Where does this time-dependent wavefunction of $\Psi(x,t)= \sum_{n=1}^\infty \psi_n(x)\exp(\frac{-in^2\pi^2\hbar t}{2mL^2})$ come from?

Question

I was reading this blog post on simulating the probability desnity of the Schrodinger equation but there was one equation that I could not quite understand.

Firstly, defining $$V(x) = \begin{cases} 0 & 0\le x\le L \\ \infty & \mathrm{otherwise} \end{cases}$$ for the classic Particle in a Box. Next, solving the time-independent Schrodinger equation for $V(x)$ you get $$\psi_n(x)=\sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right)$$ I understand up to here but the next equation is what messes me up. The post says:

The general time dependent solution can be written as a linear combination of the separable solutions along with their time dependent part, $$\Psi(x,t)= \sum_{n=1}^\infty \psi_n(x)\, \exp\left({\frac{-in^2\pi^2\hbar t}{2mL^2}}\right)$$

I do not understand where the term $\exp(\frac{-in^2\pi^2\hbar t}{2mL^2})$ comes from as the blog doesn't explain the equation and I do not know what this term is even called. It says its a combination of the seprable solutions with their time dependent part. What is the time dependant part? Why is it summed?

Answer 1

That exponential factor is time evolution factor.

$$|\psi(t)\rangle = e^{-i \hat{H} t/\hbar} |\psi(0)\rangle $$

Such exponential factor comes from unitary property of time evolution operator to maintain its normalization condition. Here, $H$ is Hamiltonian which has eigenvalues as energy.

Energy eigenvalues for given hard wall potential can be obtained from following process:

$$-\frac{\hbar^2}{2m} \frac{d\psi}{dx} = E \psi \; \rightarrow \; \psi \sim \sin kx$$

$$k= \sqrt{\frac{2mE}{\hbar^2}} = \frac{n \pi}{L}$$

$$E_n = \frac{n^2 \pi^2 \hbar}{2mL^2}$$

Above DE has solutions you written, and $k$ is quantized by boundary condition. If you put those energy eigenvalues into time evolution factor, then you can get desired solution.

Linear combination means any wave function in this system can be expanded with energy eigenstates. It's because $\sin nx/L$ has completeness. (Think Fourier series)