Topic 4: The Finite Potential Well
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1 Topic 4: The Finite Potential Well Outline: The quantum well The finite potential well (FPW) Even parity solutions of the TISE in the FPW Odd parity solutions of the TISE in the FPW Tunnelling into classically forbidded regions Comparison with the IPW QM PHY202 p. 17
2 The AlGaAs-GaAs Quantum Well QM PHY202 p. 18
3 The AlGaAs-GaAs Quantum Well Example of a potential well: Sandwich of GaAs and AlGaAs layers QM PHY202 p. 18
4 The AlGaAs-GaAs Quantum Well Example of a potential well: Sandwich of GaAs and AlGaAs layers Constrained motion along the x-axis; free motion in the y z plane. V 0 :1-dimensionalinfinitepotentialwell V 0 < : 1-dimensionalfinitepotentialwell QM PHY202 p. 18
5 The Finite Potential Well QM PHY202 p. 19
6 The Finite Potential Well Consider a particle in the potential V 0 x< L 2 V (x) = 0 L 2 x L 2 V 0 x> L 2 V 0 > 0 QM PHY202 p. 19
7 The Finite Potential Well Consider a particle in the potential V 0 x< L 2 V (x) = 0 L 2 x L 2 V 0 x> L 2 V 0 > 0 E>V 0 unboundstates,totalenergye continuous (not quantized) E<V 0 boundstates,expectdiscretestates QM PHY202 p. 19
8 The Finite Potential Well Consider a particle in the potential V 0 x< L 2 V (x) = 0 L 2 x L 2 V 0 x> L 2 V 0 > 0 E>V 0 unboundstates,totalenergye continuous (not quantized) E<V 0 boundstates,expectdiscretestates Solve the TISE: QM PHY202 p. 19
9 The Finite Potential Well Consider a particle in the potential V 0 x< L 2 V (x) = 0 L 2 x L 2 V 0 x> L 2 V 0 > 0 E>V 0 unboundstates,totalenergye continuous (not quantized) E<V 0 boundstates,expectdiscretestates Solve the TISE: L 2 x L 2 (Region I): 2 2m d 2 dx 2 ψ(x) =Eψ(x) ψ (x) = k 2 ψ(x) k 2 = 2mE 2 > 0 solutions: ψ I (x) =A sin kx + B cos kx, (as for the IPW) A, B arbitraryconstants QM PHY202 p. 19
10 x> L 2 (Region II): Note: in region II E = KE + PE = KE + V 0 <V 0 KE < 0! QM PHY202 p. 20
11 x> L 2 (Region II): Note: in region II E = KE + PE = KE + V 0 <V 0 KE < 0! 2 2m d 2 dx 2 ψ(x) +V 0 ψ(x) =Eψ(x) ψ (x) =α 2 ψ(x) α 2 = 2m(V 0 E) 2 > 0 QM PHY202 p. 20
12 x> L 2 (Region II): Note: in region II E = KE + PE = KE + V 0 <V 0 KE < 0! 2 2m d 2 dx 2 ψ(x) +V 0 ψ(x) =Eψ(x) ψ (x) =α 2 ψ(x) α 2 = 2m(V 0 E) 2 > 0 solutions ψ II (x) =Ce αx + De αx C, D arbitraryconstants put D =0,otherwiseψ(x) not square integrable (blows up at large +ve x) QM PHY202 p. 20
13 x> L 2 (Region II): Note: in region II E = KE + PE = KE + V 0 <V 0 KE < 0! 2 2m d 2 dx 2 ψ(x) +V 0 ψ(x) =Eψ(x) ψ (x) =α 2 ψ(x) α 2 = 2m(V 0 E) 2 > 0 solutions ψ II (x) =Ce αx + De αx C, D arbitraryconstants put D =0,otherwiseψ(x) not square integrable (blows up at large +ve x) x< L 2 (Region III): solutions ψ III (x) =Fe αx + Ge αx (like in region II) F, G arbitraryconstants put F =0,otherwiseψ(x) not square integrable (blows up at large -ve x) QM PHY202 p. 20
14 The potential is symmetric w.r.t. to x =0 expect symmetric (even-parity) and antisymmetric (odd-parity) states QM PHY202 p. 21
15 The potential is symmetric w.r.t. to x =0 expect symmetric (even-parity) and antisymmetric (odd-parity) states Consider even-parity solutions only: ψ I (x) =B cos kx QM PHY202 p. 21
