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E = 0
V
t
− V
s
= const (∆
′
= 0)
V
t
+ V
s
= const (Σ
′
= 0)
g V
p
= cγg tan γx m = = c = γ = 1
n
+
= 1 n
−
= 2
V
p
= cγg tan γx
ψ
+
ψ
−
m = = c = γ = 1
g = 2
V
0
V
t
= V
s
= −
V
0
cosh
2
αx
|E| = mc
2
−2V
0
+ mc
2
m =
= c = α = 1
α V
t
= V
s
= −
V
0
cosh
2
αx
mc
2
−2V
0
+ mc
2
m = = c = 1 V
0
= 0.75
|ψ
+
|
2
|ψ
−
|
2
|ψ|
2
= |ψ
+
|
2
+
|ψ
−
|
2
n = 0 V
t
= V
s
= −
V
0
cosh
2
αx
m = = c = V
0
= α = 1
E
V
0
V
t
= V
0
sec
2
αx V
s
= V
0
tan
2
αx m = =
c = α = 1
E
α V
t
= V
0
sec
2
αx V
s
= V
0
tan
2
αx m = =
c = 1
V
0
= 1
|ψ
+
|
2
|ψ
−
|
2
|ψ|
2
= |ψ
+
|
2
+
|ψ
−
|
2
|φ|
2
n = 0 V
t
=
V
0
sec
2
αx
V
s
= V
0
tan
2
αx m = = c = 1 V
0
= 3 α = 5
V
0
V
t
= V
0
sech
2
αx V
s
= V
0
tanh
2
αx
−m
eff
c
2
2V
0
− m
eff
c
2
m = = c = α = 1
α V
t
= V
0
sech
2
αx V
s
= V
0
tanh
2
αx
−m
eff
c
2
2V
0
− m
eff
c
2
m = = c = 1 V
0
= 4
|ψ
+
|
2
|ψ
−
|
2
|ψ|
2
= |ψ
+
|
2
+
|ψ
−
|
2
|φ|
2
n = 0 V
t
=
V
0
sech
2
αx
V
s
= V
0
tanh
2
αx m = = c = 1 V
0
= α = 5
g
s
= 10 g
p
Σ = −c|α|g
s
sech
2
αx
∆ = 0 V
p
= c|α|g
p
tanhαx n = 0 g
p
E = −mc
2
−1 −
g
p
g
s
(g
p
− 1)
m = = c = γ = 1
g
s
= −10 g
p
Σ = −c|α|g
s
sech
2
αx
∆ = 0 V
p
= c|α|g
p
tanhαx
m = = c = γ = 1
g
p
= 10 g
s
Σ = −c|α|g
s
sech
2
αx
∆ = 0 V
p
= c|α|g
p
tanhαx
−1 −
g
p
g
s
(g
p
− 1)
m = = c = α = 1
g
s
= 10 g
p
∆ = −c|α|g
s
sech
2
αx
Σ = 0 V
p
= c|α|g
p
tanhαx
1 −
g
p
g
s
(g
p
+ 1)
N
m = = c = γ = 1
insight
1/2
0
x p
y
= p
z
= 0
x
×
γ
5
E = 0
(V
t
= V
s
)
V
t
±V
s
= const
V
t
± V
s
= const
m p
E =
p
2
c
2
+ m
2
c
4
c
ˆ
E = i
∂
∂t
, ˆp = −i
∇
= h/(2π) Φ(r, t)
i
∂Φ(r, t)
∂t
=
−
2
c
2
∇
2
+ m
2
c
4
1/2
Φ(r, t)
∇
2
E
2
= p
2
c
2
+ m
2
c
4
[A, B] = 0 AΦ = BΦ A
2
Φ = B
2
Φ
−
2
∂
2
Φ(r, t)
∂t
2
=
−
2
c
2
∇
2
+ m
2
c
4
Φ(r, t)
=
1
c
2
∂
2
∂t
2
− ∇
2
+
m
2
c
2
2
Φ(r, t) = 0
∂ρ
∂t
+
∇ ·
J = 0
ρ =
i
2mc
2
Φ
∗
∂Φ
∂t
− Φ
∂Φ
∗
∂t
J =
i
2im
Φ
∗
(
∇Φ) − Φ(
∇Φ
∗
)
ρ
Φ(r, t) = ϕ(r, t)exp
−
i
mc
2
t
ϕ(r, t)
E mc
2
E = E − mc
2
E E ≪ mc
2
i
∂ϕ
∂t
≈ Eϕ ≪
mc
2
ϕ
∂Φ
∂t
=
∂ϕ
∂t
−
i
mc
2
ϕ
exp
−
i
mc
2
t
≈ −
i
mc
2
ϕ exp
−
i
mc
2
t
∂
2
Φ
∂t
2
=
∂
∂t

∂ϕ
∂t
−
i
mc
2
ϕ
exp
−
i
mc
2
t

≈ −
i2mc
2
∂ϕ
∂t
+
m
2
c
4
2
ϕ
exp
−
i
mc
2
t
−
2
2m
∇
2
ϕ = i
∂ϕ
∂t
m
p +
V
e
c
2
c
2
+ (mc
2
+ V
s
)
2
Φ = (E − V
t
)
2
Φ
E p
t e
s
φ
Φ = e
−iΛ
φ
Λ =
x
V
e
(y)
c
dy
p +
V
e
c
2
Φ = e
−iΛ
p
2
φ
−
2
c
2
φ
′′
+ (mc
2
+ V
s
)
2
φ = (E − V
t
)
2
φ,
′
x
−
2
2m
φ
′′
+
V
2
s
− V
2
t
2mc
2
+ V
s
+
E
mc
2
V
t
φ =
E
2
− m
2
c
4
2mc
2
φ
φ
E
|x| → ∞
(V
2
