Choice of Basis for solution space of a Hamiltonian

Question

I'm quite new to the field of Quantum chemistry, but I can't wrap my head around the following postulate of Quantum Chemistry, stated as Postulate 3 in [1]:

Postulate 3: In any measurement of the observable associated with the operator A, the only values that will ever be observed are the eigenvalues $a_n$ which satisfy the eigenvalue equation $$A\psi_n = a_n \psi_n.$$

My first thought was, that this was just a little bit fuzzy, because one result in QC is, that energies, as eigenvalues of the Hamiltonian, don't exists continiuosly but only discrete, so we would need restrict the set of possible $\psi_n$ to a set of basis vectors for the space of eigenfunctions of A. Otherwise we could use any linear combination of eigenvectors and thus realise any (energy) eigenvalue.

This is for example done, if we take a look at the 1 dimensional particle in the box, which is usually provided with the basis set $B_1 = \{\psi_n\}_{n \in \mathbb{N}}$ with \begin{equation} \psi_n(x) = \begin{cases}\left(\frac{2}{L}\right)^{1/2} \sin\left(\frac{n\pi x}{L}\right) & 0\leq x \leq L \\ 0 & else\end{cases}. \end{equation}

According to the postulate above, the choice of the basis is quite important as the corresponding energy values to the wave functions $\psi_n$ are the only ones that could be observed. However, and here is my problem, the choice of this basis set for the 1D particle in the box seems arbitrary to me, as we could also set \begin{equation} \psi'_1 = \frac{1}{\sqrt{2}}(\psi_1 + \psi_2) \text{ and } \psi'_2 = \frac{1}{\sqrt{2}}(\psi_1 - \psi_2), \end{equation}

so that $B_2 = \{\psi'_1, \psi'_2, \psi_3, \psi_4,\dots\}$ would also be an (orthonormal) basis set for the same space as $B_1$.

The important implication would now be, that the corresponding energy values for $B_1$ and $B_2$ are different and hence, depending on the choice of such a basis set, the system could be in different (discrete) states.

To me this seems as a contradiction, as I would expect that the energy states of the system should be independent of that mathematical framework, i.e. every mathematical formulation should yield the same physical properties. This leads me to the conclusion, that I made a mistake somewhere or I did understand something wrong. Any help would be much appreciated.

[1]: D. A. McQuarrie, “Quantum Chemistry,” Oxford University Press, Oxford, 1983.

Answer 1

If you can algebraically solve the Schroedinger differential equation then generally $n$ quantised solutions are obtained and are determined by the potential energy function you use and the boundary conditions. If you cannot solve the equation algebraically then using a basis set of other functions, such as sin/cos, gaussians etc, then the solution can be found by adding these functions to form a basis set of $m$ functions and $m$ can increase until sufficient accuracy is obtained for the $n^{th}$ solution to the Schroedinger equation. Generally $m \gg n$ because the basis set functions are not solutions to your problem but will approximate to it if sufficient number of terms are used. See 'Solution of the Schrödinger Equation for One-Dimensional Anharmonic Potentials' J. Chem. Educ. 2011, 88, 7, 929–931 for examples.