4.5 The Quantum Mechanical Analog
In spherical coordinates (r, θ, φ), the potential depends only on r, and the stationary
wave functions separate as (discarding spin)
ψ nlm r
ð Þ ¼ R nl r
ð ÞΨ lm θ, φ
ð
Þ; n ¼ 1, 2, 3, . . . ; l ¼ 0, 1, . . . , n À 1; m
¼ Àl, 1 À l, . . . , l À 1, l
According to Unsöld’s (1927) theorem,
X l
m¼Àl
Ψ lm θ, φ
ð
Þ
j
j
2
2l þ 1
4π
the expression
F nl r
ð Þ
1
2l þ 1
X l
m¼Àl
ψ nlm r
ð Þ
j
j
2 ¼
1
4π
R
2
nl r
ð Þ
is spherically symmetric. Following Schrödinger’s (1926, para. 7) interpretation of
|ψ nlm (r)|
2 as being the weight of configuration r in state nlm, F nl describes the
average weight of the configurations {r| |r| ¼ r} in the 2l + 1 stationary states
{nlm| À l m l}.
The additional or accidental symmetry of the quantum Kepler problem shows up
in the existence of the quantum Laplace–Runge–Lenz vector, which I propose to
term Pauli vector (Pauli 1926, Eq. (50)).
Thus, the set of all possible values of |ψ nlm (r)|
2 (n ¼ 1,2,...; l ¼ 0, 1, . . .,n; m ¼ Àl,
Àl + 1,. . ., 0, . . ., l ) forms a spherically symmetric figure. Here, no single initial state
(“initial condition”) plays any role. The same holds true for the set of all possible
values of |ψ nlm (r)|
2 under the condition of given (expectation) value(s) of
• main quantum number, n, or energy, E n , iff V(r) ~ 1/r, or
• angular quantum number, l, or modulus of angular momentum,
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
l l þ 1
ð
Þ
p
ħ, or
• main and angular quantum numbers, n and l, or energy, E nl , and/or modulus of
angular momentum,
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
l l þ 1
ð
Þ
p
ħ, in case of general spherically symmetric potentials, V(r), or
• Pauli vector.
The latter condition has been posed to the auditorium for examination in their
classes.
44
P. Enders
In spherical coordinates (r, θ, φ), the potential depends only on r, and the stationary
wave functions separate as (discarding spin)
ψ nlm r
ð Þ ¼ R nl r
ð ÞΨ lm θ, φ
ð
Þ; n ¼ 1, 2, 3, . . . ; l ¼ 0, 1, . . . , n À 1; m
¼ Àl, 1 À l, . . . , l À 1, l
According to Unsöld’s (1927) theorem,
X l
m¼Àl
Ψ lm θ, φ
ð
Þ
j
j
2
2l þ 1
4π
the expression
F nl r
ð Þ
1
2l þ 1
X l
m¼Àl
ψ nlm r
ð Þ
j
j
2 ¼
1
4π
R
2
nl r
ð Þ
is spherically symmetric. Following Schrödinger’s (1926, para. 7) interpretation of
|ψ nlm (r)|
2 as being the weight of configuration r in state nlm, F nl describes the
average weight of the configurations {r| |r| ¼ r} in the 2l + 1 stationary states
{nlm| À l m l}.
The additional or accidental symmetry of the quantum Kepler problem shows up
in the existence of the quantum Laplace–Runge–Lenz vector, which I propose to
term Pauli vector (Pauli 1926, Eq. (50)).
Thus, the set of all possible values of |ψ nlm (r)|
2 (n ¼ 1,2,...; l ¼ 0, 1, . . .,n; m ¼ Àl,
Àl + 1,. . ., 0, . . ., l ) forms a spherically symmetric figure. Here, no single initial state
(“initial condition”) plays any role. The same holds true for the set of all possible
values of |ψ nlm (r)|
2 under the condition of given (expectation) value(s) of
• main quantum number, n, or energy, E n , iff V(r) ~ 1/r, or
• angular quantum number, l, or modulus of angular momentum,
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
l l þ 1
ð
Þ
p
ħ, or
• main and angular quantum numbers, n and l, or energy, E nl , and/or modulus of
angular momentum,
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
l l þ 1
ð
Þ
p
ħ, in case of general spherically symmetric potentials, V(r), or
• Pauli vector.
The latter condition has been posed to the auditorium for examination in their
classes.
44
P. Enders