The Schrödinger equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. It is the quantum mechanical equivalent of Newton's laws of motion in classical mechanics. The equation is written as i-h-bar times the partial derivative with respect to time of the wave function equals the Hamiltonian operator acting on the wave function. Here, Psi represents the wave function, H-hat is the Hamiltonian operator representing energy, h-bar is the reduced Planck constant, and i is the imaginary unit. The wave function evolves over time according to this equation, determining the probability of finding a particle in a specific state.
For systems where the Hamiltonian operator is independent of time, we can separate the time and space variables in the Schrödinger equation. This leads to the time-independent Schrödinger equation, which gives us stationary states with definite energy. The wave function can be written as the product of a spatial function and a time-dependent phase factor. The time-independent equation is an eigenvalue equation, where the energy E is the eigenvalue of the Hamiltonian operator. A classic example is the particle in a box, where the stationary states are standing waves. The energy levels are quantized, with higher energy states having more nodes in their wave functions. For the particle in a box, the energy is proportional to the square of the quantum number n.
The wave function itself is not directly observable. Instead, the square of its magnitude gives a probability density. The probability of finding a particle in a small volume dV around position r is given by the absolute square of the wave function, |Ψ(r,t)|² dV. This is also written as the product of the wave function and its complex conjugate. For a wave function to have physical meaning, it must be normalized so that the total probability of finding the particle somewhere in space equals 1. This is expressed mathematically by integrating the probability density over all space. In this example, we see a wave function in blue and its corresponding probability density in red. Notice that where the wave function has its maximum amplitude, the probability density also peaks, and where the wave function is zero (nodes), the probability of finding the particle is zero.
The Schrödinger equation has numerous applications in quantum mechanics and modern technology. It is fundamental to understanding atomic and molecular structure, as it describes the behavior of electrons in atoms. For example, in the hydrogen atom, the Schrödinger equation gives us the electron orbitals that determine chemical bonding properties. The equation is also crucial in solid state physics, which forms the basis for semiconductor technology in modern electronics. In quantum computing, the Schrödinger equation governs how quantum bits evolve and interact. Another important application is quantum tunneling, where particles can penetrate energy barriers that would be impossible in classical physics. This phenomenon is essential for nuclear fusion in stars and is used in scanning tunneling microscopes and quantum devices.
To summarize what we've learned about the Schrödinger equation: First, it is the fundamental equation of quantum mechanics that describes how quantum states evolve over time. Second, for time-independent systems, we can find stationary states with definite energy values. Third, the square magnitude of the wave function represents probability density, telling us where we're likely to find a particle. Fourth, wave functions must be normalized so that the total probability equals one, giving them physical meaning. Finally, the Schrödinger equation has wide-ranging applications from understanding atomic structure to enabling quantum computing and explaining tunneling phenomena. This equation represents one of the most profound shifts in our understanding of the physical world.