16 The potential is symmetric w.r.t. to x =0 expect symmetric (even-parity) and antisymmetric (odd-parity) states Consider even-parity solutions only: ψ I (x) =B cos kx Apply general conditions on ψ at x = L/2: QM PHY202 p. 21
17 The potential is symmetric w.r.t. to x =0 expect symmetric (even-parity) and antisymmetric (odd-parity) states Consider even-parity solutions only: ψ I (x) =B cos kx Apply general conditions on ψ at x = L/2: ψ continuous at x = L/2: ψ I (L/2) = ψ II (L/2) B cos (kl/2) = C exp ( αl/2) QM PHY202 p. 21
18 The potential is symmetric w.r.t. to x =0 expect symmetric (even-parity) and antisymmetric (odd-parity) states Consider even-parity solutions only: ψ I (x) =B cos kx Apply general conditions on ψ at x = L/2: ψ continuous at x = L/2: ψ I (L/2) = ψ II (L/2) B cos (kl/2) = C exp ( αl/2) ψ continuous at x = L/2: ψ I (L/2) = ψ II (L/2) Bk sin (kl/2) = Cαexp ( αl/2) No new constraints at x = L/2 since we consider symmetric solutions only. QM PHY202 p. 21
19 The potential is symmetric w.r.t. to x =0 expect symmetric (even-parity) and antisymmetric (odd-parity) states Consider even-parity solutions only: ψ I (x) =B cos kx Apply general conditions on ψ at x = L/2: ψ continuous at x = L/2: ψ I (L/2) = ψ II (L/2) B cos (kl/2) = C exp ( αl/2) ψ continuous at x = L/2: ψ I (L/2) = ψ II (L/2) Bk sin (kl/2) = Cαexp ( αl/2) No new constraints at x = L/2 since we consider symmetric solutions only. divide them side-by-side: tan kl 2 = α k QM PHY202 p. 21
20 The potential is symmetric w.r.t. to x =0 expect symmetric (even-parity) and antisymmetric (odd-parity) states Consider even-parity solutions only: ψ I (x) =B cos kx Apply general conditions on ψ at x = L/2: ψ continuous at x = L/2: ψ I (L/2) = ψ II (L/2) B cos (kl/2) = C exp ( αl/2) ψ continuous at x = L/2: ψ I (L/2) = ψ II (L/2) Bk sin (kl/2) = Cαexp ( αl/2) No new constraints at x = L/2 since we consider symmetric solutions only. divide them side-by-side: tan kl 2 = α k introduce θ = kl 2 LHS: y(θ) = tan θ QM PHY202 p. 21
21 The potential is symmetric w.r.t. to x =0 expect symmetric (even-parity) and antisymmetric (odd-parity) states Consider even-parity solutions only: ψ I (x) =B cos kx Apply general conditions on ψ at x = L/2: ψ continuous at x = L/2: ψ I (L/2) = ψ II (L/2) B cos (kl/2) = C exp ( αl/2) ψ continuous at x = L/2: ψ I (L/2) = ψ II (L/2) Bk sin (kl/2) = Cαexp ( αl/2) No new constraints at x = L/2 since we consider symmetric solutions only. divide them side-by-side: tan kl 2 = α k introduce θ = kl 2 LHS: y(θ) = tan θ and θ 0 = k 0L const. wherek = 2mV 0 > 0 2 RHS: y(θ) = α k = k0 2 V 1= 0 E θ = k E θ 2 QM PHY202 p. 21
22 The even-parity solutions are determined when the curve y = tan θ intersects the curve y = α k. θ = kl 2 QM PHY202 p. 22
23 The even-parity solutions are determined when the curve y = tan θ intersects the curve y = α k. θ = kl 2 The intersection points determine k and hence E = 2 k 2 2m. QM PHY202 p. 22
24 The even-parity solutions are determined when the curve y = tan θ intersects the curve y = α k. θ = kl 2 The intersection points determine k and hence E = 2 k 2 2m. When E V 0 r α k = θ0 2 θ 2 (solid line) 1 1 θ QM PHY202 p. 22
