Central Forces in Quantum Mechanics
Radially symmetric problems appear if the interaction between two particles depends only on their separation r. We will first see how the dynamical problem of the motion of the two particles can be separated in terms of center of mass motion and relative
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Central Forces in Quantum Mechanics
Radially symmetric problems appear if the interaction between two particles depends only on their separation r. We will first see how the dynamical problem of the motion of the two particles can be separated in terms of center of mass motion and relative motion and then write the effective Hamiltonian for the relative motion of the two particles in spherical coordinates.
7.1 Separation of Center of Mass Motion and Relative Motion The separation of center of mass motion and relative motion proceeds like in classical mechanics. The Hamiltonian of the two-particle system is H =
p21 p2 p2 P2 + + V (r), + 2 + V (|x1 − x2 |) = 2m1 2m2 2M 2μ
(7.1)
where M = m1 + m2 ,
m1 m2 m1 + m2
(7.2)
r = x1 − x2
(7.3)
μ=
are the total and reduced mass, R=
m1 x1 + m2 x2 , m1 + m2
are the operators for center of mass and relative coordinates, and
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1_7
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7 Central Forces in Quantum Mechanics
˙ = p1 + p2 , P = MR
p = μ˙r =
μ μ m2 p1 − m1 p2 p1 − p2 = m1 m2 m1 + m2
(7.4)
are the momentum operators of center of mass motion and relative motion. The relative motion of the two original particles also comes with an angular momentum l = x1 × p1 + x2 × p2 − R × P = r × p.
(7.5)
The inverse transformations are x1 = R +
m2 r, M
x2 = R −
m1 r, M
p1 =
m1 P + p, M
p2 =
m2 P − p, M
(7.6)
and if we assume m2 ≥ m1 , m2 =
M+
√
M(M − 4μ) , 2
m1 =
M−
√ M(M − 4μ) . 2
(7.7)
The ∇ operators transform as ∂ ∂ ∂ = + , ∂R ∂x 1 ∂x 2
m2 ∂ ∂ m1 ∂ = − , ∂r M ∂x 1 M ∂x 2
∂ ∂ m1 ∂ + , = ∂x 1 M ∂R ∂r
∂ ∂ m2 ∂ − . = ∂x 2 M ∂R ∂r
(7.8)
(7.9)
These are the same transformations for operators as the corresponding transformations for classical coordinates and momenta in classical mechanics. From the quantum mechanics perspective this is not surprising, since the transformation equations for the operators are linear and therefore also hold for the expectation values of the operators, hence for the classical variables. What becomes particularly relevant for quantum mechanics is that the transformations preserve canonical commutation relations, [x1 , p1 ] = ih1, ¯ [x2 , p2 ] = ih1 ¯
⇔
[R, P] = ih1, ¯ [r, p] = ih1. ¯
(7.10)
Since the interaction does not depend on the center of mass coordinates, we can separate the center of mass motion with momentum P = h¯ K in the wave function for the time-independent Schrödinger equation, (x 1 , x 2 ) = √
1 2π
3
exp(iK · R)ψ(r),
(7.11)
and the energy eigenvalue problem H | = Etotal | reduces to an eigenvalue problem for the relative motion,
7.1 Separation of Center of Mass Motion and Relative Motion
Eψ(r) = −
131
h¯ 2 ψ(r) + V (r)ψ(r), 2μ
(7.12)
h¯ 2 K 2 2M
(7.13)
where Etotal = E +
is the total energy in the center of mass motion and relative motion. The discussion of separated solutions in Sect. 5.5 implies that the
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