Abstract
The usual method of calculating the diagonal matrix elements of an integral power of the radius
r
in an inverse square quantum system is that due to Waller. His procedure is based on the Schrödinger form of the theory, and utilizes in particular the generating series for the Langueree polynomials. For negative powers, Dirac's very elegant theory of "q-numbers," developed in these 'proceedings' during the early days of quantum mechanics, furnishes an interesting alternative method which appears to have been overlooked, and which we believe is easier. From his theory the following rule can be derived: Suppose that we desire the mean value (diagonal element) of 1/r
s
, where
s
is an integer greater than unity. We write down the experssion with A(α
l
+ β
l
e
-
i
x
+ γ
l
e
i
x
)
s-2
A = 16π
4
m
2
z
2
e
4
/(
l
+ ½)
n
3
h
4
, α
l
= 4π
2
m
z
e
2
/
h
2
l
(
l
+1), β
l
= 2π
2
m
Z
e
2
/
h
2
(
l
+½)(
l
+1)[1-(
l
+1)
2
/
n
2
]½, γ
l
= 2π
2
m
z
e
2
/
h
2
l
(
l
+½)[1-
l
2
/
n
2
]½.
Cited by
38 articles.
订阅此论文施引文献
订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献