s
−V
2
t
)/(2mc
2
)
|V
s
| > |V
t
| |V
s
| = |V
t
| V
s
+ V
t
E/(mc
2
) > 0
|x| → ∞
Σ = V
t
+ V
s
e ∆ = V
t
− V
s
−
2
2m
φ
′′
+
(E + mc
2
)Σ + (E − mc
2
)∆ − Σ∆
2mc
2
φ =
E
2
− m
2
c
4
2mc
2
φ
E ≈ mc
2
−
2
2m
φ
′′
+
(2mc
2
)Σ + E∆ − Σ∆
2mc
2
φ = E φ
−
2
2m
φ
′′
+
Σ +
E∆
2mc
2
−
Σ∆
2mc
2
φ = Eφ
E = E − mc
2
x
mc
2
−
2
2m
φ
′′
+ Σ φ = Eφ
φ
E
i
∂Ψ(r, t)
∂t
=
c
i
α
1
∂
∂x
1
+ α
2
∂
∂x
2
+ α
3
∂
∂x
3
Ψ(r, t) + βmc
2
Ψ(r, t) ≡ HΨ(r, t)
α
i
β
−
2
∂
2
Ψ(r, t)
∂t
2
=
−
2
c
2
∇
2
+ m
2
c
4
Ψ(r, t)
−
2
∂
2
Ψ
∂t
2
= −
2
c
2
3
i,j=1
α
j
α
i
+ α
i
α
j
2
∂
2
Ψ
∂x
i
∂x
j
+
mc
3
i
3
i=1
(α
i
β + βα
i
)
∂Ψ
∂x
i
+ β
2
m
2
c
4
Ψ
α
i
α
j
+ α
j
α
i
= 2δ
ij
α
i
β + βα
i
= 0
α
2
i
= β
2
= 1
α
i
β
α
i
β N × N
N
α
i
β
N
α
i
β = −βα
i
= −Iβα
i
I
det(AB) = detA detB
det(α
i
β) = detα
i
detβ = det(−I) detα
i
detβ
det(−I) = 1
(−1)
N
= 1
N N
2 × 2 2 × 2
4 ×4
σ
i
(i = 1, 2, 3)
α
i
=
0 σ
i
σ
i
0
, β =
1 0
0 −1
Ψ
i
∂Ψ(r, t)
∂t
=
cα · p + βmc
2
Ψ(r, t) ≡ HΨ(r, t)
α · p = α
1
p
1
+ α
2
p
2
+ α
3
p
3
∂ρ
∂t
+
∇ ·
J = 0
ρ = Ψ
†
Ψ
J
k
= cΨ
†
α
k
Ψ
Ψ
†
= (Ψ
∗
1
, Ψ
∗
2
, Ψ
∗
3
, Ψ
∗
4
) |Ψ|
2
= Ψ
†
Ψ
p → p − (e/c)
A eφ H
A
φ
i
∂Ψ(r, t)
∂t
=
cα ·
p −
e
c
A
+ eφ + βmc
2
Ψ(r, t)
Ψ =
˜ϕ
˜χ
α
i
β
i
∂
∂t
˜ϕ
˜χ
= cσ ·π
˜χ
˜ϕ
+ eφ
˜ϕ
˜χ
+ mc
2
˜ϕ
−˜χ
σ = (σ
1
, σ
2
, σ
3
) π ≡ p −(e/c)
A
˜ϕ
˜χ
= exp
−
i
mc
2
t
ϕ
χ
ϕ χ
i
∂ϕ
∂t
= cσ ·π χ + eφ ϕ
i
∂χ
∂t
= cσ ·π ϕ + eφ χ − 2mc
2
χ
r
mc
2
χ =
σ ·π
2mc
ϕ
χ ϕ
Ψ χ v/c ≪ 1 ϕ
χ
i
∂ϕ
∂t
=
(σ ·π)(σ ·π)
2m
ϕ + eφ ϕ
(σ ·a)(σ ·
b) = a ·
b + iσ · (a ×
b)
(σ ·π)(σ ·π) = π
2
+ iσ · (π ×π)
=
p −
e
c
A
2
−
e
c
σ · (∇ ×
A)
=
p −
e
c
A
2
−
e
c
σ ·
B
B
i
∂ϕ
∂t
=
p − (e/c)
A
2
2m
−
e
2mc
σ ·
B + eφ
ϕ
1/2
1/2
µ =
e
2mc
m
HΨ = EΨ, H = cα · p + βmc
2
+ V
Ψ
T
= (Ψ
1
, Ψ
2
, Ψ
3
, Ψ
4
)
2 × 2
S
i
=
2
σ
i
0
0 σ
i
i = 1, 2, 3
V = IV
t
+ α · ˆrV
e
+ βV
s
+ iβα · ˆrV
T
ˆr = r/r
T
x (x
2
= x
3
= p
2
= p
3
= 0)
H = cα
1
p + βmc
2
+ IV
t
+ α
1
V
e
+ βV
s
+ iβα
1
V
T
S
1
[H, S
1
] = 0
H S
1
S
1
H
S
1
Ψ
k, m
= mΨ
k, m
HΨ
k, m
= E
k
Ψ
k, m
m = ±1/2
Ψ
k, ±1/2
=
φ
±
±φ
±
χ
±
±χ
±
φ
±
=
Ψ
2
± Ψ
1
2
, χ
±
=
Ψ
4
± Ψ
3
2
Ψ
2
= ±Ψ
1
, Ψ
4
= ±Ψ
3
H
˜
Ψ
k, ±1/2
= E
k
˜
Ψ
k, ±1/2
H = cσ
1
p + σ
3
mc
2
+ IV
t
+ σ
1
V
e
+ σ
3
V
s
− σ
2
V
T
˜
Ψ
k, ±1/2
=
φ
±
±χ
±
˜
Ψ
k, +1/2
Ψ
1
Ψ
4
˜
Ψ
k, −1/2
Ψ
2
Ψ
3
S
1
σ
2
V
T
m V
HΨ = EΨ
H = cαp + βmc
2
+ V
× α β
Ψ V
V = IV
t
+ βV
s
+ αV
e
− iβγ
5
V
p
γ
5
= α
× p
α β
α = σ
1
β = σ
3
βγ
5
= iσ
2
H = cα
p +
V
e
c
+ βmc
2
+
I + β
2
Σ +
I − β
2
∆ − iβγ
5
V
p
ψ
Ψ = e
−iΛ
ψ
Λ
p +
V
e
c
Ψ = e
−iΛ
p ψ.
