next up [*]
Next: 10.3 Completeness and time-dependence Up: 10. Time dependent wave Previous: 10.1 correspondence between time-dependent

10.2 Superposition of time-dependent solutions

There has been an example problem, where I asked you to show ``that if $ \psi_{1}^{}$(x, t) and $ \psi_{2}^{}$(x, t) are both solutions of the time-dependent Schrödinger equation, than $ \psi_{1}^{}$(x, t) + $ \psi_{2}^{}$(x, t) is a solution as well.'' Let me review this problem
- $\displaystyle {\frac{\hbar^2}{2m}}$$\displaystyle {\frac{\partial^2}{\partial x^2}}$$\displaystyle \psi_{1}^{}$(x, t) + V(x)$\displaystyle \psi_{1}^{}$(x, t) = $\displaystyle {\frac{\hbar i\partial}{\partial t}}$$\displaystyle \psi_{1}^{}$(x, t)  
- $\displaystyle {\frac{\hbar^2}{2m}}$$\displaystyle {\frac{\partial^2}{\partial x^2}}$$\displaystyle \psi_{2}^{}$(x, t) + V(x)$\displaystyle \psi_{2}^{}$(x, t) = $\displaystyle {\frac{\hbar i\partial}{\partial t}}$$\displaystyle \psi_{2}^{}$(x, t)  
- $\displaystyle {\frac{\hbar^2}{2m}}$$\displaystyle {\frac{\partial^2}{\partial x^2}}$[$\displaystyle \psi_{1}^{}$(x, t) + $\displaystyle \psi_{2}^{}$(x, t)] + V(x)[$\displaystyle \psi_{1}^{}$(x, t) + $\displaystyle \psi_{2}^{}$(x, t)] = $\displaystyle {\frac{\hbar i\partial}{\partial t}}$[$\displaystyle \psi_{1}^{}$(x, t) + $\displaystyle \psi_{2}^{}$(x, t)], (10.4)

where in the last line I have use the sum rule for derivatives. This is called the superposition of solutions, and holds for any two solutions to the same Schrödinger equation!

Question: Why doesn't it work for the time-independent Schrödinger equation?



© 1998 Niels Walet, UMIST
Email Niels.Walet@umist.ac.uk