Schrödinger Equation
The Schrödinger equation is a fundamental equation in quantum mechanics that describes the behavior of wavefunctions and the time evolution of quantum systems. It was formulated by Erwin Schrödinger and provides a mathematical framework for understanding the wave-like nature of particles. Here are the key formulas associated with the Schrödinger equation:
1. Time-dependent Schrödinger equation:
The time-dependent Schrödinger equation is given by:
iħ ∂ψ/∂t = H ψ
where ħ is the reduced Planck's constant (h/2π), ψ is the wavefunction of the system, t is time, and H is the Hamiltonian operator representing the total energy of the system.
2. Time-independent Schrödinger equation:
The time-independent Schrödinger equation is a special case of the time-dependent equation when the wavefunction is in a stationary state. It is given by:
H ψ = E ψ
where E is the energy of the stationary state. This equation allows the determination of allowed energy levels and corresponding wavefunctions for a given system.
3. Hamiltonian operator:
The Hamiltonian operator, H, represents the total energy of the system and is given by:
H = -ħ^2/(2m) ∇^2 + V
where ∇^2 is the Laplacian operator representing the spatial derivatives, m is the mass of the particle, and V is the potential energy function that depends on the spatial coordinates.
4. Probability interpretation:
The wavefunction ψ describes the probability amplitude for finding a particle in a particular state. The probability density is given by |ψ|^2, and the integral of |ψ|^2 over all space must equal 1, representing the normalization condition.
The Schrödinger equation is a foundational equation in quantum mechanics and is used to describe a wide range of physical systems, from single particles to complex quantum systems. It allows the calculation of energy levels, wavefunctions, and the time evolution of quantum systems, providing insights into various phenomena such as atomic and molecular structure, quantum tunneling, and quantum entanglement.