ψ
ψ =
ψ
+
ψ
−
ψ
+
ψ
−
−icψ
′
−
+ mc
2
ψ
+
+ Σψ
+
− iV
p
ψ
−
= Eψ
+
−icψ
′
+
− mc
2
ψ
−
+ ∆ψ
−
+ iV
p
ψ
+
= Eψ
−
ψ
+
ψ
−
+∞
−∞
dx(|ψ
+
|
2
+ |ψ
−
|
2
) = 1
ψ
+
ψ
−
ψ
−
E = −mc
2
+ ∆
ψ
−
= −i
cψ
′
+
− V
p
ψ
+
E + mc
2
− ∆
ψ
+
−
2
c
2
ψ
′′
+
+c∆
′
V
p
ψ
+
− cψ
′
+
E + mc
2
− ∆
+[V
2
p
+cV
′
p
−(E −mc
2
−Σ)(E + mc
2
−∆)]ψ
+
= 0
ψ
+
E = mc
2
+ Σ
ψ
+
= −i
cψ
′
−
+ V
p
ψ
−
E − mc
2
− Σ
ψ
−
−
2
c
2
ψ
′′
−
−cΣ
′
V
p
ψ
−
+ cψ
′
−
E − mc
2
− Σ
+[V
2
p
−cV
′
p
−(E −mc
2
−Σ)(E + mc
2
−∆)]ψ
−
= 0
∆ = 0 E = −mc
2
ψ
−
= −i
cψ
′
+
− V
p
ψ
+
E + mc
2
−
2
2m
ψ
′′
+
+
(E + mc
2
)Σ
2mc
2
+
V
2
p
2mc
2
+
V
′
p
2mc
ψ
+
=
E
2
− m
2
c
4
2mc
2
ψ
+
Σ = 0 E = mc
2
ψ
+
= −i
cψ
′
−
+ V
p
ψ
−
E − mc
2
−
2
2m
ψ
′′
−
+
(E − mc
2
)∆
2mc
2
+
V
2
p
2mc
2
−
V
′
p
2mc
ψ
−
=
E
2
− m
2
c
4
2mc
2
ψ
−
∆ = 0 E = −mc
2
Σ = 0 E = mc
2
∆ = 0 E = −mc
2
Σ = 0 E = mc
2
∆ = 0 E = −mc
2
ψ
+
= ψ
(0)
+
exp
+
x
dy
V
p
(y)
c
ψ
′
−
+
V
p
c
ψ
−
= −
i
c
(Σ + 2mc
2
)ψ
+
Σ = 0 E = mc
2
ψ
−
= ψ
(0)
−
exp
−
x
dy
V
p
(y)
c
ψ
′
+
−
V
p
c
ψ
+
= −
i
c
(∆ − 2mc
2
)ψ
−
ψ
(0)
+
ψ
(0)
−
Σ = 0 E = mc
2
∆ = 0 E = −mc
2
−
2
2m
ψ
′′
∓
+
V
2
p
2mc
2
∓
V
′
p
2mc
ψ
∓
=
E
2
− m
2
c
4
2mc
2
ψ
∓
V
p
V
2
p
/(2mc
2
) ∓V
′
p
/(2mc)
b
a
dx|ψ
±
|
2
= |N
±
|
2
N
+
N
−
(E ± mc
2
)
b
a
dx |ψ
∓
|
2
=
(c)
2
ψ
∗
±
ψ
′
±
∓ cV
p
|ψ
±
|
2
x=b
x=a
+2c
2
b
a
dx ψ
∗
±
−
2
2
d
2
dx
2
+
V
2
p
2c
2
±
2c
V
′
p
ψ
±
ψ
±
(b) = ψ
±
(a) = 0
b
a
dx |ψ
±
|
2
=
E ± mc
2
E ∓ mc
2
, para E = ±mc
2
−
2
2
ψ
′′
∓
+
V
2
p
2c
2
∓
2c
V
′
p
ψ
∓
= 0, para E = ±mc
2
.
