Schrödinger Equation

The Schrödinger equation was proposed by physicist Erwin Schrödinger in 1926. It succeeded the quantum theory ideas of Planck which stated the quantization of energy and the great Einstein. It sparked the quantum mechanical era and disproves many concepts from classical mechanics.

History and Background

The Schrödinger equation integrates both classical mechanics and optics. It uses conservation of Energy from classical mechanics written in terms of its wavefunction. It is the basic equation used to solve wave functions of atomic particles such as electrons, protons, and atoms. Some of these examples include the particle-in-a box, the Bohr Hydrogen model, an electron in a force field, the harmonic oscillator, and Spectroscopy (infrared, electronic).

There are two types of Schrödinger equations, a time-dependent and a time-independent. The time-independent Schrödinger equation is used when dealing with stationary states because they do not change over time. In chemistry (general chemistry and physical chemistry) we are mostly dealing with stationary states. The wavefunction is a function of position. In the time-dependent Schrödinger equation, the wavefunction is a function of position and time.

The Time-Independent Schrödinger Equation

From classical mechanics: T + V = E. Kinetic Energy + Potential Energy = Total Energy, as from the Law of Conservation of Energy. The Schrödinger equation uses this fundamental principle in terms of its wavefunction:

$\hat{H}\psi_n = E_n\psi_n$

In the time-independent Schrödinger equation, the Hamiltonian operator is equivalent to the total energy operator. This means that the total energy is also an operator and a scalar quantity. The Hamiltonian operator consists of kinetic energy and potential energy. The energy has an eigenvalue, and the wavefunctions are eigenfunctions. Since the time-independent Schrödinger equation is in terms of its wavefunction, the Energy derived from the Schrödinger equation must be quantized. This equation is only useful for functions that represent waves such as sine, cosine, exponential functions, etc. The second derivative of the function must equal the original function. The following representation Schrödinger equation is readily expandable to three dimensions, though it is more typical to expand to spherical polar coordinates rather than Cartesian.

$-\dfrac{h^2}{8\pi^2m} \times \dfrac{d^2\psi}{dx^2}+ V \psi = E \psi$

where $$-\dfrac{h^2}{8\pi^2m} \times \dfrac{d^2\psi}{dx^2} = KE$$

$V\psi = PE$

$E = Total Energy = V \; + \; E$

Solutions to the Schrödinger Equation

1. The wavefunctions are termed eigenfunctions and each corresponding energy is the eigenvalue.
2. When two or more eigenfunctions have the same eigenvalues, they are said to be degenerate.
3. If a wavefunction from the independent - Schrödinger equation is a solution, any constant multiple of that wavefunction is also a solution.

Atomic Orbitals

Examples of the solutions to the Schrödinger equations are atomic orbitals. If two atomic orbitals are degenerate solutions with the same eigenvalue, then their linear combination is also a solution. The linear combination of the two atomic orbitals would result in a molecular orbital. If two wavefunctions or atomic orbitals are not degenerate and do not have the same eigenvalue, then their linear combination would equal 0. They are then said to be orthogonal:

$\displaystyle \int^{\infty}_{-\infty}\psi^*_{j}\psi_{i}dx = 0$

How to Use the Time-Independent Schrödinger Equation

1. The first step to solving the Schrödinger Equation is to define the potential.

2. The third step is to get the wave functions. If the Schrödinger Equation applies then three things must be true about the wave function.

3. The third step is to solve the Schrödinger Equation.

1. It must be a single value at any point.
2. It must be finite.
3. It must be continuous and its first derivative must also be continuous.
1. The fourth step is to get the Energies.
2. The fifth step is to apply this to real - life situations such as:
1. Transition State Theory
2. Chemical Reactions
3. Optical Properties (colors)
4. Vibrations
5. Bonding

References

1. Chang, Raymond. 2005. Physical Chemistry for the Biosciences. Sausalito (CA): University Science Books. pg. 416 - 426
2. Harris, Daniel C. 1948. Symmetry and Spectroscopy, an Introduction to Vibrational and Electronic Spectroscopy. New York: Dover Publications. pg. 70 - 75.

Contributors

• Michelle Towles (UCD)