Time-Independent Schrodinger Equation

The wave function for de Broglie waves must satisfy an equation developed by Schrodinger. If we simply consider a particle moving in one dimension (e.g. along the x-axis), we can write Schrodinger equation as

$\frac{d^{2} \psi}{dx^{2}} = – \frac{2m}{h^{2}} \left(E – U \right) \psi$

where E is the total energy and U is the potential energy.

This is the famous time-independent Schrodinger eqution (TISE), and its solution ψ(x) are often referred to as time-dependent wave functions. In principle, if the potential energy U(x) is known for the system, we can solve the TISE and obtain the wave functions and allowed energy states. Since U(x) may vary with position, it may be required to solve the equation piece-wise. In doing so, we require the wave function ψ(x) to satisfy several conditions.

  1. ψ(x) and $\frac{d \psi}{dx}$ must be continuous in all regions
  2. For ψ(x) to obey the normalisation condition, ψ(x) must approach zero as x approaches ±∞.
  3. ψ(x) must be single-valued.


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