N
±
=
E ± mc
2
2E
E = −mc
2
E = +mc
2
V
p
= 0
−
2
c
2
ψ
′′
+
−
2
c
2
∆
′
E + mc
2
− ∆
ψ
′
+
− (E − mc
2
− Σ)(E + mc
2
− ∆)] ψ
+
= 0
−
2
c
2
ψ
′′
−
−
2
c
2
Σ
′
E − mc
2
− Σ
ψ
′
−
− (E − mc
2
− Σ)(E + mc
2
− ∆)] ψ
−
= 0
∆
′
= 0 E = −mc
2
+ const ψ
+
φ
Σ
′
= 0 E = mc
2
+ const ψ
−
φ
∆
′
= 0 Σ
′
= 0
−
2
c
2
ψ
′′
+
+c∆
′
V
p
ψ
+
− cψ
′
+
E + mc
2
− ∆
+[V
2
p
+cV
′
p
−(E −mc
2
−Σ)(E + mc
2
−∆)]ψ
+
= 0
E ∼ mc
2
−
2
c
2
ψ
′′
+
+ c∆
′
V
p
ψ
+
− cψ
′
+
2mc
2
− ∆
+ [V
2
p
+ cV
′
p
− (E − Σ)(2mc
2
− ∆)]ψ
+
= 0
E 2mc
2
− ∆
2
c
2
2mc
2
− ∆
−ψ
′′
+
−
∆
′
2mc
2
− ∆
(ψ
′
+
−
V
p
c
ψ
+
) +
V
′
p
c
ψ
+
+
V
2
p
2
c
2
ψ
+
+ Σψ
+
= Eψ
+
x
|∆(x)| ≪ mc
2
ψ
−
= −i
cψ
′
+
− V
p
ψ
+
E + mc
2
− ∆
E ∼ mc
2
ψ
−
= −i
cψ
′
+
− V
p
ψ
+
2mc
2
− ∆
2mc
2
− ∆ ∼ 2mc
2
V
p
≪ 2mc
2
ψ
−
= −
i
2mc
ψ
′
+
=
p
2mc
ψ
+
ψ
−
v/c ≪ 1 ψ
+
2mc
2
−∆ ∼ 2mc
2
∆
′
ψ
+
−
2
2m
ψ
′′
+
+ Σψ
+
= Eψ
+
E
∆ = 0
−
2
2m
ψ
′′
+
+
Σ +
V
2
p
2mc
2
+
V
′
p
2mc
ψ
+
= Eψ
+
Σ = 0 |∆|/(mc
2
)
−
2
2m
ψ
′′
+
+
V
2
p
2mc
2
+
V
′
p
2mc
ψ
+
= Eψ
+
ψ
+
ψ
−
x → −x
V
p
V
t
V
s
Σ ∆ x → −x
P = exp(iη)P
0
σ
3
η P
0
x −x
α βγ
5
I β P
V
t
V
s
Σ ∆
Ψ
c
= C
¯
Ψ
T
= αΨ
∗
.
C = αβ.
H
c
ψ
c
= −Eψ
c
,
ψ
c
= αψ
∗
=
ψ
∗
−
ψ
∗
+
H
c
H
c
= cαp + βmc
2
− IV
t
+ βV
s
+ iβγ
5
V
p
∆ Σ
H
c
= cαp + βmc
2
−
I + β
2
∆ −
I − β
2
Σ + iβγ
5
V
p
V
t
V
p
Σ → −∆ ∆ → −Σ
E → −E V
t
→ −V
t
γ
5
ψ
χ
= γ
5
ψ H
χ
= γ
5
Hγ
5
γ
5
β
H
χ
ψ
χ
= Eψ
χ
,
ψ
χ
=
ψ
−
ψ
+
H
χ
H
χ
= cαp − βmc
2
+ IV
t
− βV
s
+ iβγ
5
V
p
H
χ
= cαp − βmc
2
+
I + β
2
∆ +
I − β
2
Σ + iβγ
5
V
p
Σ → ∆
V
s
V
p
m → −m V
s
→ −V
s
E → E + const V → V + const
E = 0
E = 0
α β α = σ
1
β = σ
3
E = 0
α = σ
2
β = σ
1
V
2
s
V
2
p
∆
V
p
= 0 Σ = 0
E = ±mc
2
(∼ tan γx)
V
p
= cγg tan γx
γ g
x = ±π/(2|γ|) |x|
π/(2|γ|)
γ → −γ
|γ| V
p
g
g
g > 0
−
2
2m
ψ
′′
±
+ V
[±]
eff
ψ
±
= E
eff
ψ
±
,
V
[±]
eff
(x) =
2
γ
2
2m
[g(g ± 1) tan
2
γx ± g]
ψ
±
= 0 |x| = π/(2|γ|)
|x| > π/(2|γ|) ψ
+
ψ
−
g > 1
g(g ± 1) = λ(λ − 1)
E
eff
=
E
2
− m
2
c
4
2mc
2
=
2
γ
2
2m
(n
2
±
+ 2n
±
λ
±
+ λ
±
± g)
λ
+
= g + 1, λ
−
= g, n
±
= 0, 1, 2, . . .
V
[±]
eff
x → −x
|x| = ±π/(2|γ|)
|E| >
m
2
c
4
+ (cγ)
2
g
V
[+]
eff
V
[−]
eff
n
−
= n
+
+ 1
n
−
= 1, 2, 3, . . . V
[−]
eff
V
[+]
eff
E = ±
m
2
c
4
+ (cγ)
2
[n
2
+
+ 2n
+
(g + 1) + 2g + 1].
