Quantum Theory - Particle Physics Group

CHAPTER 2. WAVE MECHANICS AND THE SCHRÖDINGER EQUATION 24V 0 > 0V(x)✻✛-a a✲x✎☞✍✌ IV 0 < 0✎☞✍✌ II✎☞✍✌ III❄Figure 2.3: One-dimensional square potential well and barrierContinuity conditionsWe first need to study the behavior of the wave function at a discontinuity of the potential.Integrating the time-independent Schrödinger equation (2.40) in the formu ′′ (x) = 2m (V − E)u(x) (2.48)2 over a small interval [a − ε,a + ε] about the position a of the jump we obtain∫a+εa−εu ′′ (x)dx = u ′ (a + ε) − u ′ (a − ε) = 2m 2∫a+εa−ε(V − E)u(x)dx. (2.49)Assuming that u(x) is continuous (or at least bounded) the r.h.s. vanishes for ε → 0 andwe conclude that the first derivative u ′ (x) is continuous at the jump and only u ′′ (x) has adiscontinuity, which according to eq. (2.48) is proportional to u(a) and to the discontinuity ofV (x). With u(a ± ) = lim ε→0 u(a ± ε) the matching condition thus becomesu(a + ) = u(a − ) and u ′ (a + ) = u ′ (a − ), (2.50)confirming the consistency of our assumption of u being continuous. Even more unrealisticpotentials like an infinitely high step for which finiteness of (2.49) requires{V 0 for x < aV (x) =⇒ u(x) = 0 for x ≥ a , (2.51)∞ for x > aor δ-function potentials, for which (2.49) implies a discontinuity of u ′{u(a+ ) − u(a − ) = 0V (x) = V cont. + Aδ(a) ⇒u ′ (a + ) − u ′ (a − ) = A 2m u(a) , (2.52) 2are used for simple and instructive toy models.