Shrödinger's Equation

The current fundamental physical model of the atom was created by the Austrian physicist Erwin Schödinger while he was a young professor at the University of Zurich. Schrödinger's colleague Victor Henri gave him a copy of de Broglie's doctoral thesis on the wave properies of an electron. Schödinger was not impressed with the thesis and began looking in other directions. It was several months later while reading a section of the thesis dealing with Bohr's quantization rules that he recognized the connection between Bohr's stationary states of the hydrogen atom and deBroglie's wave properties of an electron. Schödinger realized that waves are macroscopic systems that behave like atoms. A standing wave like and atom can only absorb or release energy in quantitized amounts. This bold decision was the beginning of quantum mechanics. The results of Schrödinger's doctoral thesis was a single general equation first published in 1926 (Ann. Physik, 79,361). The invention of quantum mechanics, which Schödinger shares with Heisenberg and Dirac, has been one the most important advancements in science - to be compared with the contributions of Galileo, Newton and Einstein.

Like the equations comprising Newton's laws of motion, Schrödinger's equation can not be derived. His equation is a generalizations of the world as we observe it and is validated by how well it describes experimental observations. What follows is not a derivation but merely a procedure by which Schrödinger's equation can be constructed. Schödinger developed his equation using analogies to the behavior of light. He reasoned that the classical equations used to describe light waves could be used to describe matter waves if the equations were modified to include newly discovered quantum properties of photons. The energy of a photon of light is related to its frequency by Planck's constant, Elight = hn and the momentum of a photon of light is related to its wavelength by Planck's constant, plight = h/l. Rather than using a metaphor based on light, as Schödinger did, I am going to construct his equation by the analogy to a standing wave such as the vibration of a guitar string.

There are many similarities between the motion of a guitar string and the motions of an electron trapped in an atom. Both are described by wavefunctions that oscillate in time and space. The waves are characterized by stationary points called nodes where the wavefunction goes through zero. The guitar string is fixed at the bridge and neck of the guitar and hence must have a node at these positions. Attachment of the guitar string at the bridge and neck place a restraint or boundary condition on the guitar string's movement. The boundary conditions limits the motion of the string to certain special vibrations with fixed energies. The first three special vibrations or overtones of a guitar string are depicted above with nodes represented as black dots. These are the principle overtones your hear when you pluck a string and the reason that the notes C and G harmonize with each other. These special vibrations in a quantum mechanical system would be called eigen states of the system. The sound coming from a single guitar string produces a line spectrum much like the hydrogen atom. The composite tone that we hear is a superposition of the eigen states of the system.

We will begin our construction of Shrödinger's equation with the mathematical equation for a standing wave:

The result of differentiating twice in respects to x is the second order differential equation for a wave:

We can begin the transformation of this classical equation to a quantum mechanical wave equation by using the deBroglie relation p=h/l for momentum.

Momentum also plays a central role in classical equations of motion. A standing wave is a conservative system in which the potential energy does not depend on momentum. In such a system the total energy (kinetic plus potential) is a constant of motion:

The total energy must also be a constant of our bound quantum mechanical system. Applying this classical relationship to our wave equation gives:

This is Schrödinger's time independent wave equation. The same equation that is presented at the top of this page. Schrödinger's equation is a second-order differential equation whose solution is the wavefunction for the system. The energy of the system will depend on how fast the wavefunction bends (second derivative of the wavefunction) and the potential of the system (V).

It is instructive to write this equation as an operator equation.

Classically .H, the Hamiltonian of the system is defined simply as the sum of the kinetic energy T and potential energy V of the system. The same Hamiltonian applies to quantum mechanical systems but any terms that are associated with kinetic energy and momentum must be replaced by their equivalent quantum mechanical operator:

These equations can easily be generalized to three dimensions by using the dell operator:

This ends our construction of Schrödinger's time independent equation. His equation is a simple operator eigen value equation. The Hamiltonian operating on the wavefunction gives the wavefunction back again times the constant energy of system. Such an equation is called a eigen equation after the German word for "self". The wavefunction is called the eigen function and the energy is called the eigen value. For a given Hamiltonian there are many possible eigen functions, but not all of these functions will have physical significance. It is a central postulate of wave mechanics that all of the measurable information about a system is contained in its wavefunction. To correspond to physical observations we must put some constraints on the wavefunctions for a system.
  1. The wavefunction should be single valued
  2. The wavefunction should be continuous so we can take its derivatives
  3. The wavefunction should be finite so that we can take its integral
  4. The wavefunction for a bound electron should vanish at the boundaries of the system

These constraints like the attachments on a guitar limit the energies of the system to certain quantitized values that we can count with a quantum number n.

Schödinger guessed that the line spectrum of the hydrogen atom indicated that the equations of motion must be wave equations with boundary conditions that fix the possible energy levels. The quantum numbers that describe an atom are a natural consequence of Schrödinger's wave equation that describes the atom.

In summary we can write down the steps we need take in order to apply Schrödinger's equation:

  1. Determine the appropriate potential for the system
  2. Write down the classical Hamiltonian for the system
  3. Form the quantum mechanical Hamiltonian by replacing the momentum and kinetic energy in the classical Hamiltonian with their quantum operators
  4. Establish the boundary conditions for the wavefunction
  5. Solve Schrödinger's eigen value equation to determine the eigen functions and eigen values of the system.