g γ = 1
ψ
±
= N
±
|γ|(n
±
+ λ
±
)
Γ
2λ
±
+ n
±
Γ
n
±
+ 1
1 − z
2
1/4
P
1/2−λ
±
n
±
+λ
±
−1/2
(z)
z = senγx P
µ
ν
(z)
C
(a)
n
(z)
n [−1, +1]
ψ
±
= N
±
2
−λ
±
2|γ|(n
±
+ λ
±
)
Γ
n
±
+ 1
Γ
n
±
+ 2λ
±
Γ (2λ
±
)
Γ (λ
±
+ 1/2)
1 − z
2
λ
±
/2
C
λ
±
n
±
(z)
C
(a)
n
(−z) = (−)
n
C
(a)
n
(z) C
(a)
n
(z) n ψ
+
ψ
−
ψ
+
ψ
−
±1
ψ
+
ψ
−
n
+
= 1 n
−
= 2
|γ| g
4.4
m
eff
=
m
2
+
γ
c
2
g
|E| > m
eff
c
2
λ
eff
= /(m
eff
c)
(∆x)
mim
= λ
eff
/2 (∆p)
m ax
= m
eff
c
Σ = 0 ∆ = 0
ψ
∓
= N
∓
exp
∓
x
dy
V
p
(y)
c
ψ
′
±
∓
V
p
c
ψ
±
= ±i
2mc
ψ
∓
N
±
=
(E ± mc
2
)/(2E) E = +mc
2
ψ ∼ (1 − z
2
)
−g/2
1
0
E = −mc
2
ψ ∼ (1 − z
2
)
g/2
0
1
E = −mc
2
ψ = 2
−g+1/2
|γ|g
Γ (2g)
Γ (g + 1/2)
1 − z
2
g/2
0
1
g
m = 0 g = 1.001
△x = 0.5680 λ
eff
△p = 0.9995 m
eff
c γ
E = −mc
2
2m
eff
c
2
E = 0
E = 0 2
√
3c|γ|
σ
3
ψ
−E
= σ
3
ψ
E
−iµβα ·
∇φ(r) µ
φ
σ
2
µφ
′
ln(cos γx)
g
n-=4
n+=3
n-=3
n+=2
n-=2
n+=1
n-=1
n+=0
–10
–5
0
5
10
E
2 4 6 8 10 12 14
g
g V
p
= cγg tan γx m = = c = γ = 1
–0.6
–0.4
–0.2
0
0.2
0.4
0.6
ψ
–1.5
–1
–0.5
0
0.5
1
1.5
Χ
n
+
= 1 n
−
= 2
V
p
= cγg tan γx ψ
+
ψ
−
m = = c = γ = 1 g = 2
V
t
= V
s
∆ = 0
V
t
= −V
s
Σ = 0
V
t
= V
s
V
p
= 0
ψ
−
= −
ic
E + mc
2
ψ
′
+
V
p
= 0
−
2
2m
ψ
′′
+
+
E + mc
2
mc
2
V
s
ψ
+
=
E
2
− m
2
c
4
2mc
2
ψ
+
E =
−mc
2
V
eff
=
(E + mc
2
)V
s
/(mc
2
)
E
eff
= (E
2
− m
2
c
4
)/(2mc
2
)
V
s
= −
V
0
cosh
2
αx
, V
0
> 0.
(V
eff
)
min
< E
eff
< 0 V
s
→ 0 x → ±∞
−(E + mc
2
)V
0
<
E
2
− m
2
c
4
2
< 0.
|E| > mc
2
−mc
2
< E < +mc
2
V
0
mc
2
V
0
< mc
2
−2V
0
+ mc
2
< E < +mc
2
−mc
2
mc
2
− 2V
0
√
m
2
c
4
− E
2
= c|α|a
n
,
n a
n
= s − n > 0
s =
1
2
−1 +
1 +
8(E + mc
2
)V
0
(cα)
2
.
E > −mc
2
s > 0
V
0
α = 1 V
0
E = −mc
2
−2V
0
+ mc
2
E = −mc
2
V
0
α V
0
V
0
< mc
2
E = −2V
0
+ mc
2
α
n = 0
α → ∞
C
(a)
n
(z)
n
ψ
+
= N
1 − z
2
a
n
/2
C
(a
n
+1/2)
n
(z) , z = tanh αx,
N
ψ
−
= N
icα
E + mc
2
(1 − z
2
)
a
n
/2
(2n + 2a
n
+ 1)
×
(n + a
n
)(n + 1)C
(a
n
+1/2)
n+1
(z) − (n + 2a
n
)(n + a
n
+ 1)C
(a
n
+1/2)
n−1
(z)
.
|ψ
+
|
2
|ψ
−
|
2
|ψ|
2
= |ψ
+
|
2
+ |ψ
−
|
2
n = 0
V
0
≡ cg|α|/2 g > 0 V
s
|α| → ∞
V
s
= −
cg|α|
2 cosh
2
αx
−→
|α|→∞
−cgδ(x).
α → ∞ s
s −→
|α|→∞
γ =
E + mc
2
c|α|
g,
γ −n > 0
n = 0
√
m
2
c
4
− E
2
= c|α|γ
E = mc
2
(1−g
2
)/(1+ g
2
)
ψ
+
= N
1 − z
2
s/2
−→
|α|→∞
ψ
+
δ
= N e
−γ|αx|
,
ψ
−
δ
= iN g
x
|x|
e
−γ|αx|
.
