How Quantum Mechanics reconciles with Classical Mechanics?

Question

Imagine we have to charged particles. The kinetic energy of the system is:

$$ T = \frac{1}{2}(m_1 + m_2) \mathbf{\dot{R}}_{cm}^2 + \frac{1}{2} \mu \dot{R}^2 + \frac{L^2}{2 \mu R^2} $$

and its potential energy is:

$$ V = -\frac{q^2}{4\pi \varepsilon_0 |\mathbf r_1 - \mathbf r_2|} = -\frac{q^2}{4\pi \varepsilon_0 |\mathbf R|} $$

The total energy of the system is: $$ E = \frac{1}{2}(m_1 + m_2) \mathbf{\dot{R}}_{cm}^2 + \frac{1}{2} \mu \dot{R}^2 + \frac{L^2}{2 \mu R^2} - \frac{q^2}{4\pi \varepsilon_0 R} $$

Where $\mathbf{R}_{cm}$ is the position of the center of mass $\mathbf R = \mathbf r_1 - \mathbf r_2$ and $\mathbf L$ is the angular momentum of the system.

If we assume the center of mass, and $\mathbf r$ do not change with time, we have: $$ E(r) = \frac{L^2}{2 \mu r^2} - \frac{q^2}{4\pi \varepsilon_0 r} $$

If we find the minimum energy possible:

$$ E'(r) = 0 $$ $$ r_0 = \frac{4 \pi \varepsilon_0 L^2}{\mu q^2} $$

$$ E_{\text{min}} = E(r_0) $$ Finally we get: $$ E_{\text{min}} = -\frac{\mu q^4}{32 \pi^2 \varepsilon_0^2 L^2} $$

which reminds me a little too much to the energy eigenvalues of the hydrogen atom: $$ E_n = -\frac{\mu e^4}{32 \pi^2 \varepsilon_0^2 n^2 \hbar^2} $$ more specifically to the ground state of the energy levels. Is this a mere coincidence? I know that classical quantum mechanics was built upon classical mechanics, but this shoked me while I was deriving the result by myself.

I know this classical model breaks up when you take into account radiation, but could you reconocile this with the Bohr model somehow?

Deriving from this two equations that: $$ L = n \hbar $$ makes sense?

Answer 1

Yep! It does.

Take that as $\hbar$ goes to zero, also $n$ goes to $\infty$ at the "just right" rate so that $n\hbar$ approaches the desired classical $L$.

In terms of the underlying quantum state, you don't wanna use the Hilbert formalism, but a formalism like the Wigner function. (In some sense, the Hilbert formalism on the usual space "assumes $\hbar > 0$".) What I believe, at least, will happen is that the Wigner function will develop finer and finer oscillations that limit to a classical smooth, non-negative probability distribution reflecting an agent whose knowledge is incomplete but mathematically coherently understandable as arising from an underlying "hidden variable" of a pair of charges of well-defined positions and motion, just that said agent doesn't know where and in what way.

Answer 2

Classical mechanics and quantum mechanics are connected through a hamiltonian formulation of classical dynamics known as the "Poisson Bracket" formulation. According to J.J. Sakurai (Modern Quantum Mechanics, 50), it was Dirac who first noted that quantum mechanical relations can be obtained from classical ones by simply replacing the classical poisson bracket in the classical variables by a corresponding quantum bracket multiplied by -i2pi/h. Thus, the classical equation of motion in the poisson bracket formulation looks identical to the Heisenberg equation of motion, apart from the afore mentioned factor (Sakurai, 84). The similarities in these formulae are superficial in the sense that the position and momentum in the classical equations are numbers, while in the quantum formulation they are operators. The analogous form of the equations is due to the fact that the poisson bracket and quantum mechanical commutator are both examples of Lie algebras, thus there are deep fundamental connections between classical and quantum mechanics that go beyond coincidence. Now, historically, Bohr did calculations very similar to the one that you performed using the "correspondence principle" (allowing Bohr to ignore radiation) and what was then known as a "quantum condition"; which is really equivalent to the quantum canonical commutation relation. Only Bohr quantized the angular momentum via "quantum action" using the Hamilton-Jacobi theory of action angle variables (another exotic formulation of Classical dynamics cf. Goldstein). The end result is that the action of an electron in a high energy circular orbit around the nucleus is equal to nh; a result later generalized by Sommerfeld to elliptical orbits. At any rate, however, it is the aforementioned principle of Dirac that is the underlying reason that semi-classical calculations like the one's performed by Bohr work in the first place, no matter how ill-conceived or incorrect the justifications for his calculations were at the time.