README.tex.md

Infinite-square-well-Schrodinger-equation

Analytical solution to the Time-Dependent Schrodinger equation for a particle in an infinite square well.

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The Theory

\usepackage{physics} \usepackage{math}

Solving energy eigenvalues and eigenstates are used to understand the future of a physical system in quantum mechanics. The Schrödinger equation is the tool that allows for time evolution where the equation is,

\begin{equation} \ket{\psi} = i \hbar \frac{d}{dt} \ket{\psi}. \end{equation}

When the Schrödinger equation is time-dependent, the solution becomes

\begin{equation} \tag{2} \ket{\psi(t)} = \Sigma_n C_{n}e^{-iE_{n}/\hbar} \ket{E_n}. \end{equation}

Therefore, when $t = 0$ and $\psi(0)$ becomes

\being{equation} \tag{3} \ket{\psi(0)} = \Sigma_n C_{n} \ket{E_n} \end{equation}

since $e^{0} = 1$.

The initial $\psi(x,0)$ is generated by a complex gaussian wave packet. Once the parameters in the program are set or the default parameters are used the following equations are executed. The first step is to normalize the wave function given by the equation below,

\begin{equation} \tag{4} 1 = \int_{-\infty}^{\infty} A^{2}|\psi(x,0)|^{2} dx. \end{equation}

Once $\psi(x,0)$ is normalized as the first step, the energy eigenstate wave functions are then found. Again, the energy eigenstate wave function need to be normailized and the following equation is derived,

\begin{equation} \tag{5} \varphi_n(x) = \sqrt{\frac{2}{L}}*sin\Big(\frac{n\pi x}{L}\Big). \end{equation}

Discrete quantized wave vectors are required to solve the Time-Dependent equation, $\psi(x,t)$. The energy quantization for the system becomes,

\begin{equation} \tag{6} E_n = \frac{n^2\pi^2 \hbar^2}{2mL^2}. \end{equation}

Finally, the last item the time dependent Schrödinger equation depends on is $C_n$. $C_n$ is the expansion coefficient that chages the probability amplitude. The indefinite integral of the complex conjugate of eigenstate wave function and the initial wave function with respect to the spatial dimension x,

\begin{equation} \tag{7} c_n = \int_{-\infty}^{\infty} \varphi_n(x)^{*} \psi(x,0) dx. \end{equation}

From equation (2): we bring together, $\varphi_n(x)$, $E_n$, and $C_n$. The result is

\begin{equation} \tag{8} \psi(x,t) = \sum_{n}^{500} c_{n} e^{-iE_nt/\hbar}\varphi_{n}(x). \end{equation}

Now, as t increase the future of wave equation is determined as shown below in the animation.

Animation real time

The Code