25 The even-parity solutions are determined when the curve y = tan θ intersects the curve y = α k. θ = kl 2 The intersection points determine k and hence E = 2 k 2 2m. When E V 0 r α k = θ0 2 θ 2 (solid line) 1 1 θ When E V 0 θ θ 0 α k = r θ 2 0 θ (dashed line) QM PHY202 p. 22
26 The even-parity solutions are determined when the curve y = tan θ intersects the curve y = α k. θ = kl 2 The intersection points determine k and hence E = 2 k 2 2m. When E V 0 r α k = θ0 2 θ 2 (solid line) 1 1 θ When E V 0 θ θ 0 α k = r θ 2 0 θ (dashed line) bound states with discrete (quantized) energy QM PHY202 p. 22
27 Comments: larger V 0 more bound states; smaller V 0 less bound states QM PHY202 p. 23
28 Comments: larger V 0 more bound states; smaller V 0 less bound states when θ 0 <π V 0 < 2π2 2 ml 2 only one symmetric state exists In the FPW there is always at least one bound state. Even in a very shallow well (V 0 0). QM PHY202 p. 23
29 Comments: larger V 0 more bound states; smaller V 0 less bound states when θ 0 <π V 0 < 2π2 2 ml 2 only one symmetric state exists In the FPW there is always at least one bound state. IPW: wavenumbers k n = nπ L,orθ n = nπ 2 (n =1, 3, 5,... for symmetric states). Even in a very shallow well (V 0 0). The wavenumber and energy of the nth state is less than in the IPW. QM PHY202 p. 23
30 The FPW odd-parity solutions QM PHY202 p. 24
31 The FPW odd-parity solutions Consider odd-parity solutions only: ψ I (x) =B sin kx QM PHY202 p. 24
32 The FPW odd-parity solutions Consider odd-parity solutions only: ψ I (x) =B sin kx Apply general conditions on ψ at x = L/2: QM PHY202 p. 24
33 The FPW odd-parity solutions Consider odd-parity solutions only: ψ I (x) =B sin kx Apply general conditions on ψ at x = L/2: ψ continuous at x = L/2: ψ I (L/2) = ψ II (L/2) B sin (kl/2) = C exp ( αl/2) QM PHY202 p. 24
34 The FPW odd-parity solutions Consider odd-parity solutions only: ψ I (x) =B sin kx Apply general conditions on ψ at x = L/2: ψ continuous at x = L/2: ψ I (L/2) = ψ II (L/2) B sin (kl/2) = C exp ( αl/2) ψ continuous at x = L/2: ψ I (L/2) = ψ II (L/2) Bk cos (kl/2) = Cαexp ( αl/2) No new constraints at x = L/2 since we consider antisymmetric solutions only. QM PHY202 p. 24
35 The FPW odd-parity solutions Consider odd-parity solutions only: ψ I (x) =B sin kx Apply general conditions on ψ at x = L/2: ψ continuous at x = L/2: ψ I (L/2) = ψ II (L/2) B sin (kl/2) = C exp ( αl/2) ψ continuous at x = L/2: ψ I (L/2) = ψ II (L/2) Bk cos (kl/2) = Cαexp ( αl/2) No new constraints at x = L/2 since we consider antisymmetric solutions only. divide them side-by-side: cot kl 2 = α k QM PHY202 p. 24
36 The FPW odd-parity solutions Consider odd-parity solutions only: ψ I (x) =B sin kx Apply general conditions on ψ at x = L/2: ψ continuous at x = L/2: ψ I (L/2) = ψ II (L/2) B sin (kl/2) = C exp ( αl/2) ψ continuous at x = L/2: ψ I (L/2) = ψ II (L/2) Bk cos (kl/2) = Cαexp ( αl/2) No new constraints at x = L/2 since we consider antisymmetric solutions only. divide them side-by-side: cot kl 2 = α k (as before) introduce θ = kl 2 LHS: y(θ) = cot θ QM PHY202 p. 24
