Schrödinger equation

Examples of Schrödinger equation in the following topics:

• Description of the Hydrogen Atom

  • Bohr's results for the frequencies and underlying energy values were confirmed by the full quantum-mechanical analysis which uses the Schrödinger equation, as was shown in 1925–1926.
  • The solution to the Schrödinger equation for hydrogen is analytical.
  • The Schrödinger equation also applies to more complicated atoms and molecules, albeit they rapidly become impossibly difficult beyond hydrogen or other two-body type problems, such as helium cation He+.
  • The solution of the Schrödinger equation (wave equations) for the hydrogen atom uses the fact that the Coulomb potential produced by the nucleus is isotropic (it is radially symmetric in space and only depends on the distance to the nucleus).
  • The energy levels of hydrogen are given by solving the Schrödinger equation for the one-electron atom:

• Wave Equation for the Hydrogen Atom

  • The hydrogen atom is the simplest one-electron atom and has analytical solutions to the Schrödinger equation.
  • The solution of the Schrödinger equation for the hydrogen atom uses the fact that the Coulomb potential produced by the nucleus is isotropic—it is radially symmetric in space and only depends on the distance to the nucleus.
  • Using a three-dimensional approach, the following form of the Schrödinger equation can be used to describe the hydrogen atom:
  • Explain how the solution of the Schrödinger equation for the hydrogen atom yields the four quantum numbers and use these to identify degenerate states

• Separable Equations

  • Separable differential equations are equations wherein the variables can be separated.
  • A separable equation is a differential equation of the following form:
  • The original equation is separable if this differential equation can be expressed as:
  • Integrating such an equation yields:
  • A wave function which satisfies the non-relativistic Schrödinger equation with $V=0$.

• The Wave Function

  • The laws of quantum mechanics (the Schrödinger equation) describe how the wave function evolves over time.
  • The wave function behaves qualitatively like other waves, such as water waves or waves on a string, because the Schrödinger equation is mathematically a type of wave equation.

• Quantum Tunneling

  • Imaginary space is not real, but it is explicitly referenced in the time-dependent Schrödinger equation, which has a component of $i$ (the square root of $-1$, an imaginary number):
  • The solution for the Schrödinger equation in such a medium is:

• Particle in a Box

  • The solution to the particle in a box can be found by solving the Schrödinger equation:
  • Separating the variables reduces the problem to one of simply solving the spatial part of the equation:
  • The above equation establishes a direct relationship between the second derivative of the the wave function and the kinetic energy of the system.
  • The best way to visualize the time-independent Schrödinger equation is as a stationary snapshot of a wave at particular moment in time.
  • The size or amplitude of the wave function at any point determines the probability of finding the particle at that location, as given by the equation:

• Quantum Numbers

  • In three dimensions, the solutions of the Schrödinger equation provided a set of three additional quantum numbers that could be used to describe electron behavior even in more complicated many-electron atoms.
  • Formally, the dynamics of any quantum system are described by a quantum Hamiltonian (H) applied to the wave equation.
  • For particles in a time-independent potential, per the Schrödinger equation, it also labels the nth eigenvalue of Hamiltonian (H) (i.e. the energy E with the contribution due to angular momentum, the term involving J2, left out).

• The de Broglie Wavelength

  • In 1926, Erwin Schrödinger published an equation describing how a matter wave should evolve—the matter wave equivalent of Maxwell's equations—and used it to derive the energy spectrum of hydrogen.
  • The de Broglie equations relate the wavelength (λ) to the momentum (p), and the frequency (f) to the kinetic energy (E) (excluding its rest energy and any potential energy) of a particle:
  • The two equations can be equivalently written as
  • He related this to the principal quantum number n through the equation:
  • Use the de Broglie equations to determine the wavelength, momentum, frequency, or kinetic energy of particles

• de Broglie and the Bohr Model

  • By assuming that the electron is described by a wave and a whole number of wavelengths must fit along the circumference of the electron's orbit, we have the equation:
  • By different reasoning, another form of the same theory, wave mechanics, was discovered independently by Austrian physicist Erwin Schrödinger.
  • Schrödinger employed de Broglie's matter waves, but instead sought wave solutions of a three-dimensional wave equation.

• Quantum-Mechanical View of Atoms

  • Adopting Louis de Broglie's proposal of wave-particle duality, Erwin Schrödinger, in 1926, developed a mathematical model of the atom that described the electrons as three-dimensional waveforms rather than point particles.
  • Modern quantum mechanical view of hydrogen has evolved further after Schrödinger, by taking relativistic correction terms into account.
  • Identify major contributions to the understanding of atomic structure that were made by Niels Bohr, Erwin Schrödinger, and Werner Heisenberg