Ch 5: Section 6 The Finite Well
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Ch 5: Section 6 The Finite Well Finite Well ∗ Inside U(x) = 0 ∗ Outside U(x) = U0 Inside the Finite Well β2 ππ 2 ππ π₯π₯ − + ππ π₯π₯ ππ π₯π₯ = πΈπΈππ π₯π₯ 2 2ππ ππππ In the well U = 0, β2 ππ 2 ππ π₯π₯ − = πΈπΈππ π₯π₯ 2 2ππ ππππ Re-write as we did for the infinite well: ππ 2 ππ π₯π₯ πππ₯π₯ 2 = −ππ 2 ππ π₯π₯ where ππ 2 = 2ππππ β2 Inside the Finite Well General ππ 2 ππ π₯π₯ solution to πππ₯π₯ 2 = −ππ 2 ππ π₯π₯ is ππ π₯π₯ = π΄π΄ sin ππππ + π΅π΅ cos(ππππ) (For infinite well, we said there wave function outside must be zero. We cannot get the cosine part to be zero at x = 0. So we dismissed the cosine part in the case of an infinite well.) Outside the Finite Well Outside U(x) = U0 β2 ππ 2 ππ π₯π₯ − + ππ π₯π₯ ππ π₯π₯ = πΈπΈππ π₯π₯ 2 2ππ ππππ β2 ππ 2 ππ π₯π₯ − + ππ0 ππ π₯π₯ = πΈπΈππ π₯π₯ 2 2ππ ππππ Rewrite as ππ 2 ππ π₯π₯ πππ₯π₯ 2 ππ 2 ππ π₯π₯ ππππ 2 = 2ππ(ππ0 −πΈπΈ) ππ β2 π₯π₯ and = πΌπΌ 2 ππ π₯π₯ where πΌπΌ 2 = 2ππ(ππ0 −πΈπΈ) β2 Outside the Finite Well ππ2 ππ π₯π₯ 2 = πΌπΌ ππ π₯π₯ 2 ππππ The MATHMATICAL SOLUTION is ππ π₯π₯ = πΆπΆππ πΌπΌπ₯π₯ + πΊπΊππ −πΌπΌπΌπΌ where C and G are constants (see p. 161) Physically, the wave function cannot diverge as x →±∞. So ππ π₯π₯ = πΆπΆππ πΌπΌπ₯π₯ in II ππ π₯π₯ = πΊπΊππ −πΌπΌπΌπΌ in III Applying Boundary Conditions So far, we have the form of solution in three regions, and we have 4 constants. We find information about these constants as well as about the energy levels by applying the boundary conditions. The wave function and its first derivative must be smooth. ππ π₯π₯ = πΆπΆππ πΌπΌπ₯π₯ ππ π₯π₯ = π΄π΄ sin ππππ + π΅π΅ cos(ππππ) ππ π₯π₯ = πΊπΊππ −πΌπΌπΌπΌ Energy Levels in Finite Well β2 ππ 2 ππππ 2 sec ππ = 1, 3, 5 … 2 ππ0 = 2ππ β2 ππ 2 ππππ 2 csc ππ = 2, 4, 6 … 2 2ππ where ππ − 1 ππ < ππππ < ππππ ∗ U0 is the depth of the well (a constant) ∗ Only certain values of k (or E) are allowed (where U0 crosses). ∗ Depth of well determines how many energy states are allowed. (At least one.) Compare Finite to Infinite well Finite ∗ Energy is quantized. ∗ Ground state is NOT zero. ∗ Can have finite number of energy levels determined by depth (U0). ∗ Levels are lower (for same n) ∗ Wave function is exponential decaying outside Infinite ∗ Energy is quantized. ∗ Ground state is NOT zero. ∗ Can have infinite number of energy levels. ∗ Levels are higher (for same n) ∗ Wave function is zero outside Penetration depth ∗ The fact that the wave function is an exponential decay outside the finite well (as opposed to being zero) means the particle penetrates the walls of the well. ∗ The penetration depth is 1 β πΏπΏ ≡ = πΌπΌ 2ππ(ππ0 − πΈπΈ) Summary of Finite Well Equations (memorize) 2ππππ (same as infinite well) β2 2ππ(ππ0 −πΈπΈ) 2 πΌπΌ = β2 1 β πΏπΏ ≡ = πΌπΌ 2ππ(ππ0 −πΈπΈ) ∗ ππ 2 = ∗ ∗ ππ π₯π₯ = πΆπΆππ πΌπΌπ₯π₯ ππ π₯π₯ = π΄π΄ sin ππππ + π΅π΅ cos(ππππ) ππ π₯π₯ = πΊπΊππ −πΌπΌπΌπΌ Compare to infinite well (memorize) Standing waves with quantized λ and k: ππππ = 2ππ ππππ = ππππ πΏπΏ and ππ 2 = 2ππππ β2 2 ππππ π₯π₯ = sin ππππ π₯π₯ ππππππππππππ πΏπΏ ππ π₯π₯ = 0 (ππππππππππππππ) And quantized energy 1 πππππ πΈπΈππ = 2ππ πΏπΏ 2 Homework ∗ Ch 5: 34 due Thursday (05NOV15) ∗ Ch 5: 36, 37 and 42 in class
