Chapter 1: Problem 19

Discuss Schrodinger wave equation for hydrogen atom.

Short Answer

Expert verified

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 \( (r, \theta, \phi) \)

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Discuss Schrodinger wave equation for hydrogen atom.

Short Answer

Step by step solution

Formulation of Schrödinger Equation

Defining Potential Energy

Substitution into Schrödinger Equation

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