The Simple Step Potential and How it Explains the All Paths Feynman Approach to the Double Slit results

It took me forever to understand why we could just ‘take all possible paths’ in the double slit experiment.

It wasn’t till I re-solved (30 years after my initial graduate school QM course), the problem of the STEP Potential Barrier. This simple problem has a revealating solution – it contains a REFLECTED part of the particle.

Classically, reflection should not be possible, since the energy of the particle is greater than the energy of the step. Still, reflection does occur. This means, the particle follows a path that isn’t classically allowed. This also starts to explain Feynman’s ALL POSSIBLE PATHs approach to the double slit experiment.

1) Step Potential with E > V_0: Partial Reflection is Real

Consider a particle of mass m incident from the left on a potential step:

    \[ V(x)= \begin{cases} 0, & x < 0 \\ V_0, & x \ge 0 \end{cases} \quad \text{with } E > V_0 \]

Define region-dependent wavenumbers:

    \[ k_1=\frac{\sqrt{2mE}}{\hbar}, \quad k_2=\frac{\sqrt{2m(E-V_0)}}{\hbar}. \]

Time-independent Schrödinger equation solutions:

    \[ \psi_I(x)=e^{ik_1 x}+r\,e^{-ik_1 x}, \quad (x<0) \]

    \[ \psi_{II}(x)=t\,e^{ik_2 x}, \quad (x>0) \]

Applying continuity of \psi and \psi' at x=0:

    \[ 1+r=t, \quad ik_1(1-r)=ik_2 t. \]

Solving for reflection and transmission coefficients:

    \[ r=\frac{k_1-k_2}{k_1+k_2}, \quad t=\frac{2k_1}{k_1+k_2}. \]

The probability current for a plane wave Ae^{ikx} is j=\frac{\hbar k}{m}|A|^2. Hence,

    \[ R=|r|^2=\left(\frac{k_1-k_2}{k_1+k_2}\right)^2 > 0, \]

    \[ T=\frac{k_2}{k_1}|t|^2=\frac{4k_1k_2}{(k_1+k_2)^2}, \]

and R+T=1.

Why reflection without a barrier?
Classically, with E>V_0 there’s no turning point, so reflection is impossible. Quantum mechanically, the de Broglie wavelength jumps at the interface (from 2\pi/k_1 to 2\pi/k_2). To satisfy boundary conditions for both \psi and \partial_x\psi, a left-moving component is required—an impedance mismatch effect analogous to Fresnel reflection at an optical interface. Reflection is thus an interference requirement, not an energy shortfall.

2) From “Reflection at a Step” to “All Paths” (Feynman’s Picture)

Feynman’s path integral expresses the amplitude for a particle to go from x_a at time t_a to x_b at time t_b as a coherent sum over all paths x(t):

    \[ K(x_b,t_b;x_a,t_a)=\int \mathcal{D}[x(t)]\,\exp\!\left\{\frac{i}{\hbar}S[x(t)]\right\}, \]

where S[x]=\int (\tfrac{1}{2}m\dot{x}^2 - V(x))\,dt.

  • In the classical limit (\hbar \to 0), phases from wildly different paths cancel; only those near stationary action (classical paths) dominate.
  • At finite \hbar, non-classical paths contribute with phases e^{iS/\hbar} and interfere constructively or destructively.

How That Explains Step Reflection

At a sharp step, many paths “sample” the region x>0 (where kinetic energy is E-V_0) before returning to x<0. Their actions differ due to the potential term. When all contributions are added coherently, two dominant families of paths appear at the detector on the left:

  • Those that keep momentum +k_1 (forward-going)
  • Those that reverse to -k_1 (reflected)

The relative phase between these families depends on the action difference tied to the k_1 \to k_2 mismatch—precisely what boundary conditions captured. The constructive interference of the “returning” family yields a nonzero R.

This “impedance mismatch ⇒ coherent back-sum” corresponds directly to the path integral description.

And Now the Double Slit

In the double-slit experiment, the amplitude at a screen point P is a sum over all paths passing through each slit:

    \[ \mathcal{A}(P)=\sum_{\text{paths through slit A}} e^{\frac{i}{\hbar}S[x]} +\sum_{\text{paths through slit B}} e^{\frac{i}{\hbar}S[x]}. \]

Within each slit’s sum are countless non-classical trajectories—zigzags, edge grazes, small detours—each contributing a slightly different phase. The envelopes of these sums are dominated by near-straight (stationary-action) routes. Coherent addition across the two slits gives the interference pattern, with phase difference \Delta\phi \approx \frac{2\pi}{\lambda}\Delta L. Blocking one slit removes one entire family of paths, eliminating interference.

Unifying Intuition

  • Step reflection with E>V_0: coherence across paths entering the step forces a back-propagating component—nonzero R without a classical turning point.
  • Double-slit fringes: coherence across paths through both apertures yields the interference pattern.

In both cases, classically forbidden effects (reflection without a barrier, interference without waves) naturally emerge once we accept that all paths contribute with phase e^{iS/\hbar}.
Classical motion reappears only when non-stationary paths’ phases cancel out. Wherever discontinuities (steps, slit edges) create phase mismatches between path families, quantum interference manifests macroscopically—as reflection, diffraction, or fringes.