The Schrödinger Wave Equation
The Schrödinger equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes with time. It was formulated by Austrian physicist Erwin Schrödinger in 1926.
Time-Dependent Schrödinger Equation
\[ i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r},t) = \hat{H}\Psi(\mathbf{r},t) \]
Where:
- \( \Psi(\mathbf{r},t) \) is the wave function of the system
- \( \hat{H} \) is the Hamiltonian operator (representing the total energy of the system)
- \( i \) is the imaginary unit
- \( \hbar \) is the reduced Planck constant
Time-Independent Schrödinger Equation
For stationary states where the Hamiltonian doesn’t depend explicitly on time, we can separate variables and obtain:
\[ \hat{H}\psi(\mathbf{r}) = E\psi(\mathbf{r}) \]
Where:
- \( \psi(\mathbf{r}) \) is the time-independent wave function
- \( E \) is the energy eigenvalue
Derivation of the Schrödinger Equation
The Schrödinger equation cannot be derived from first principles, but we can motivate its form using analogies from classical physics and quantum principles.
Step 1: Wave-Particle Duality
De Broglie’s hypothesis relates a particle’s momentum to its wavelength:
\[ p = \hbar k \]
where \( k = 2\pi/\lambda \) is the wave number.
Step 2: Energy Relations
For a free particle, the total energy is kinetic energy:
\[ E = \frac{p^2}{2m} \]
Combining with de Broglie’s relation:
\[ E = \frac{\hbar^2 k^2}{2m} \]
Step 3: Wave Function and Operators
Assume a plane wave solution:
\[ \Psi(x,t) = Ae^{i(kx-\omega t)} \]
Note that:
\[ -i\hbar\frac{\partial}{\partial x}\Psi = \hbar k\Psi = p\Psi \]
\[ i\hbar\frac{\partial}{\partial t}\Psi = \hbar\omega\Psi = E\Psi \]
This suggests the operator correspondences:
\[ \hat{p} = -i\hbar\nabla \]
\[ \hat{E} = i\hbar\frac{\partial}{\partial t} \]
Step 4: Constructing the Hamiltonian
For a particle in a potential \( V(x) \), the total energy is:
\[ E = \frac{p^2}{2m} + V(x) \]
Replacing with operators:
\[ i\hbar\frac{\partial}{\partial t} = -\frac{\hbar^2}{2m}\nabla^2 + V(x) \]
Acting on the wave function \( \Psi \), we recover the time-dependent Schrödinger equation.
Interpretation of the Wave Function
The wave function \( \Psi \) contains all information about a quantum system. According to the Born rule:
\[ P(\mathbf{r},t) = |\Psi(\mathbf{r},t)|^2 \]
gives the probability density of finding the particle at position \( \mathbf{r} \) at time \( t \).
Applications
- Solving for atomic orbitals in atoms
- Predicting molecular structure in chemistry
- Semiconductor physics and electronics
- Quantum computing and information theory