SME
Quantum Mechanics · Problem type
Tunneling through a barrier
Use this for “An electron meets a barrier 0.3 eV above its energy and 1.2 nm wide. Estimate the chance that it tunnels through.”
Diagram
Inside, ψ falls as e−κx: across a it shrinks to e−κa = 0.0772 of its size, so T ≈ e−2κa = 0.00596.
Heights are not to scale; the decay inside the barrier is.
This is an example. Type your own values to start a new problem.
2
×
m (kg)
(
U − E (eV)
)
ħ²
=
κ (nm⁻¹)
T
≈
e
−2
×
×
=
Formulas
κ
=
√(2
×
m
×
(U
−
E)
×
1.602 ×
10
-19
)
÷
(1.0546
×
10
-34
)
×
10
-9
5.123
=
√(2
×
(9.109
×
10
-31
)
×
(1)
×
1.602 ×
10
-19
)
÷
(1.0546
×
10
-34
)
×
10
-9
T
=
e
−2 ×
κ
×
a
0.00596
=
e
−2 × 5.123 × 0.5
Assumptions
•
Inside the barrier the particle’s energy E is below U, so ψ decays as e−κx with κ = √(2m(U − E)) ÷ ħ instead of waving.
•
The estimate is for a thick barrier (2κa well above 1): T ≈ e−2κa, leaving out a prefactor near 1.
•
ħ = 1.0546 × 10⁻³⁴ J·s and 1 eV = 1.602 × 10⁻¹⁹ J; masses: electron 9.109 × 10⁻³¹ kg, proton 1.673 × 10⁻²⁷ kg.
Step-by-step
WE KNOW
m = 9.109 × 10⁻³¹ kg, U − E = 1 eV, a = 0.5 nm
FIND
decay constant (κ), transmission (T)
Step 1 · Find decay constant (κ)
Use κ = √(2 × m × (U − E) × 1.602 × 10⁻¹⁹) ÷ (1.0546 × 10⁻³⁴) × 10⁻⁹
U
−
E
in
joules,
then
√
2
m
(U − E)
ħ
is
κ
per
meter;
×
10
-9
gives
it
per
nm.
κ
=
√(2
×
(9.109
×
10
-31
)
×
1
×
1.602 ×
10
-19
)
÷
(1.0546
×
10
-34
)
×
10
-9
κ
=
√((1.8218
×
10
-30
)
×
(1.602
×
10
-19
))
÷
(1.0546
×
10
-34
)
×
10
-9
κ
=
√
2.9185 ×
10
-49
1.0546 ×
10
-34
×
10
-9
κ
=
5.4023 ×
10
-25
1.0546 ×
10
-34
×
10
-9
κ
=
(5.123
×
10
9
)
×
10
-9
κ
=
5.123
nm⁻¹
Step 2 · Find transmission (T)
Use T = e−2 × κ × a
ψ
shrinks
by
e
−
κa
across
the
barrier,
and
the
chance
goes
as
ψ
2
,
so
as
e
−2
κa
.
T
=
e
−2 × 5.123 × 0.5
T
=
e
−10.246 × 0.5
T
=
e
−5.123
T
=
0.00596
Check
Put the numbers back in. Both sides should match.
5.123
=
√(2
×
(9.109
×
10
-31
)
×
1
×
1.602 ×
10
-19
)
÷
(1.0546
×
10
-34
)
×
10
-9
✓
0.00596
=
e
−2 × 5.123 × 0.5
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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