10.5: Exercise:Solve Schrödinger equation by variable separation method Consider a particle with the mass $m$ in the box (i.e., the closed interval $[0,2]$) in the one dimensional space ${\mathbb R}$. The motion of this particle (i.e., the wave function of the particle) is represented by the following Schrödinger equation

\begin{align} i \hbar \frac{\partial}{\partial t } \psi (q ,t) = - \frac{\hbar^2 \partial^2}{2 m \partial q^2 }\psi (q ,t) + V_0(q)\psi (q ,t) \qquad (\mbox{ in } H = L^2 ({\mathbb R} ) ) \end{align}

where

\begin{align} V_0(q) = \left\{\begin{array}{ll} 0 \quad & (0 \le q \le 2) \\ \infty & (\mbox{ otherwise }) \end{array}\right. \end{align} Figure 10.5: Particle in a box

Put

\begin{align} \phi(q,t) = T(t) X(q) \qquad (0 \le q \le 2). \end{align}

And consider the following equation:

\begin{align} i \hbar \frac{\partial}{\partial t } \phi (q ,t) = - \frac{\hbar^2 \partial^2}{2 m \partial q^2 }\phi (q ,t). \end{align}

Then, we see

\begin{align} \frac{i T' (t) }{T(t)} =- \frac{ X'' (q)}{2 m X(q) } = K (= \mbox{ constant }). \end{align}

Then,

\begin{align} \phi (q, t) = T(t)X(q) = C_3 \exp (iKt) \Big(C_1 \exp(i \sqrt{2mK/\hbar} \; q) + C_2 \exp(-i \sqrt{2mK/\hbar} \; q). \Big) \end{align}

Since $X(0)=X(2)=0$ (perfectly elastic collision), putting $K= \frac{n^2 \pi^2 \hbar}{8 m}$, we see

\begin{align} \phi (q, t) = T(t)X(q) = C_3 \exp (\frac{ i n^2 \pi^2 \hbar t}{8 m}) \sin(n \pi q / 2) \qquad (n=1,2,...). \end{align}

Assume the initial condition:

\begin{align} \psi(q,0) = c_1 \sin (\pi q / 2) + c_2 \sin (2 \pi q / 2) + c_3 \sin (3 \pi q / 2) + \cdots. \end{align}

where $\int_{\mathbb R} | \psi(q , 0)|^2 dq = 1$. Then we see

\begin{align} & \psi(q,t) \\ = & c_1 \exp (\frac{ i \pi^2 \hbar t}{8 m}) \sin (\pi q / 2) + c_2 \exp (\frac{ i 4 \pi^2 \hbar t}{8 m}) \sin (2 \pi q / 2) + c_3 \exp (\frac{ i 9 \pi^2 \hbar t}{8 m}) \sin (3 \pi q / 2) + \cdots. \end{align}

And thus, we have the time evolution of the state by

\begin{align} \rho_t = | \psi(\cdot , t ) \rangle \langle \psi(\cdot , t ) | \;\; \quad ( \in {\frak S}^p(Tr (H)) \subseteq B(H) ) \qquad \qquad (\forall t \ge 0 ) \end{align}