37 The FPW odd-parity solutions Consider odd-parity solutions only: ψ I (x) =B sin kx Apply general conditions on ψ at x = L/2: ψ continuous at x = L/2: ψ I (L/2) = ψ II (L/2) B sin (kl/2) = C exp ( αl/2) ψ continuous at x = L/2: ψ I (L/2) = ψ II (L/2) Bk cos (kl/2) = Cαexp ( αl/2) No new constraints at x = L/2 since we consider antisymmetric solutions only. divide them side-by-side: cot kl 2 = α k (as before) introduce θ = kl 2 LHS: y(θ) = cot θ and θ 0 = k 0L const. wherek = 2mV 0 > 0 2 RHS: y(θ) = α k = k0 2 θ 1= 2 k θ 2 QM PHY202 p. 24
38 The odd-parity solutions are determined when the curve y = cot θ intersects the curve y = α k. θ = kl 2 QM PHY202 p. 25
39 The odd-parity solutions are determined when the curve y = cot θ intersects the curve y = α k. θ = kl 2 larger (smaller) V 0 more (less) bound states QM PHY202 p. 25
40 The odd-parity solutions are determined when the curve y = cot θ intersects the curve y = α k. θ = kl 2 larger (smaller) V 0 more (less) bound states when θ 0 < π 2 V 0 < π2 2 2mL 2 no antisymmetric state exists QM PHY202 p. 25
41 The odd-parity solutions are determined when the curve y = cot θ intersects the curve y = α k. θ = kl 2 larger (smaller) V 0 more (less) bound states when θ 0 < π 2 V 0 < π2 2 2mL 2 no antisymmetric state exists IPW: wavenumbers k n = nπ L,orθ n = nπ 2 (n =2, 4, 6,...for antisymmetric states). The wavenumber and energy of the nth state is less than in the IPW. QM PHY202 p. 25
42 Compare the FPW and the IPW QM PHY202 p. 26
43 Compare the FPW and the IPW Infinite well: ψ(x) confined to the well k n = nπ L infinite tower of states Finite well: ψ(x) spreads out beyond the well k n and energies lower finite tower of states no unbound states unbound states when E>V 0 QM PHY202 p. 26
44 Compare the FPW and the IPW Infinite well: ψ(x) confined to the well k n = nπ L infinite tower of states Finite well: ψ(x) spreads out beyond the well k n and energies lower finite tower of states no unbound states unbound states when E>V 0 The energy levels in the FPW are lower because the wavefunction spreads out (by penetrating the classically forbidden region) and therefore reduces its KE. QM PHY202 p. 26
45 Quantum Tunnelling QM PHY202 p. 27
46 Quantum Tunnelling At x>l/2 the wavefn ψ(x) e αx ; at x< L/2 the wavefn ψ(x) e αx. α 2 = 2m(V 0 E) 2 > 0 QM PHY202 p. 27
47 Quantum Tunnelling At x>l/2 the wavefn ψ(x) e αx ; at x< L/2 the wavefn ψ(x) e αx. α 2 = 2m(V 0 E) 2 > 0 when V 0 α 1/α 0 QM PHY202 p. 27
48 Quantum Tunnelling At x>l/2 the wavefn ψ(x) e αx ; at x< L/2 the wavefn ψ(x) e αx. α 2 = 2m(V 0 E) 2 > 0 when V 0 α 1/α 0 when E V 0 α 0 1/α QM PHY202 p. 27
49 Quantum Tunnelling At x>l/2 the wavefn ψ(x) e αx ; at x< L/2 the wavefn ψ(x) e αx. α 2 = 2m(V 0 E) 2 > 0 when V 0 α 1/α 0 when E V 0 α 0 1/α The depth of tunneling is determined by α thepenetrationdepth QM PHY202 p. 27
50 Quantum Tunnelling At x>l/2 the wavefn ψ(x) e αx ; at x< L/2 the wavefn ψ(x) e αx. α 2 = 2m(V 0 E) 2 > 0 when V 0 α 1/α 0 when E V 0 α 0 1/α The depth of tunneling is determined by α thepenetrationdepth Non-zero wavefunction in classically forbidden regions (KE < 0!) is apurelyquantummechanicaleffect.itallowstunnellingbetween classically allowed regions. QM PHY202 p. 27
