SME
Physical Chemistry II: Quantum & Spectroscopy
Schrödinger equation
Use this for “A free electron has a wavelength of 0.500 nm. Find its wavenumber, momentum and kinetic energy.”
Chart
Period 500 pm
This is an example. Type your own values to start a new problem.
m
Mass
example
λ
Wavelength
example
k
Wavenumber
calculated · m⁻¹
p
Momentum
calculated · kg·m/s
E
Energy
calculated · J
EeV
Energy in electronvolts
worked out (not typed) · eV
Formulas
k
=
2
×
π
λ
×
10
-12
1.26 ×
10
10
=
2
×
π
500 ×
10
-12
p
=
6.626 ×
10
-34
λ
×
10
-12
1.33 ×
10
-24
=
6.626 ×
10
-34
500 ×
10
-12
E
=
p
2
2 ×
m
9.64 ×
10
-19
=
(1.33 ×
10
-24
)
2
÷
(2
×
(9.109
×
10
-31
))
EeV
=
E
1.602 ×
10
-19
6.02
=
9.64 ×
10
-19
1.602 ×
10
-19
Assumptions
•
V = 0, so ψ = eikx solves −(ħ² ÷ 2m)ψ″ = Eψ with E = ħ²k² ÷ 2m, all of it kinetic.
•
The graph draws the real part, cos(kx): one full wave in each wavelength λ.
•
h = 6.626 × 10⁻³⁴ J·s and e = 1.602 × 10⁻¹⁹ C; an electron’s mass is 9.109 × 10⁻³¹ kg.
Step-by-step
WE KNOW
m = 9.109 × 10⁻³¹ kg, λ = 500 pm
FIND
wavenumber (k), momentum (p), energy (E), energy in electronvolts (EeV)
Step 1 · Find wavenumber (k)
Use k = 2 × π ÷ (λ × 10⁻¹²)
One
wave
is
2π
radians:
divide
2π
by
the
wavelength
in
meters
(pm
×
10
-12
).
k
=
2
×
π
500 ×
10
-12
k
=
1.26 ×
10
10
m
-1
Step 2 · Find momentum (p)
Use p = 6.626 × 10⁻³⁴ ÷ (λ × 10⁻¹²)
de
Broglie:
Planck’s
constant
over
the
wavelength
in
meters
(pm
×
10
-12
),
the
same
as
p
=
ħk.
p
=
6.626 ×
10
-34
500 ×
10
-12
p
=
1.325 ×
10
-24
kg·
m
/s
Step 3 · Find energy (E)
Use E = p² ÷ (2 × m)
With
V
=
0
the
energy
is
all
kinetic:
p
2
2
m
,
the
same
as
ħ
2
k
2
2
m
.
E
=
(1.325 ×
10
-24
)
2
÷
(2
×
(9.109
×
10
-31
))
E
=
(1.325 ×
10
-24
)
2
÷
(1.8218
×
10
-30
)
E
=
1.7556 ×
10
-48
1.8218 ×
10
-30
E
=
9.64 ×
10
-19
J
Step 4 · Find energy in electronvolts (EeV)
Use EeV = E ÷ (1.602 × 10⁻¹⁹)
One
electronvolt
is
1.602 ×
10
-19
J.
EeV
=
9.64 ×
10
-19
1.602 ×
10
-19
EeV = 6.02 eV
Check
Put the numbers back in. Both sides should match.
1.26 ×
10
10
=
2
×
π
500 ×
10
-12
✓
1.325 ×
10
-24
=
6.626 ×
10
-34
500 ×
10
-12
✓
9.64 ×
10
-19
=
(1.325 ×
10
-24
)
2
÷
(2
×
(9.109
×
10
-31
))
✓
6.02
=
9.64 ×
10
-19
1.602 ×
10
-19
✓
Topic check
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