- #1

Yourong Zang

- 5

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**1. Homework Statement**

Consider a potential field

$$V(r)=\begin{cases}\infty, &x\in(-\infty,0]\\\frac{\hslash^2}{m}\Omega\delta(x-a), &x\in(0,\infty)\end{cases}$$

The eigenfunction of the wave function in this field suffices

$$-\frac{\hslash^2}{2m}\frac{d^2\psi}{dx^2}+\frac{\hslash^2}{m}\Omega\delta(x-a)\psi=E\psi$$

A textbook gives the following solution:

$$\psi(x)=\begin{cases}Asin(kx), &x\in(0,a)\\ sin(kx+\phi), &x\in(a,\infty)\end{cases}$$

where

$$k^2=\frac{2mE}{\hslash^2}$$

**2. Homework Equations**

__I can clearly understand the first part but in the second part, why does the amplitude of the function equal to 1 and why is there a phase angle?__

__And is this wave__

$$\psi(x)=\sin(kx+\phi)$$

called something like the "excitation mode"?$$\psi(x)=\sin(kx+\phi)$$

called something like the "excitation mode"?

**3. The Attempt at a Solution**

A solution about delta potential is not what I want. There is an infinite potential on the left+a delta potential.

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