9.3 KiB
9.3 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Bohr–Sommerfeld model | 2/2 | https://en.wikipedia.org/wiki/Bohr–Sommerfeld_model | reference | science, encyclopedia | 2026-05-05T16:28:10.422419+00:00 | kb-cron |
d
2
u
d
φ
2
=
−
(
1
−
k
2
Z
2
e
4
c
2
p
φ
2
)
u
+
m
0
k
Z
e
2
p
φ
2
(
1
+
W
m
0
c
2
)
=
−
ω
0
2
u
+
K
{\displaystyle {\frac {d^{2}u}{d\varphi ^{2}}}=-\left(1-k^{2}{\frac {Z^{2}e^{4}}{c^{2}p_{\mathrm {\varphi } }^{2}}}\right)u+{\frac {m_{\mathrm {0} }kZe^{2}}{p_{\mathrm {\varphi } }^{2}}}\left(1+{\frac {W}{m_{\mathrm {0} }c^{2}}}\right)=-\omega _{\mathrm {0} }^{2}u+K}
with solution
u
=
1
r
=
K
+
A
cos
ω
0
φ
{\displaystyle u={\frac {1}{r}}=K+A\cos \omega _{\mathrm {0} }\varphi }
The angular shift of periapsis per revolution is given by
φ
s
=
2
π
(
1
ω
0
−
1
)
≈
4
π
3
k
2
Z
2
e
4
c
2
n
φ
2
h
2
{\displaystyle \varphi _{\mathrm {s} }=2\pi \left({\frac {1}{\omega _{\mathrm {0} }}}-1\right)\approx 4\pi ^{3}k^{2}{\frac {Z^{2}e^{4}}{c^{2}n_{\mathrm {\varphi } }^{2}h^{2}}}}
With the quantum conditions
∮
p
φ
d
φ
=
2
π
p
φ
=
n
φ
h
{\displaystyle \oint p_{\mathrm {\varphi } }\,d\varphi =2\pi p_{\mathrm {\varphi } }=n_{\mathrm {\varphi } }h}
and
∮
p
r
d
r
=
p
φ
∮
(
1
r
d
r
d
φ
)
2
d
φ
=
n
r
h
{\displaystyle \oint p_{\mathrm {r} }\,dr=p_{\mathrm {\varphi } }\oint \left({\frac {1}{r}}{\frac {dr}{d\varphi }}\right)^{2}\,d\varphi =n_{\mathrm {r} }h}
we will obtain energies
W
m
0
c
2
=
(
1
+
α
2
Z
2
(
n
r
+
n
φ
2
−
α
2
Z
2
)
2
)
−
1
/
2
−
1
{\displaystyle {\frac {W}{m_{\mathrm {0} }c^{2}}}=\left(1+{\frac {\alpha ^{2}Z^{2}}{\left(n_{\mathrm {r} }+{\sqrt {n_{\mathrm {\varphi } }^{2}-\alpha ^{2}Z^{2}}}\right)^{2}}}\right)^{-1/2}-1}
where
α
{\displaystyle \alpha }
is the fine-structure constant. This solution (using substitutions for quantum numbers) is equivalent to the solution of the Dirac equation. Nevertheless, both solutions fail to predict the Lamb shifts.
== See also ==
Bohr model Old quantum theory
== References ==