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.


Back To Quantum Mechanics

Mini Physics

As the Administrator of Mini Physics, I possess a BSc. (Hons) in Physics. I am committed to ensuring the accuracy and quality of the content on this site. If you encounter any inaccuracies or have suggestions for enhancements, I encourage you to contact us. Your support and feedback are invaluable to us. If you appreciate the resources available on this site, kindly consider recommending Mini Physics to your friends. Together, we can foster a community passionate about Physics and continuous learning.

Leave a Comment

This site uses Akismet to reduce spam. Learn how your comment data is processed.