# Meaning of eigenfunctions in quantum mechanics

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?

• I think there is a typo in your argument of the $\sin$. – ZeroTheHero May 25 '17 at 13:35

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.
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: $$\mathbf{u}=u_i\mathbf{g}_i$$.