Let us consider a toy quantum mechanical problem: a particle with mass $m$ is in a potential of the following form:
\[U(x) = \begin{cases}
0, \quad &|x| > a, \\
-U_0, \quad &|x| < a
\end{cases}.\]
Now let us imagine that we reduce the width of the potential $a$ to zero, leaving the product $U_0 a$ constant and equal to $\hbar^2 \varkappa_0/m$.

In such a system, it turns out that there is only one bound stationary state with the energy:
\[E=-\frac{\hbar^2 \varkappa_0^2}{m}\]

A1 Find the units of $\varkappa_0$.

A2 Using the one-dimensional Schrödinger equation
\[ -\frac{\hbar^2}{2m} \Psi'' + U(x) \Psi = E \Psi,\]
obtain solutions for the wave function $\Psi_-$ at $x<0$ and $\Psi_+$ at $x>0$ for the bound state. Remember that

• The probability density $\rho(x)$ of finding a particle at point $x$ is equal to $\Psi^*(x) \Psi (x)$ and therefore \[1 = \int\limits_{-\infty}^0 \Psi_-^* \Psi_- \, dx + \int\limits_0^{+\infty} \Psi_+^* \Psi_+ \, dx\]
• The wave function is continuous $\Psi_-(0)=\Psi_+(0)$
• The wave function is defined up to phase factor $e^{i \varphi}$, where $\varphi$ is a real number

Sketch the graph showing the dependence of $|\Psi|$ on $x$.

A3 Calculate the probability $p(b)$ that the particle will be found at the distance less than $b$ from the center of the well.

Verify that $p(0)=0$ and $p(\infty)=1$.