SME
Quantum Mechanics · Problem type
The quantum harmonic oscillator
Use this for “A molecule vibrates at ω = 5.6 × 10¹⁴ rad/s. Find its zero-point energy, the energy of level n = 2 and the wavelength of the photon of one step down.”
Diagram
Eₙ = (n + ½)ħω: the levels are evenly spaced by ħω, starting from ½ħω, never 0. ψ₁ has 1 node; each ψ leaks past its turning points.
Emitted:
λ
=
hc
ΔE
.
This is an example. Type your own values to start a new problem.
Eₙ (eV)
=
(
n
+
½)ħ
ω (rad/s)
ħω
Level spacing
calculated · eV
λ
Photon wavelength
calculated · μm
Formulas
ħω
=
1.0546 ×
10
-34
×
ω
1.602 ×
10
-19
0.06583
=
1.0546 ×
10
-34
×
1 ×
10
14
1.602 ×
10
-19
Eₙ
=
(
n
+
1
2
)
×
ħω
0.03292
=
(0
+
1
2
)
×
0.06583
λ
=
2
×
π
×
3 ×
10
8
ω
×
10
6
18.85
=
2
×
π
×
3 ×
10
8
1 ×
10
14
×
10
6
Assumptions
•
The potential is ½mω²x², so the levels are Eₙ = (n + ½)ħω for n = 0, 1, 2, …: evenly spaced, and the lowest, ½ħω, is never 0.
•
ħ = 1.0546 × 10⁻³⁴ J·s, which is 6.583 × 10⁻¹⁶ eV·s.
•
A step down one level gives a photon of energy ħω, so of angular frequency ω: λ = 2πc ÷ ω, with c = 3.00 × 10⁸ m/s.
Step-by-step
WE KNOW
ω = 1 × 10¹⁴ rad/s, n = 0
FIND
level spacing (ħω), level’s energy (Eₙ), photon wavelength (λ)
Step 1 · Find level spacing (ħω)
Use ħω = 1.0546 × 10⁻³⁴ × ω ÷ (1.602 × 10⁻¹⁹)
ħω
in
joules,
then
divided
by
1.602 ×
10
-19
for
eV.
ħω
=
1.0546 ×
10
-34
×
10
14
1.602 ×
10
-19
ħω
=
1.0546 ×
10
-20
1.602 ×
10
-19
ħω = 0.06583 eV
Step 2 · Find level’s energy (Eₙ)
Use Eₙ = (n + ½) × ħω
Count
n
steps
of
ħω
up
from
the
zero-point
energy
1
2
ħω.
Eₙ
=
(0
+
1
2
)
×
0.06583
Eₙ = 0.03292 eV
Step 3 · Find photon wavelength (λ)
Use λ = 2 × π × 3 × 10⁸ ÷ ω × 10⁶
A
step
down
one
level
gives
off
ħω,
a
photon
of
angular
frequency
ω
:
λ
=
2π
c
ω
in
meters;
×
10
6
gives
μm.
λ
=
2
×
π
×
3 ×
10
8
10
14
×
10
6
λ
=
2π
×
3 ×
10
8
10
14
×
1,000,000
λ
=
1.885 ×
10
9
10
14
×
1,000,000
λ
=
(1.885
×
10
-5
)
×
1,000,000
λ
=
18.85
μm
Check
Put the numbers back in. Both sides should match.
0.06583
=
1.0546 ×
10
-34
×
10
14
1.602 ×
10
-19
✓
0.03292
=
(0
+
1
2
)
×
0.06583
✓
18.85
=
2
×
π
×
3 ×
10
8
10
14
×
10
6
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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