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If we consider the solution of the Schrodinger equation for an infinite square well we can deduce that the solution to this situation is as follows:

$$\Psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi}{L}x\right)\qquad n=1,2,3,\ldots$$

The above is an eigenfunction and I know how to arrive at this solution however am still wondering as to what this function is supposed to tell me, physically speaking.

Also I am aware that any superposition of these wavefunctions or any solution to the wavequation is also a solution, however I am confused as to:

  1. What do the different values of $n$ tell me and why I should be interested in them (what different physical situations do they represent if any)?

  2. Why is a superposition of states important, as in, what is it supposed to represent?

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  • $\begingroup$ I think there is a typo in your argument of the $\sin$. $\endgroup$ – ZeroTheHero May 25 '17 at 13:35
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Because it is a solution to the time-independent Schrodinger equation $$ \hat H\Psi_n(x)=E_n\Psi_n(x)\tag{1} $$ and it satisfies the boundary condition of your problem, this solution describes a state of specified energy, in the sense that the fluctuation of the energy $\Delta E_n=0$. This is easy to show since \begin{align} \Delta E^2&= \langle \Psi_n\vert H^2 \vert\Psi_n\rangle - \langle \Psi_n\vert H \vert\Psi_n\rangle^2\, , \\ &=E_n^2-(E_n)^2=0\, . \end{align} (If there is no fluctuation in energy you can assign a definite energy to this solution.)

Because only certain values of $E$ can lead to solutions of (1) that satisfy the boundary conditions, and because these values can be labelled by $n$, the various $E_n$ values thus give you a sequence of solutions for which the energy $E_n$ associated with this solution does not fluctuate: in other words, you can speak of solution $\psi_n(x)$ "having definite energy $E_n$". This is quite useful because energy is conserved, so speaking of a solution $\psi_n(x)$ which has definite energy is something that remains true in time.

  1. In this way the value of $E_n$ functions a time-independent label, and you can "name" solutions using energy, i.e. you can say that your system is in the state having energy $E_n$. Given $E_n$ is then enough to completely identify the state $\Psi_n(x)$ and use $\Psi_n(x)$ for whatever purpose you need. (In higher dimensional problems you need to specify more than just $E_n$ to uniquely identify the state, but in 1d this is enough.)

  2. Superpositions are solutions which satisfy the time-dependent Schrodinger equation, and in particular satisfy the boundary condition of the problem. They are legitimate solutions but do NOT have definite energy, i.e. for those solutions $\Delta E\ne 0$, in constradistinction with the states $\Psi_n(x)$ which are solution to the time-independent Schrodinger equation. In general, there is nothing to suggest that your system will be in a state of definite energy (in the sense that there is no reason to suggest $\Delta E=0$) and it will be a superposition of $\Psi_n(x)$. The precise superposition depends on the initial condition of your problem, i.e. on the initial wavefunction at $t=0$, much like a string clamped at both ends that has some weird initial shape.

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Think of the different n as the subscript i of a vector's components $u_i$. The total wave function is the linear superposition of these components: $\mathbf{u}=u_i\mathbf{g}_i$.

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