PHY202 { Quantum Mechanics Summary of Topic 4: The Finite Potential Well The quantum well In a sandwich of two semiconductors, e.g. AlGaAs-GaAs, in the conduction zone one can have an effective potential well. When V0 is very large, we have the limit of the infinite potential well. Otherwise we have a finite potential well. The Finite Potential Well A particle in the potential 8 L V0 x < − 2 < L L V (x) = 0 − 2 ≤ x ≤ 2 : L V0 x > 2 When E > V0: unbound states, total energy E is continuous (not quantized). When E < V0 { bound states. Find them by solving the TISE: • −L=2 < x < L=2 (Region I): 00 2 2 2mE (x) = −k (x), where k = 2 > 0. A general solution is I (x) = A sin kx + B cos kx ~ (A; B { arbitrary constants). • x > L=2 (Region II): 00 2 2 2m(V0−E) −αx αx (x) = α (x), where α = 2 > 0. A general solution is II (x) = Ce + De ~ (C; D { arbitrary constants). Must put D = 0, otherwise (x) not square integrable (blows up at large +ve x). L • x < − 2 (Region III): −αx αx like in region II. A general solution is III (x) = F e + Ge Must put F = 0, otherwise (x) not square integrable (blows up at large -ve x). Note in regions II and III, total E = PE + KE = V0 + KE, but E < V0 hence KE < 0! The potential is symmetric w.r.t. to x = 0 ) expect symmetric (even-parity) and antisym- metric (odd-parity) states Even parity solutions Consider even-parity solutions only: I (x) = B cos kx. Apply general conditions on at x = L=2: • continuous at x = L=2: ) I (L=2) = II (L=2) ) B cos (kL=2) = C exp (−αL=2) 0 0 0 • continuous at x = L=2: ) I (L=2) = II (L=2) ) −Bk sin (kL=2) = −Cα exp (−αL=2) kL α • divide side-by-side: ) The quantisation condition reads tan 2 = k . kL k0L • introduce θ = 2 ) LHS: y(θ) = tan θ. Introduce and θ0 = 2 { const., where q 2 q q 2 2 2mV0 α k0 V0−E θ0 k = 2 > 0, ) RHS: y(θ) = = 2 − 1 = = 2 − 1 0 ~ k k E θ q 2 q 2 α θ0 1 α θ0 • when E V0 ) k = θ2 − 1 / θ . When E % V0 ) θ % θ0 ) k = θ2 − 1 & 0. Odd-parity solutions Consider odd-parity solutions only: I (x) = B sin kx, proceed as before. ) The quantisation kL α condition reads cot 2 = − k . Comments: • The even-parity solutions are determined when the curve y = tan θ intersects the curve α y = k . The odd-parity solutions are determined when the curve y = cot θ intersects the 2 2 α ~ k curve y = − k . The intersection points determine k and hence E = 2m . • larger V0 ) more bound states; smaller V0 ) less bound states • there is always at least one (symmetric) bound state, even in a very shallow well (V0 & 0). • the wavenumber and energy of the nth state is less than in the IPW for which the wavenum- nπ nπ bers kn = L , or θn = 2 , n = 1; 3; 5; ::: for symmetric states, n = 2; 4; 6; ::: for antisym- metric states. 2 2 π π ~ • when θ0 < 2 , V0 < 2mL2 ) no antisymmetric states exist A comparison of the FPW and the IPW Infinite well: Finite well: (x) confined to the well (x) spreads out beyond the well nπ kn = L kn and energies lower infinite tower of states finite tower of states no unbound states unbound states when E > V0 The energy levels in the FPW are lower because the wavefunction spreads out (by penetrating the classically forbidden region) and therefore reduces its KE. Quantum Tunnelling At x > L=2 the wavefunction (x) / e−αx; at x < −L=2 (x) / eαx. When V0 % 1 ) α % 1 ) 1/α & 0, when E % V0 ) α & 0 ) 1/α % 1. • The depth of tunnelling is determined by α { the penetration depth . • Non-zero wavefunction in classically forbidden regions (KE < 0!) is a purely quantum mechanical effect. It allows tunnelling between classically allowed regions. • It follows from requiring that both (x) and 0(x) are continuous! ) Requiring a \reasonable behaviour" of the wavefunction leads to a (classically) \crazy" phenomenon of tunnelling. Quantum states in potential wells Some general properties: • quantum (discrete) energy states are a typical property of any well-type potential • the corresponding wavefunctions (and probability) are mostly confined inside the potential but exhibit non-zero \tails" in the classically forbidden regions of KE < 0 (except when V (x) ! 1 where the tails are not allowed). • both properties result from requiring the wavefunction (x) and its derivative 0(x) to be continuous everywhere (except when V (x) ! 1 where 0(x) is not continuous). • quantum states in symmetric potentials (w.r.t. reflections x ! −x) are either symmetric (i.e., even parity), with an even number of nodes, or else antisymmetric (i.e, odd parity), with an odd number of nodes • the lowest energy state (ground state) is always above the bottom of the potential and is symmetric (A consequence of the HUP.) • the wider and/or more shallow the potential, the lower the energies of the quantum states (A consequence of the HUP.) • inside FPW-type of potentials the number of quantum states is finite • when the total energy E is larger than the height of the potential, the energy becomes continuous, i.e., we have continuous states • when V = V (x), both bound and continuous states are stationary, i.e, the time-dependent wavefunctions are of the form Ψ(x; t) = (x) exp − i Et ~ Further reading: See, e.g., Phillips, Ch. 5.1..