2.2: Schrodinger equation, particle in a box, wavefunctions

• Erwin Schrodinger and Werner Heisenberg proposed wave properties to electrons and developed the theories of Wave Mechanics.
• Schrodinger's equation describes the behavior of the electron in three dimensions.
• Schrodinger's equation defined the electron's position, mass, total energy, and potential energy.
• The Hamiltonian operator is a function over three-dimensional space that corresponds the sum of kinetic energies and potential energies of the particles in a system.
• The kinetic energy of the electron is the energy due to motion of the electron.
• The potential energy depends on the attractive electrostatic force between the electron and the nucleus, which is essentially the same as the electrostatic force defined by Coulomb's law.
• Coulomb's law states that when two opposite charges are attracted to one another, the potential energy of the force is negative.
• When an electron is close to the nucleus, the potential is a large negative number corresponding to a strong attractive force.
• When an electron is farther from the nucleus, the potential energy is still negative but with a smaller magnitude, corresponding to a weaker attractive force.
• If the electron is very far from the nucleus, then the attractive force, and the potential energy, is zero.
• Wavefunction of an electron describes the electron's position in space, relative to the nucleus.
• The square of the wavefunction describes an atomic orbital.
• Generally, in a one-electron atom, the electron wavefunction is defined by the wave's distance from the nucleus and its angle with respect to the x, y, and z axes of the atom's Cartesian coordinates.
• The wavefunction is defined by three of the quantum numbers and the radial variation, R, depends on the electron's distance from the nucleus.
• The quantum numbers n (energy level) and l (orbital type) define R.
• The angular variation, Y, depends on the angle with respect to the x, y, and z coordinates, and depends on the quantum numbers l and m1.
• The values of x, y, and z for the Hamiltonian are limited by the allowed positions of electrons according to the wavefunction, which is limited by integer values of n.
• The wavefunction describes the wave properties of an electron.
• The probability of finding the electron somewhere in space is the square of the wavefunction and describes the shape and size of an electron's orbital.
• There is only one possible value for the wavefunction for any set of the three quantum numbers n, l, m1.
• The wavefunction approaches zero as r approaches infinity, and so the square of the wavefunction also approaches zero as r appproaches infinity.
• The wavefunction must be normalized. The total probability of finding the electron in all of space must be 1.
• Any two orbitals must not occupy the same space, they are orthogonal.
• The probability of finding the electron anywhere in infinite space must be defined. The wavefunctions and their first derivatives must be continuous.