N =
|α|γ/(1 + g
2
)
g
E = mc
2
(1 − 2g
2
)
E = mc
2
E = −mc
2
n=2
n=1
n=0
–1
–0.5
0
0.5
1
E
2 4
6
8 10 12
V0
V
0
V
t
= V
s
= −
V
0
cosh
2
αx
|E| = mc
2
−2V
0
+ mc
2
m = = c = α = 1
E
n=2
n=1
n=0
–0.5
0
0.5
1
1 2 3 4
5
6
α
α V
t
= V
s
= −
V
0
cosh
2
αx
mc
2
−2V
0
+ mc
2
m = = c = 1 V
0
= 0.75
V
t
±V
s
= const
V
t
±V
s
= const
0
0.1
0.2
0.3
–2 0 2
x
|ψ
+
|
2
|ψ
−
|
2
|ψ|
2
= |ψ
+
|
2
+|ψ
−
|
2
n = 0 V
t
= V
s
= −
V
0
cosh
2
αx
m = = c = V
0
= α = 1
V
t
− V
s
= const (∆
′
= 0)
V
t
−V
s
= V
0
V
0
E = E − V
0
, E
eff
=
E
2
− m
2
c
4
2mc
2
, V
eff
=
E + mc
2
mc
2
V
s
−
2
2m
φ
′′
+ V
eff
φ = E
eff
φ
E = −mc
2
φ
ψ
+
ψ
−
= −icψ
′
+
/(E + mc
2
) E = −mc
2
ψ
+
= const., ψ
−
=
2ψ
+
ic
x
dx
V
s
+ mc
2
ψ
+
ψ
+
V
0
φ ψ
+
V
0
ψ
−
V
t
= V
0
sec
2
αx V
s
= V
0
tan
2
αx
ψ
−
φ ψ
+
V
eff
= U
0
tan
2
αx, U
0
=
E + mc
2
mc
2
V
0
V
eff
α → −α |α|
E < −mc
2
V
0
> 0 E > −mc
2
V
0
< 0 U
0
< 0
E > −mc
2
V
0
> 0 E < −mc
2
V
0
< 0 U
0
> 0
x = ±π/(2|α|) |x| ≤
π/(2|α|)
|x| = ±π/(2|α|)
E > mc
2
V
0
> 0 E < −mc
2
V
0
< 0
|E| < mc
2
V
0
> 0
φ = ψ
+
= 0 |x| = π/(2|α|)
E
2
− m
2
c
4
2mc
2
=
2
α
2
2m
n
2
+ 2nλ + λ
, n = 0, 1, 2, . . .
λ =
1
2
1 +
1 +
8mU
0
2
α
2
2
E
2
− m
2
c
4
+ 2 (E + mc
2
) V
0
−
2
c
2
α
2
+ 8V
0
(E + mc
2
) = c|α|(2n + 1)
E > mc
2
V
0
>
0
E < −mc
2
V
0
< 0
V
0
> 0 E ≈ mc
2
V
0
≪ mc
2
V
0
< 0 E ≈ −mc
2
|V
0
| ≪ mc
2
V
0
α
V
0
> 0
V
0
< 0
V
0
φ = ψ
+
= N 2
−λ
2|α|(n + λ)
Γ (n + 1)
Γ (n + 2λ)
Γ (2λ)
Γ
λ +
1
2
1 − z
2
λ/2
C
(λ)
n
(z)
z = sen αx N
+∞
−∞
dx (|ψ
+
|
2
+ |ψ
−
|
2
) = 1
|ψ
+
|
2
|ψ
−
|
2
|ψ|
2
= |ψ
+
|
2
+ |ψ
−
|
2
|φ|
2
n = 0
|ψ
+
|
2
|ψ
−
|
2
ψ
−
ψ
+
V
0
> 0 V
0
< 0
|φ|
2
| |ψ|
2
△x = 0.089λ
c
△x = 0.121λ
c
m = c = = 1 V
0
= 3 α = 5
m
eff
=
m
2
+
α
c
2
λ
V
t
+ V
s
= const (Σ
′
= 0)
V
t
+ V
s
= V
0
m
eff
= |m +
V
0
c
2
|, V
0
= −mc
2
E
eff
=
E
2
− m
2
eff
c
4
2m
eff
c
2
V0
n=3
n=2
n=1
n=0
0
2
4
6
ε
0.5
1
1.5
2
E
V
0
V
t
= V
0
sec
2
αx V
s
= V
0
tan
2
αx m = = c = α = 1
V
eff
=
E − m
eff
c
2
sgn (mc
2
+ V
0
)
m
eff
c
2
V
t
−
2
2m
eff
φ
′′
+ V
eff
φ = E
eff
φ
E = m
eff
c
2
sgn (mc
2
+ V
0
)
φ ψ
−
ψ
+
= −
icψ
′
−
E − m
eff
c
2
sgn (mc
2
+ V
0
)
E = m
eff
c
2
sgn (mc
2
+ V
0
)
ψ
−
= const., ψ
+
=
2iψ
−
c
x
dx
V
s
+ mc
2
ψ
−
ψ
−
V
0
φ ψ
−
V
0
ψ
+
V
t
= V
0
sech
2
αx V
s
= V
0
tanh
2
αx
n=3
n=2
n=1
n=0
0
2
4
6
ε
0.5
1
1.5
2
α
E
α V
t
= V
0
sec
2
αx V
s
= V
0
tan
2
αx m = = c = 1
V
0
= 1
ψ
+
φ ψ
−
V
eff
= −U
0
sech
2
αx, U
0
=
m
eff
c
2
sgn
mc
2
+ V
0
− E
V
0
m
eff
c
2
V
eff
α → −α |α|
E > m
eff
c
2
V
0
> 0 E < m
eff
c
2
−mc
2
< V
0
