic energy of an electron becomes E/4, then its de-Broglie wavelength will be:

1. √2λ

2. 2λ

3. λ/2

4. λ/√2

Solution:

We know that

λ = h/√(2mK.E)

λ√(K.E) = h/√2m = constant

Therefore,

λ√(E) = √(E/4)λ'

λ' = 2λ

Hence, the correct answer is 2λ.

Problem-2: (JEE Main 2023)

An electron accelerated through a potential difference V1 has a de-Broglie wavelength of . When the potential is changed to V2, its de-Broglie wavelength increases by 50%. The value of (V1V2) is equal to

1. 3/2

2. 4

3. 3

4. 9/4

Solution:

We know that

K.E = P²/2m

λ = h/p ⇒ p = h/λ

eV₁ = {(h/λ)²}/2m……………….(1)

eV₂ = {(h/1.5λ)}²/2m…………….(2)

Using equation (1) and (2)

V₁/V₂ = (1.5)² = 9/4

Hence, Option (d) 9/4 is correct.

Now, You have a basic idea of how mathematical principles and concepts play a crucial role in understanding Wave-Particle Duality.