Question

Discuss Schrodinger wave equation for hydrogen atom.

Short answer

The Schrödinger equation for the hydrogen atom, \(-\frac{\hbar^2}{2m} \nabla^2 \psi -\frac{k e^2}{r} \psi = E \psi \), is solved by substituting the potential energy of the system.

Step-by-step solution

Formulation of Schrödinger Equation

The Time-Independent Schrödinger equation for a hydrogen atom is given by \(-\frac{\hbar^2}{2m} \nabla^2 \psi + V \psi = E \psi \) where \( \nabla^2 \) is the Laplacian operator that represents a second spatial derivative, \( \psi \) is the wave function of the atom which we are trying to find, \( V \) is the potential energy, \( m \) is the mass of the electron, \( \hbar \) has a value of Planck’s constant divided by \( 2\pi \) and \( E \) is the total energy.

Defining Potential Energy

The potential energy \( V \) in a hydrogen atom is defined by the equation \( -\frac{k e^2}{r} \), where \( r \) is the distance between the electron and the nucleus, \( e \) is the charge of an electron and \( k \) is Coulomb's constant.

Substitution into Schrödinger Equation

Substitute the potential energy function into the Schrödinger equation, the equation becomes \(-\frac{\hbar^2}{2m} \nabla^2 \psi -\frac{k e^2}{r} \psi = E \psi \). This equation is solved using spherical polar coordinates \( (r, \theta, \phi) \)

Key concepts

Hydrogen Atom

The hydrogen atom is one of the simplest and most studied systems in quantum mechanics. It consists of a single proton acting as the nucleus and a single electron orbiting around it. Despite its simplicity, the hydrogen atom plays a crucial role in understanding atomic structures and quantum mechanics fundamentals. Its simplicity allows scientists to apply quantum theories and equations, such as the Schrödinger wave equation, to explore the behavior of electrons in more complex atoms. This system serves as a fundamental example for testing and validating quantum theories over time.

Quantum Mechanics

Quantum mechanics is a foundational theory in physics that explains the behavior of matter and energy on atomic and subatomic scales. Unlike classical physics, which can accurately describe macroscopic phenomena, quantum mechanics reveals a world where particles behave as both particles and waves. Central to quantum mechanics is the concept of quantization, implying that energy, momentum, and other properties can only take discrete values. This theory is necessary to understand the hydrogen atom, as it accounts for the electron's wavelike nature and energy levels, which are not explained by classical physics.

Wave Function

The wave function, denoted by the symbol \( \psi \), is a mathematical function that describes the quantum state of a particle or system. In the context of the hydrogen atom, the wave function provides information about the electron's probability distribution around the nucleus. These distributions help predict where an electron is likely to be found at any given time. The square of the absolute value of the wave function, \( |\psi|^2 \), gives the probability density of finding the electron at a particular position. Solving the Schrödinger equation for the hydrogen atom involves finding this wave function, which results in a set of quantized energy levels.

Potential Energy

In quantum mechanics and particularly in the study of the hydrogen atom, potential energy plays a critical role. The potential energy of the electron in the hydrogen atom is mainly determined by its interaction with the proton in the nucleus. This interaction is described by Coulomb's law, which results in a potential energy expression \( V(r) = -\frac{k e^2}{r} \). Here, \( r \) is the distance between the electron and the nucleus, \( e \) is the electron charge, and \( k \) is Coulomb's constant. This negatively valued potential energy indicates an attractive force between the negatively charged electron and positively charged nucleus, keeping the electron bound in orbit and defining the characteristic energy levels of the atom.

Coulomb's Law

Coulomb's law is fundamental among the principles governing electromagnetic forces. It explains the force between two charged particles, such as the proton and electron in a hydrogen atom. According to Coulomb's law, the force \( F \) is proportional to the product of the charges, and inversely proportional to the square of the distance \( r \) between them: \( F = k \frac{|q_1 q_2|}{r^2} \). In the context of the hydrogen atom, it is instrumental in determining the potential energy \( V \) of the electron as it defines how the electron is held in its orbit around the nucleus. This attraction is central to the functioning of the Schrödinger wave equation for the hydrogen atom.