51 Quantum Tunnelling At x>l/2 the wavefn ψ(x) e αx ; at x< L/2 the wavefn ψ(x) e αx. α 2 = 2m(V 0 E) 2 > 0 when V 0 α 1/α 0 when E V 0 α 0 1/α The depth of tunneling is determined by α thepenetrationdepth Non-zero wavefunction in classically forbidden regions (KE < 0!) is apurelyquantummechanicaleffect.itallowstunnellingbetween classically allowed regions. It follows from requiring that both ψ(x) and ψ (x) are continuous! QM PHY202 p. 27
52 Quantum Tunnelling At x>l/2 the wavefn ψ(x) e αx ; at x< L/2 the wavefn ψ(x) e αx. α 2 = 2m(V 0 E) 2 > 0 when V 0 α 1/α 0 when E V 0 α 0 1/α The depth of tunneling is determined by α thepenetrationdepth Non-zero wavefunction in classically forbidden regions (KE < 0!) is apurelyquantummechanicaleffect.itallowstunnellingbetween classically allowed regions. It follows from requiring that both ψ(x) and ψ (x) are continuous! Requiring a reasonable behaviour of the wavefunction leads to a(classically) crazy phenomenonoftunnelling QM PHY202 p. 27
53 Quantum states in potential wells QM PHY202 p. 28
54 Quantum states in potential wells Some general properties: QM PHY202 p. 28
55 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) where the tails are not allowed. QM PHY202 p. 28
56 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) where the tails are not allowed. both properties result from requiring the wavefunction ψ(x) and its derivative ψ (x) to be continuous everywhere Except when V (x) where ψ (x) is not continuous. QM PHY202 p. 28
57 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) where the tails are not allowed. both properties result from requiring the wavefunction ψ(x) and its derivative ψ (x) to be continuous everywhere Except when V (x) where ψ (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 QM PHY202 p. 28
58 the lowest energy state (ground state) is always above the bottom of the potential and is symmetric AconsequenceoftheHUP. QM PHY202 p. 29
59 the lowest energy state (ground state) is always above the bottom of the potential and is symmetric AconsequenceoftheHUP. the wider and/or more shallow the potential, the lower the energies of the quantum states AconsequenceoftheHUP. QM PHY202 p. 29
60 the lowest energy state (ground state) is always above the bottom of the potential and is symmetric AconsequenceoftheHUP. the wider and/or more shallow the potential, the lower the energies of the quantum states AconsequenceoftheHUP. inside FPW-type of potentials the number of quantum states is finite QM PHY202 p. 29
61 the lowest energy state (ground state) is always above the bottom of the potential and is symmetric AconsequenceoftheHUP. the wider and/or more shallow the potential, the lower the energies of the quantum states AconsequenceoftheHUP. 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 QM PHY202 p. 29
62 the lowest energy state (ground state) is always above the bottom of the potential and is symmetric AconsequenceoftheHUP. the wider and/or more shallow the potential, the lower the energies of the quantum states AconsequenceoftheHUP. 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), bothboundandcontinuousstatesarestationary, i.e, the time-dependent wavefunctions are of the form Ψ(x, t) =ψ(x)exp ( i Et) QM PHY202 p. 29
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