< 0 E < −m
eff
c
2
V
0
< −mc
2
U
0
< 0
E > m
eff
c
2
V
0
> 0 E < −m
eff
c
2
V
0
< 0 U
0
> 0
E < −m
eff
c
2
V
0
> 0 E > m
eff
c
2
V
0
< 0
−m
eff
c
2
< E < 2V
0
− m
eff
c
2
V
0
> 0 |E| < m
eff
c
2
V
0
< −mc
2
φ = ψ
−
= 0 |x| → ∞
0
1
2
3
4
–0.2 0 0.2
x
|ψ
+
|
2
|ψ
−
|
2
|ψ|
2
= |ψ
+
|
2
+|ψ
−
|
2
|φ|
2
n = 0 V
t
= V
0
sec
2
αx V
s
=
V
0
tan
2
αx
m = = c = 1 V
0
= 3 α = 5
E
eff
= −
2
α
2
a
2
n
2m
eff
a
n
= s − n, n = 0, 1, 2, . . . < s
s =
1
2
−1 +
1 +
8m
eff
U
0
2
α
2
|V
0
|
|α|
2
c
2
α
2
+ 8V
0
[m
eff
c
2
sgn (mc
2
+ V
0
) − E]
−2
m
2
eff
c
4
− E
2
= c|α|(2n + 1)
−m
eff
c
2
< E < 2V
0
− m
eff
c
2
V
0
> 0 |E| < m
eff
c
2
V
0
<
−mc
2
V
0
> 0 E ≈ −mc
2
V
0
≪ mc
2
−V
0
< E < V
0
V
0
≫ mc
2
V
0
< −mc
2
E ≈ 0 V
0
≈ −mc
2
−|V
0
| < E < |V
0
| |V
0
| ≫ mc
2
V
0
α
V
0
α
α → ∞
α
V
0
< −mc
2
φ = ψ
−
= N 2
a
n
Γ
a
n
+
1
2
|α|a
n
π
Γ (n + 1)
Γ (n + 1 + 2a
n
)
×
1 − z
2
a
n
/2
C
(a
n
+1/2)
n
(z)
z = tanh αx N
+∞
−∞
dx (|ψ
+
|
2
+ |ψ
−
|
2
) = 1
|ψ
+
|
2
|ψ
−
|
2
|ψ|
2
= |ψ
+
|
2
+|ψ
−
|
2
|φ|
2
n = 0
|ψ
+
|
2
|ψ
−
|
2
ψ
+
ψ
−
V
0
< −mc
2
|φ|
2
| |ψ|
2
△x = 0.160λ
c
△x = 0.179λ
c
m = c = = 1 V
0
= α = 5
V
t
±V
s
= V
0
V
0
n=3
n=2
n=1
n=0
–5
0
5
E
2 4
6
8 10
V0
V
0
V
t
= V
0
sech
2
αx V
s
= V
0
tanh
2
αx
−m
eff
c
2
2V
0
− m
eff
c
2
m = = c = α = 1
V
t
= ±V
s
Σ = 0 ∆ = 0
Σ = 0
∆ = 0
V
s
= ±V
t
n=3
n=2
n=1
n=0
E
–4
–2
0
2
2 4
6
8
α
α V
t
= V
0
sech
2
αx V
s
= V
0
tanh
2
αx
−m
eff
c
2
2V
0
− m
eff
c
2
m = = c = 1 V
0
= 4
∆ = 0 Σ = 0
∆ = 0 Σ = 0
0
0.5
1
1.5
2
2.5
–0.4 –0.2 0 0.2 0.4
x
|ψ
+
|
2
|ψ
−
|
2
|ψ|
2
= |ψ
+
|
2
+|ψ
−
|
2
|φ|
2
n = 0 V
t
= V
0
sech
2
αx V
s
=
V
0
tanh
2
αx
m = = c = 1 V
0
= α = 5
Σ = −c|α|g
s
sech
2
αx, ∆ = 0, V
p
= c|α|g
p
tanhαx.
∆ → Σ Σ → ∆ m → −m
V
p
→ −V
p
γ
5
Σ ∆ = 0
∆ = −c|α|g
s
sech
2
αx Σ = 0 V
p
= c|α|g
p
tanhαx
m g
p
E = −mc
2
ψ
−
E = −mc
2
−
2
2m
eff
ψ
′′
+
− U
0
sech
2
αx ψ
+
= E
eff
ψ
+
,
U
0
=
2
α
2
2m
eff
g
p
(g
p
− 1) +
g
s
c|α|
(E + mc
2
)
,
E
eff
=
E
2
− m
2
eff
c
4
2m
eff
, m
eff
=
m
2
+
αg
p
c
2
U
0
ψ
+
= N
1 − z
2
a
n
/2
C
(a
n
+1/2)
n
(z) , z = tanh αx,
m
2
eff
c
4
− E
2
= c|α|(s − n),
n a
n
= s −n > 0 N C
(a)
n
(z)
n
s =
1
2
−1 +
1 +
8m
eff
U
0
(cα)
2
.
c|α|
1 + 4
g
p
(g
p
− 1) +
g
s
c|α|
(E + mc
2
)
−2
m
2
eff
c
4
− E
2
= c|α|(2n + 1)
g
s
g
p
U
0
> 0
E > −mc
2
−
g
p
(g
p
− 1)
g
s
c|α| , for g
s
> 0
E < −mc
2
+
g
p
(g
p
− 1)
|g
s
|
c|α| , for g
s
< 0
g
s
= 0 g
p
= 0
∆ = −c|α|g
s
sech
2
αx
Σ = 0 V
p
= c|α|g
p
tanhαx
c|α|
1 + 4
g
p
(g
p
+ 1) +
g
s
c|α|
(E − mc
2
)
−2
m
2
eff
c
4
− E
2
= c|α|(2N + 1) .
