Schrödinger equation

Examples of Schrödinger equation in the following topics:

• 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:

• Perturbation Theory

  • We didn't resolve Schrodinger's equation, but rather we used the spherical harmonic solutions to understand how various additional terms like the interaction between the electrons would affect the energies of the states.
  • Through this process we built up a picture of the structure of atoms from two simple ideas: Schrodinger's equation and that the wavefunction of a bunch of electrons is odd under interchange of any pair of electrons.
  • Let's start with the time-dependent Schrodinger equation and add a small extra time-dependent term in the potential.
  • Solutions to equations like the above equation form a complete set.
  • This means that you can use a sum of them to represent any function, so let's imagine that the real solution to the first equation is the sum of the solutions to the second equation but let's allow the coefficients to be a function of time

• 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

• Applications

  • There are four kinematic equations that describe the motion of objects without consideration of its causes.
  • Notice that the four kinematic equations involve five kinematic variables: $d$, $v$, $v_0$, $a$, and $t$.
  • Each of these equations contains only four of the five variables and has a different one missing.
  • Step two - Find an equation or set of equations that can help you solve the problem.
  • Choose which kinematics equation to use in problems in which the initial starting position is equal to zero

• Particle Current

  • We can prove this simply by integrating the fourth equation over $d^3 {\bf p}$.
  • The first two terms yield the left-hand side of the equation above.
  • Let's define ${\bf V}=\langle {\bf v} \rangle$ and write out the equation above by components,
  • This is the continuity equation.
  • Because the fourth equation is consistent with the Lioville equation (seventh equation) and more generally with the Boltzmann equation (sixth equation) and $J^\mu_{;\mu}=0$ if particles are conserved, the Lioville and Boltzmann equations cannot hold if $\nabla_{\bf p} \cdot F \neq 0$ and particles are conserved.

• Problem-Solving Techniques

  • Identify the problem and solve the appropriate equation or equations for the quantity to be determined.
  • Solve the appropriate equation or equations for the quantity to be determined (the unknown).
  • We cannot use any equation that incorporates t to find ω, because the equation would have at least two unknown values.
  • The equation $\omega 2=\omega 02+2$ will work, because we know the values for all variables except ω.
  • Taking the square root of this equation and entering the known values gives

• Atomic Structure

  • The wavefunction evolves forward in time according to the time-dependent Schrodinger equation
  • If the Hamiltonian is independent of time we can solve this equation by
  • This realization allows us to write the equation that the wavefunction of an atom must satisfy
  • For most atomic states, these effects can be treated at perturbations.We can simplify these equations by using
  • This gives the following dimensionless equation

• Problem Solving

  • The Ideal Gas Law is the equation of state of a hypothetical ideal gas.
  • Variations of the ideal gas equation may help solving the problem easily.
  • Substitute the known values into the equation.
  • Choose a relevant gas law equation that will allow you to calculate the unknown variable: We can use the general gas equation to solve this problem: $\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}$.
  • Substitute the known values into the equation.