 Solution

ID: past-exam-of-the-short-options-paper-shared-by-prelims-and-parts-a-and-b-of-the-physics-course-of-the-university-of-oxford/2018/s18/2/c/solution

Factorizing the projector found in part b gives the normalized eigenfunctions

φ_{n} (x) = \frac{2}{a} sin \frac{nπ ( x + a /2 )}{a}, - a /2 < x < a /2, n \geq 1,

with zero continuation outside. Their squared integrals are one and different positive integers give orthogonal functions. These are the familiar infinite-square-well states shifted so that the well centre is at zero. Each eigenfunction has an arbitrary overall constant phase, which cancels from the projector.

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