Question

By Solving one-dimensional Schordingers time-independent wave equation for a particle in the well given ?
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Answer 1

The Time independent Schrödinger equation it is just a simple consequence of the dependent one, that tells us that:

$$\frac{\partial \Psi}{\partial t}= I \hbar \hat{H}\Psi$$

This function is extremely easy to solve if we have our wave function expressed in a set for which applying the Hamiltonian operator $$\hat{H}$$ only has the effect of multiplying the function by a constant, that is, the eigenfunctions of the observable. Consequently, egeinvalues of the time independent Schrödinger problem become extremely important, as they are the perfect building blocks for the time dependent wave function, in which each eigenfunction will be multiplied by a phase factor of the form:

$$\Psi= \Sum \limits_{I=0}^{N}\c_{I,0} Psi_i(X)exp (i\hbarE_i t)$$

So you just need to calculate the projection of your original wave function into the eigenfunction set of the particle inside an infinitely feel well, functions that have a simple analytical expression, and once you have all these projections they will be $$c_{i,0}^2$$ for each eigenfunction, also the eigenvalue for the energy will give you the frequency in the imaginary exponential. Just go on until you have the whole lot and you're golden.

General rule in solving the TDSE. It is generally a good idea to divide your problem in small steps so that you can apply a numerical integration and at each of those points to calculate what are the eigenfunctions of the Hamiltonian at that point in time, since those will be very easy to propagate.