On the Designing of Quantum Potential Wells with Given Spectral Properties

Abstract

In this article, a method is described for modifying a 1D square potential well in a way that converts it to an effective n-level system with a large energy gap between the n:th eigenvalue of the Hamiltonian and the higher eigenvalues, with n varying from 2 to 5. The method is based on forming potential energy perturbation terms that have a large overlap integral with the square modulus of some energy eigenfunction in the position representation, and therefore have most effect on the eigenvalue of that particular state. The procedure is conceptually simpler to understand than the solution of the full Inverse Scattering Problem, and its possible applications are in the production of semiconductor-based quantum wells with predetermined energy spacings. It is also likely that the results can be extended for multidimensional potential wells and systems of several electrons in a straightforward way.

Full text

En
|ψniˆ
H
ˆ
H
E1≤E2≤E3≤. . .
V(x)x
V(x)
V(x)7→ V(x+c)V(x)7→ V(−x)

V(x)V(x)
Enen
Γ(n)
E1, E2E1, E2, E3
exp(−En/kT )
V(x) = 0,|x| ≤ L
2
V0,|x|>L
2
.
V0→ ∞
ψn(x) = Cncos nπx
L, n
Cnsin nπx
L, n .
Cn
En=h2n2
8mL2=π2~2n2
2mL2,
V0n
n=n
|x| ≤ L/2
V0(x)
n En
E0
n=hψn|ˆ
V0|ψni=Z∞
−∞
|ψn(x)|2V0(x)dx.
|ψni
ψn(x) = hx|ψni
V0(x)
V0(x) = Aexp h−(x/δ)2i,

δ V (x)
Enψn(x)
x= 0 n
V(x)
Enψn
V0(x)
x
V0(x)
|ψn(x)|2V0(x)|ψn(x)|2
E3
x= 0 V(x)
V0(x) = Ccos23πx
L,
C
V0(x) = Ccos43πx
L
V0V0
n
Imn =Z∞
−∞
|ψm(x)|2|ψn(x)|2dx
m=n m 6=n
N×N A
|φ1i,|φ2i, . . . |φNiλ1, λ2. . . λNA A 7→ A+C|φjihφj|
C j λj
ˆxˆp
V(x)V(0)(x)
n ψ(0)
n(x)
V(1)(x)
V(1)(x) = V(0) (x) +
∞
X
k=1
C(0)
k|ψ(0)
k(x)|2.

C(0)
k
C(0)
k
V(1)(x)
|ψ(1)
n(x)|2
V(2)(x) = V(1) (x) +
∞
X
k=1
C(1)
k|ψ(1)
k(x)|2,
C(1)
k
ψ(2)
n(x)E(2)
n
V0= 200
V0= 600 L= 2
V0L
R2=E3−E2
E2−E1
, R3=E4−E3
E3−E1
.
R2
R3
n
H

E1E5
−~2
2m
2ψ(x)
x2+V(x)ψ(x) = Eψ(x)
m=~= 1
−1
2
2ψ(x)
x2+V(x)ψ(x) = Eψ(x).
x, E, V (x)
2ψ(x)
x2= 2(V(x)−E)ψ(x).
V(x) = V(−x)x E1
x= 0 ψ(0) = 1
ψ0(0) = 0 |x|=L/2
E < E1E > E1
E1
Enn > 1n ψ(0) = 0 ψ0(0) = 1
En
ˆ
H x y ψn(x, y) = X(x)Y(y)
x y
V(x, y) = Vx(x) + Vy(y)
n
(n−1)

V0= 200 L= 2
x= 0
V0(x) = θ(L/2− |x|)A1cos2πx
2+A3cos23πx
2+A4sin2(2πx),
n= 1 n= 3 n= 4
θ
A1A3A40≤A1≤10 −27 ≤A3≤0
0≤A4≤20 36
E2−E1E3−E2
E4−E3E3−E1
A1A3A4
V0(x) = θ(L/2− |x|)A1cos2πx
2+A3cos23πx
2+A4sin2(2π)
A1A3A4A1A3A4
E5−E4

E4−E1
V0= 200 L= 2
V0(x) = θ(L/2− |x|)A1cos4πx
2+A3cos43πx
2+A4sin4(2πx),
A1A3A4
A1A3A4
V0(x) = θ(L/2− |x|)A1cos4πx
2+A3cos43πx
2+A4sin4(2π)
A1A3A4A1A3A4
R3R21
E2−E1E3−E2E4−E3E5−E4
E6−E5

V0= 200 L= 2 x= 0
Ck|ψ(0)
k(x)|2
ψ(1)
k(x)
|ψ(1)
k(x)|2−1.05L/2≤x≤1.05L/2R
1V(2)(x)|x|= 1.05L/2
0
E(2)
kk≤6
V(2)(x)|ψ(2)
k(x)|2
E1E5
R2R3

En
V0(x) = θ(L/2− |x|)A1cos2πx
2+A3cos23πx
2+A4sin2(2π)
A1A3A4L V0
E1E2E3E4E5R2R3

En
V0(x) = θ(L/2− |x|)A1cos2πx
2+A3cos23πx
2+A4sin2(2π)
E1E2E3E4E5R2R3
V(x)

V(x)
V0= 600 L= 2
1
V0(x) = θ(L/2− |x|)A1cos2πx
2+A2sin2(πx) + A3cos23πx
2
+A4sin2(2πx) + A5cos25πx
2,
A1= 13.1A2= 14.0A3= 8.6A4=−38.2A5= 20.5
E2−E1E3−E2E4−E3
E5−E4>3.5(E4−E1)
V(x) + V0(x)

V(x)
L= 2 V0= 600
R2R3

EnV0(x) = θ(L/2−
|x|)A1cos4πx
2+A3cos43πx
2+A4sin4(2π)
A1A3A4L V0
E1E2E3E4E5R2R3
V(x)

V(x)
|ψ(1)
n(x)|2
|x|= 1.05L/2
C(1)
k
C(1)
1=−0.425 C(1)
3=−0.95 C(1)
4= 10.0E(2)
k
|ψ(2)
n(x)|2
R310.0
C(1)
4= 10.0
E4C(1)
1C(1)
3

V(2)(x)
x= 0 V0= 200
L= 2 C(0)
kC(0)
1= 8.0C(0)
2= 10.0C(0)
3= 18.0
C(0)
4= 3.0C(0)
5=−37.0V(1)(x)ˆ
H
|ψ(1)
k(x)|2
|x|= 1.05L/2
C(1)
kC(1)
1= 3.7C(1)
2= 2.4C(1)
3= 2.55
C(1)
4= 3.8C(1)
5=−28.0C(1)
6= 4.0V(2)(x)E(2)
k
|ψ(2)
k(x)|2k= 1 k= 3
|E(2)
2−E(2)
1| |E(2)
3−E(2)
2| |E(2)
4−E(2)
3| |E(2)
5−E(2)
4|
|E(2)
6−E(2)
5|
C(l)
k

V(1)(x)

V(2)(x)

[V0(x)]2
V(x)
V0
ρn(x, y, z)n