91影视

To show that the infinite well鈥檚 standing-wave function can be expressed as the sum of two travelling waves, we add two waves moving in opposite directions.

The equation for each wave is

$蠄(x,t)=A{e}^{i(kx-蝇t)}$

Since our two waves are moving in opposite directions, they have opposite momentum(so our second wave is negative) and an opposite sign for K. This gives us equation for the sum of the travelling waves:

$蠄(x,t)=A{e}^{(kx-蝇t)}$

So,role="math" localid="1658425779738" $\begin{array}{rcl}蠄(x,t)& =& A\left(e^{i(kx-蝇t)}-A{e}^{i(kx-蝇t)}\right)\\ & =& A{e}^{i(kx-蝇t)}\left(e^{ikx}-e^{-ikx}\right)\\ & =& 2Ai\left(\frac{e^{ikx}-e^{-ikx}}{2i}\right)e^{iex}\end{array}$

We know that,

$\begin{array}{rcl}& & \left(\frac{e^{ikx}-e^{-ikx}}{2i}=\sin kx\right)\end{array}$

Thus,

$蠄(x,t)=2Ai\sin \left(kx\right)e^{-iex}$

What we ended up with is the indefinite-well鈥檚 standing-wave function, thus proving it is just the sum of two waves moving in opposite directions.