quantum mechanics - Feynman's path integral and energy discretization?

Feynman’s path integral formulation of quantum mechanics is based on the following two postulates:

1. 
   

   If an ideal measurement is performed to determine whether a particle has a path lying in a region of spacetime, the probability that the result will be affirmative is the absolute square of a sum of complex contributions, one from each path in the region.

   
   


2. 
   

   The paths contribute equally in magnitude but the phase of their contribution is the classical action, i.e. the time integral of the Lagrangian taken along the path.

   
   
   

Suppose that those postulates are as natural as it can be, i.e. they aren't a distilled version of something more elementary. If this is the case, then how to explain, in layman's terms, that how the discrete energy levels in, e.g. the harmonic oscillator arise from those two postulates? Are there qualitative answers?

Answer

I'm not sure what you expect for qualitative examples in "layman's terms". The time-evolution amplitude from one point to another, $$\langle x_f|U(T)|x_i\rangle=K(x_f,x_i;T)~,$$ evaluated from the path integral is the propagator, which, for the oscillator, happens to be the celebrated 1866 Mehler kernel: a causal Green's function of the oscillator equation. Not coincidentally, this was available 60 years before QM. The point being that the path integral propagator is mostly classical. The quantization of energy levels is actually a feature of the compact time domain, as @Qmechanic comments.

Specifically, the classical action for the oscillator, from the above WP reference, amounts to $$ \begin{align} S_\text{cl} & = \int_{t_i}^{t_f} \mathcal{L} \,dt = \int_{t_i}^{t_f} \left(\tfrac12 m\dot{x}^2 - \tfrac12 m\omega^2 x^2 \right) \,dt \\[6pt] & = \frac 1 2 m\omega \left( \frac{(x_i^2 + x_f^2) \cos\omega(t_f - t_i) - 2 x_i x_f}{\sin\omega(t_f - t_i)} \right)~. \end{align} $$

The propagator, then, the above amplitude, can be evaluated from the functional integral as $$K(x_f, x_i;T) = \Large e^\frac{i S_{cl}}{\hbar} \small ~ \sqrt{ \frac{m\omega}{2\pi i \hbar \sin\omega(t_f - t_i)}}~~, $$ where $T=t_f-t_i$. Effectively, it is the exponential of the classical action, with a minor normalization correction due to quantum fluctuations, which, nevertheless is not that important.

• The quantized energy levels are already in the discrete harmonics predicated by the periodicity of the classical action--themselves oblivious of $\hbar$.

This expression also equals to the conventional Hilbert space propagator in terms of Hermite functions,
$$ \begin{align} K(x_f, x_i;T ) & = \left( \frac{m \omega}{2 \pi i \hbar \sin\omega T } \right)^\frac12 \exp{ \left( \frac{i}{2\hbar} m \omega \frac{ (x_i^2 + x_f^2) \cos \omega T - 2 x_i x_f }{ \sin \omega T } \right) } \\[6pt] & = \sum_{n = 0}^\infty \exp{ \left( - \frac{i E_n T}{\hbar} \right) } \psi_n(x_f) ~\psi_n(x_i)^{*}~, \end{align} $$ with which you'd propagate your ket to the bra of the amplitude in conventional Hilbert space.(This is the expression Mehler summed in 1866.) But you might wish to pretend you are a Martian, unaware of that formulation, or Mehler's wonderful, "prescient", formula.

Rewrite this as $$ = \left( \frac{m \omega}{\pi \hbar} \right)^\frac12 e^\frac{-i \omega T} 2 \left( 1 - e^{-2 i \omega T} \right)^{-\frac12} \exp{ \left( - \frac{m \omega}{2 \hbar} \left( \left(x_i^2 + x_f^2\right) \frac{ 1 + e^{-2 i \omega T} }{ 1 - e^{- 2 i \omega T}} - \frac{4 x_i x_f e^{-i \omega T}}{1 - e^{ - 2 i \omega T} }\right) \right) }\\ \equiv \left( \frac{m \omega}{\pi \hbar} \right)^\frac12 e^\frac{-i \omega T } 2 ~ R(T)~. $$

The $e^{-in\omega T }$ Fourier modes of R(T) then multiplying this 0-point energy prefactor may be compared to the standard Hilbert space eigenstate expansion of the resolvent, to reassure you of the standard quantized spectrum of the quantum oscillator, $E_n = \left( n + \tfrac12 \right) \hbar \omega~. $

Here, faced with discreteness, you only need appreciate the essential periodicity of the system, the compactness that forces you to a harmonic structure: the waviness of the system; and that most of it is traceable to the classical action in this (somewhat exceptional) quadratic hamiltonian paradigm.

In their elementary textbook, Feynman and Hibbs work it out nicely in Probs 2-2, 3-8, (eqns 2-9,3-59) and "spike" in eqns (8-12), (8-13). (They even go ludicrously further than that, trying to make you "see" the Mehler kernel deconstruct itself to Hermite polynomials, taking requests of your type too far, in my opinion.) In any case, following the simple math is actually less obscure than summarizing it in "code" verbiage.