 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/2017/s18/2/1/solution

For

E < 0

, impose decay and continuity on

G = C e^{- κ ∣ x - x^{'} ∣}

- 2 κ C

, so

- ℏ^{2} (G_{+}^{'} - G_{-}^{'}) / (2 m) = 1

requires

G (x, x^{'}) = \frac{m}{κ ℏ ^{2}} e^{- κ ∣ x - x^{'} ∣}, κ = - 2 m E /ℏ.

There is no nonzero free solution decaying at both infinities that can be added without spoiling the derivative condition.

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