SME
University Physics III: Waves, Optics & Modern · Problem type
The size and tilt of L
Use this for “For l = 3, find |L|, the allowed Lz, and the angle between L and the z-axis when m = 2.”
Diagram
|L|
=
√
3 × 4
ħ
=
3.464ħ,
and
its
z-part
is
mħ
=
2ħ.
cos
θ
=
2
3.464
,
so
θ
=
54.74°
from
the
z-axis.
2ℓ + 1 = 7 cones; L is never straight along z, and its x- and y-parts stay unknown round the cone.
This is an example. Type your own values to start a new problem.
|L|
ħ
=
(
+
1)
ħ
m
Magnetic quantum number
example
Lz
z-part of L (in ħ)
calculated
θ
Angle from the z-axis
worked out (not typed) · °
tilts
Number of orientations
worked out (not typed)
Formulas
|L|
=
√(
l
×
(
l
+
1))
3.4641 = √(3 × (3 + 1))
Lz
=
m
2 = 2
θ
=
cos⁻¹(
m
÷
|L|)
54.7356
=
cos⁻¹(
2
3.4641
)
tilts
=
2
×
l
+
1
7 = 2 × 3 + 1
Assumptions
•
An electron in an orbital with quantum number l (s, p, d, f for l = 0, 1, 2, 3); sizes are in units of ħ.
•
Only |L| and its z-part Lz = mħ are sharp; m is a whole number from −l to l, so L takes 2l + 1 tilts.
•
Since |m| < √(l(l + 1)), L never lies along z; an s electron (l = 0) has no orbital angular momentum.
Step-by-step
WE KNOW
l = 3, m = 2
FIND
size of L (in ħ) (|L|), z-part of L (in ħ) (Lz), angle from the z-axis (θ), number of orientations (tilts)
Step 1 · Find size of L (in ħ): |L|
Use |L| = √(l × (l + 1))
The
size
of
L
in
ħ
is
the
root
of
l
(
l
+
1):
0
for
an
s
electron,
otherwise
a
little
more
than
l
.
|L| = √(3 × (3 + 1))
|L|
=
√
3 × 4
|L|
=
√
12
|L| = 3.4641
Step 2 · Find z-part of L (in ħ): Lz
Use Lz = m
The
z-part
of
L
is
m
units
of
ħ.
Lz = 2
Step 3 · Find angle from the z-axis (θ)
Use θ = cos⁻¹(m ÷ |L|)
The z-part over the size is the cosine of the tilt from z.
θ
=
cos⁻¹(
2
3.4641
)
θ
=
54.7356°
Step 4 · Find number of orientations (tilts)
Use tilts = 2 × l + 1
Count
m
from
−
l
to
l
:
l
below
0,
l
above
it,
and
0
itself.
tilts = 2 × 3 + 1
tilts = 6 + 1
tilts = 7
Check
Put the numbers back in. Both sides should match.
3.4641 = √(3 × (3 + 1)) ✓
2 = 2 ✓
54.7356
=
cos⁻¹(
2
3.4641
)
✓
7 = 2 × 3 + 1 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.