Schrodingers Equation

Schrodinger’s Equation is a fundamental formula in quantum mechanics that describes how the quantum state of a physical system changes over time. Formulated by Erwin Schrodinger in 1925, this equation provides a mathematical framework for understanding the behavior of particles at microscopic scales, such as electrons in atoms.

Schrodingers Equation

Schrodinger’s Equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system evolves over time. It uses a wave function, ψ, to represent the probability amplitude of a particle’s position and state. The equation links the wave function with the energy and potential experienced by the particle, guiding predictions of particle behavior in quantum systems.

Formula of Schrodinger’s Equation

The time-dependent Schrödinger Equation can be expressed as:

𝑖ℏ∂𝜓/∂𝑡 = 𝐻^𝜓

Where:

• 𝑖 is the imaginary unit,
• ℏ (h-bar) is the reduced Planck constant,
• ∂𝜓/∂𝑡 is the partial derivative of the wave function 𝜓ψ with respect to time,
• 𝐻^ is the Hamiltonian operator, which represents the total energy (kinetic + potential) of the system.

Derivation of Schrodinger Wave Equation

The derivation of Schrödinger’s Wave Equation involves concepts from classical physics, quantum theory, and mathematical physics. It blends these areas to describe how the quantum state of a physical system changes over time or remains unchanged in the stable state. Here, we focus on deriving the time-dependent Schrödinger equation for a non-relativistic particle.

Starting from Classical Concepts

Energy and Wave Properties:
In classical mechanics, the total energy E of a particle is the sum of its kinetic energy K and potential energy V. For a particle of mass m and velocity v, this is given by:

E = K+V = 1/2mv²+V

In wave mechanics, any particle can be described as a wave. The relationship between energy and wave properties is given by the Planck-Einstein relation:
where h is Planck’s constant and f is the frequency of the wave.
De Broglie Hypothesis:
Louis de Broglie proposed that particles have wave-like properties. The wavelength λ of a particle is related to its momentum p by:

λ = h/p

Rewriting momentum in terms of wavelength, we have:

p = h/λ

Moving to Quantum Mechanics
Wave Function:
A particle’s quantum state is described by a wave functionψ(x,t), which carries all information about the particle’s state. The wave function is complex-valued and its square modulus ∣ψ(x,t)∣²
gives the probability density of finding the particle at position x at time t.
Expressing Energy and Momentum as Operators:
In quantum mechanics, observables like energy and momentum are represented by operators. The momentum operator in one dimension is given by:

p^ = −iℏ ∂/∂x

Substituting p in the classical kinetic energy expression, the kinetic energy operator becomes:

𝐾^ = 𝑝^²/ 2m = -ℏ²/ 2m ∂²/∂x²

The total energy operator, known as the Hamiltonian 𝐻^, is:𝐻

𝐻^ = 𝐾^ + 𝑉(𝑥) = -ℏ²/ 2m ∂²/∂x² + 𝑉(𝑥)

Formulating the Time-Dependent Schrödinger Equation:
We postulate that the energy operator acting on the wave function equals the energy of the wave (from Planck-Einstein relation) times the wave function:

𝐻^ψ = Eψ

Using the wave property

E = iℏ ∂/∂t

and substituting in the above equation:

iℏ = ∂ψ/∂t = 𝐻^𝜓

This is the time-dependent Schrodinger equation.

Different Schrodingers Equation

Time-Dependent Schrodinger Equation

This is the most general form of Schrödinger’s Equation and is used when dealing with quantum systems that evolve over time. It is expressed as:

𝑖ℏ ∂𝜓/∂𝑡 = 𝐻^𝜓

𝑖 is the imaginary unit. ℏ is the reduced Planck constant. 𝜓 is the wave function of the system. 𝐻^ is the Hamiltonian operator that represents the total energy (kinetic plus potential) of the system.

Time-Independent Schrodinger Equation

When the Hamiltonian does not depend on time, the Schrödinger equation can be simplified to a time-independent form. This version is particularly useful for studying stationary states, such as the energy levels of an atom. It is typically obtained by separating the time-dependent part from the spatial part of the wave function in the full time-dependent equation:

𝐻^𝜓 = 𝐸𝜓

Here, 𝐸 represents the energy eigenvalues corresponding to each stationary state described by the eigenfunctions 𝜓.

Non-Relativistic Schrodinger Equation

This form applies to quantum systems moving much slower than the speed of light and ignores relativistic effects. Both the time-dependent and time-independent equations mentioned above generally refer to the non-relativistic case, where the kinetic energy operator is given by:

𝐾^ = −ℏ²/ 2m ∂²/∂x²

Relativistic Schrodinger Equation

For particles moving at or near the speed of light, relativistic effects become significant, and the Schrödinger equation must be modified. This leads to equations like the Klein-Gordon equation for spin-0 particles and the Dirac equation for spin-1/2 particles. These are more complex and take into account the principles of special relativity.

Multi-Particle Schrodinger Equations

In systems with more than one particle, the Schrödinger equation must account for each particle’s motion and interactions. The wave function 𝜓ψ becomes a function of the coordinates of all the particles, and the Hamiltonian includes terms for the kinetic energy of each particle and their potential interactions:

𝑖ℏ∂𝜓(𝑟1,𝑟2,…,𝑟𝑁,𝑡)/∂𝑡 = 𝐻^𝜓(𝑟1,𝑟2,…,𝑟𝑁,𝑡)

FAQs

Who developed Schrodinger’s Equation?

The Austrian physicist Erwin Schrödinger developed the equation in 1926 as part of the development of quantum mechanics.

What does Schrodinger’s Equation describe?

It describes the behavior of quantum systems, such as electrons in atoms or particles in potential wells, by providing a way to calculate their wave functions and corresponding probabilities.

How is Schrodinger’s Equation solved?

The equation is typically solved using various mathematical techniques, such as separation of variables, perturbation theory, and numerical methods.

Does Schrodinger’s Equation apply only to microscopic systems?

While it was originally developed for microscopic systems, Schrodinger’s Equation has been successfully applied to a wide range of systems, from atoms and molecules to larger quantum systems.

How does Schrodinger’s Equation relate to the wave-particle duality of quantum mechanics?

It provides a mathematical framework for understanding the wave-like behavior of particles, as described by their wave functions, and the particle-like behavior observed in experiments, such as the discrete energy levels of atoms.