N
∆ = 0 Σ = 0
g
p
g
s
= 10
g
s
= −10 g
p
g
p
= 10 g
s
Σ = 0 ∆ =
0
E → −E
∆ → −Σ V
p
→ −V
p
g
s
→ −g
s
g
p
→ −g
p
Σ = 0 g
s
= 10
g
p
g
s
= −10
g
p
Σ = 0 ∆ = 0
∆ = 0
Σ = 0
∆ = 0 Σ = 0
n=0
n=1
n=0
n=0
–10
0
10
E
–20 0 20
gp
g
s
= 10 g
p
Σ = −c|α|g
s
sech
2
αx ∆ = 0 V
p
=
c|α|g
p
tanhαx
n = 0 g
p
E = −mc
2
−1 −
g
p
g
s
(g
p
− 1)
m = = c = γ = 1
n=1
n=0
n=0
n=0
–10
0
10
E
–20 0 20
gp
g
s
= −10 g
p
Σ = −c|α|g
s
sech
2
αx ∆ = 0
V
p
= c|α|g
p
tanhαx
m = = c = γ = 1
n=0 n=0
–10
–5
0
5
10
E
–100
–50
0
50
100
gs
g
p
= 10 g
s
Σ = −c|α|g
s
sech
2
αx ∆ = 0
V
p
= c|α|g
p
tanhαx −1 −
g
p
g
s
(g
p
− 1)
m = = c = α = 1
N=0
N=0
N=0
N=1
–10
0
10
E
–20 0 20
gp
g
s
= 10 g
p
∆ = −c|α|g
s
sech
2
αx Σ = 0
V
p
= c|α|g
p
tanhαx 1 −
g
p
g
s
(g
p
+ 1)
N m = = c = γ = 1
E = 0
E = −mc
2
2m
eff
c
2
E = 0
E = 0 2
√
3c|γ|
σ
3
ψ
−E
= σ
3
ψ
E
−iµβα ·
∇φ(r) µ
φ
σ
2
µφ
′
ln(cos γx)
g
V
t
= V
s
E = mc
2
E = −mc
2
V
t
± V
s
= V
0
Σ = V
0
∆ = V
0
V
0
Σ = 0 ∆ = 0
∆ = 0 Σ = 0
C
(λ)
n
(x)
n
1 − x
2
y
′′
+ (2λ + 1) xy
′
+ n (n + 2λ) y = 0 , para y(x) = C
(λ)
n
(x)
(−1, 1) w(x) =
(1 − x
2
)
λ−1/2
1
−1
w(x)C
(λ)
n
(x) C
(λ)
m
(x) dx =
π2
(1−2λ)
Γ(n + 2λ)
(n + λ)Γ
2
(λ)Γ(n + 1)
δ
mn
, para λ > −1/2
1
(1 − 2xt + t
2
)
λ
=
∞
n=0
C
(λ)
n
(x) t
n
C
(λ)
0
(x) = 1
C
(λ)
1
(x) = 2λx
C
(λ)
2
(x) = −λ + 2λ(1 + λ)x
2
2λ(1 − x
2
)C
(λ+1)
n−1
(x) = (2λ + n − 1)C
(λ)
n−1
(x) − nxC
(λ)
n
(x)
= (n + 2λ)xC
(λ)
n
(x) − (n + 1)C
(λ)
n+1
(x)
(1 − x
2
)
d
dx
C
(λ)
n
(x) = −nxC
(λ)
n
(x) + (n + 2λ − 1)C
(λ)
n−1
(x)
C
(λ)
n
(x) =
Γ(λ + 1/2) Γ(2λ + n)
n! Γ(2λ)
1
4
(x
2
− 1)
1/4−λ/2
P
1/2−λ
n+λ−1/2
(x)
λ = 0
P
m
l
(x)
l m = 0 . . . l
1 − x
2
y
′′
− 2xy
′
+
l(l + 1) −
m
2
1 − x
2
y = 0 , para y(x) = P
m
l
(x)
m
P
−m
l
(x) = (−1)
m
(l − m)!
(l + m)!
P
m
l
(x)
(−1, 1) l
w(x) = 1
1
−1
w(x)P
m
l
(x) P
m
l
′
(x) dx =
2
2l + 1
(l + m)!
(l − m)!
δ
ll
′
(−1, 1) m (1 − x
2
)
−1
1
−1
w(x)P
m
l
(x) P
m
′
l
(x) dx =
(l + m)!
m(l − m)!
δ
mm
′
(l − m)P
m
l
(x) = x(2l − 1)P
m
l−1
(x) − (l + m − 1)P
m
l−2
(x)
−
2
2m
d
2
ψ(x)
dx
2
+ V (x)ψ(x) = Eψ(x)
x
V (x) = λ(λ − 1)U
0
tan
2
z, |x| π/(2a), λ > 1
z ≡ ax, U
0
=
2
a
2
/(2m)
P
l
m
ψ
n
=
a (n + λ)
Γ (2λ + n)
Γ (n + 1)
(cos z)
1/2
P
1/2−λ
n+λ−1/2
(senz)
E
n
= U
0
2nλ + n
2
+ λ
n = 0, 1, 2, . . .
V (x) = −V
0
sechz, V
0
> 0, z ≡ ax
P
l
m
ψ
n
=
a(s − n)Γ(2s − n + 1)
Γ(n + 1)
P
n−s
s
s =
1
2
−1 +
1 +
4V
0
U
0
, U
0
=
2
a
2
/(2m)
E
n
= −U
0
(s − n)
2
n = 0, 1, 2, . . . < s
1/|x|
B
0
tanh z −
U
0
cosh
−2
z
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