Quantum Mechanical Picture of Hydrogen Atom Assignment Help
3.13 Quantum Mechanical Picture of Hydrogen Atom
Newtonian or classical mechanics successfully describes the motion of all macroscopic objects like a falling stone, moving planet which have essentially a particle like behaviour. However the theory is not applicable for microscopic substances  because of the duality in the behaviour of microscopic substances.
The quantum mechanical picture of the hydrogen atom is a fundamental and wellunderstood example in quantum mechanics. It describes the behavior of a single electron in orbit around a nucleus consisting of a single proton. In this picture, the electron's properties, such as its energy levels and probability distributions, are determined by solving the Schrödinger equation for the hydrogen atom.
Here are some key aspects of the quantum mechanical picture of the hydrogen atom:

Hamiltonian Operator: The starting point for solving the hydrogen atom's quantum mechanics is the Hamiltonian operator, which represents the total energy of the system. It consists of the kinetic energy of the electron and the potential energy due to the electrostatic attraction between the electron and the proton.

Schrödinger Equation: The timeindependent Schrödinger equation for the hydrogen atom is a partial differential equation that must be solved to determine the allowed energy levels (quantization of energy) and the corresponding wave functions (probability distributions) of the electron. The equation is typically written as:
HΨ = EΨ
Here, H is the Hamiltonian operator, Ψ represents the wave function, and E represents the energy of the system.

Energy Levels: Solving the Schrödinger equation yields a discrete set of energy levels for the electron in the hydrogen atom. These energy levels are often labeled by a principal quantum number (n), with higher values of n corresponding to higher energy levels. The energy levels are quantized, meaning that only specific energy values are allowed.

Wave Functions: The wave functions (Ψ) describe the probability distribution of finding the electron at various positions around the nucleus. These wave functions are represented by mathematical functions, often expressed in terms of spherical coordinates. The solutions to the Schrödinger equation for the hydrogen atom are called atomic orbitals, and they have different shapes (spherical, dumbbell, etc.) corresponding to different quantum numbers.

Quantum Numbers: In addition to the principal quantum number (n), there are two other quantum numbers that describe the hydrogen atom's electron behavior: the angular momentum quantum number (l) and the magnetic quantum number (m). These quantum numbers determine the specific energy level and the shape of the orbital.

Energy Spectra: The energy levels of the hydrogen atom give rise to its spectral lines, such as the Balmer series and Lyman series. When an electron transitions from one energy level to another, it emits or absorbs photons of specific energies, leading to the observed spectral lines.

Probability Distributions: The square of the absolute value of the wave function (Ψ^2) represents the probability density of finding the electron at a particular location in space. The probability distributions for different atomic orbitals have distinct shapes, reflecting the electron's behavior around the nucleus.
The quantum mechanical picture of the hydrogen atom provides a detailed and accurate description of its behavior and serves as a foundational example in quantum physics. It also laid the groundwork for our understanding of more complex atoms and molecules.
The branch of science which deals with this behaviour is said to be quantum mechanics. The fundamental equation of quantum mechanics is Schrodinger Equation.
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