Quantum Physics Foundations: Tunneling and Barriers as Probability Flow

1) Classical expectation: a barrier is a hard “no” when energy is too low

Consider a particle moving toward a region of higher potential energy (a “barrier”). Classically, the rule is simple: if the particle’s total energy E is less than the barrier height V, the particle cannot enter the barrier region. It reaches a turning point where its kinetic energy would have to become negative, and it reflects back.

A concrete setup you can picture is a one-dimensional barrier: a flat region of potential V(x)=0 on the left, then a barrier of height V(x)=V₀ over a finite width, then back to 0 on the right. Classically:

• If E > V0, the particle crosses (maybe slowed down inside the barrier region if it is a “hill”).
• If E < V0, the particle reflects with probability 1; transmission probability is 0.

This classical picture treats the particle as a point with a definite trajectory and forbids any presence in a region where the kinetic energy would be negative.

2) Quantum prediction: nonzero transmission and exponential decay inside the barrier

Quantum mechanically, a particle approaching a barrier is described by a wave-like state. When this state meets a change in potential, it does not simply “stop”; instead, it must satisfy boundary matching conditions at the edges of the barrier. The result is that the state generally splits into a reflected part and a transmitted part.

For a barrier with E < V0, the key qualitative feature is what happens inside the barrier: the wave-like state does not oscillate as it does in free space. Instead, its amplitude decays approximately exponentially with distance into the barrier. That means the probability density becomes smaller and smaller as you go deeper into the barrier, but it is not forced to be exactly zero immediately at the boundary.

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An intuitive picture: the barrier region is “classically forbidden,” but quantum states are constrained by smoothness and continuity. The state cannot abruptly drop to zero at the boundary without violating the required matching behavior. So it leaks into the barrier, decaying as it goes. If the barrier is not infinitely wide, a small but nonzero amplitude can reach the far side, producing a transmitted component.

In probability language, tunneling is best thought of as probability flow: probability current can enter the barrier region, diminish as it propagates through the decaying amplitude, and a reduced current can emerge on the other side. Reflection is still typically dominant when E < V0, but transmission is not strictly forbidden.

Boundary matching (physical meaning)

At each boundary (where the potential changes), the state must match in a way that avoids unphysical discontinuities. Practically, this means the “shape” of the state on one side constrains the “shape” on the other side. Even if the barrier region supports only decaying solutions, those solutions must connect to the incoming and outgoing solutions in the free regions. That connection is what mathematically produces a nonzero transmitted amplitude.

3) What controls tunneling probability: width, height, mass, and energy (penetration depth)

Tunneling probability is extremely sensitive to barrier parameters. Qualitatively, transmission becomes smaller when the barrier is “harder to penetrate,” and larger when the barrier is “thinner” or “lower” relative to the particle’s energy.

Penetration depth as the key intuition

Inside the barrier, the amplitude decays roughly like an exponential. The distance over which the amplitude drops by a large factor is often summarized by a penetration depth: a characteristic length scale that tells you how far the state significantly extends into the barrier before becoming negligibly small.

Qualitatively:

• Higher barrier height (larger V0 - E) → faster decay → shorter penetration depth → much smaller transmission.
• Wider barrier → more distance for the exponential decay to act → dramatically smaller transmission.
• Larger particle mass → faster decay in the barrier → smaller transmission.
• Higher particle energy (closer to V0 from below) → slower decay → longer penetration depth → larger transmission.

Because the decay is exponential, small parameter changes can cause huge changes in transmission. For example, increasing barrier width by a modest amount can reduce tunneling by orders of magnitude.

Step-by-step qualitative reasoning for “will tunneling be noticeable?”

1. Compare energy to barrier height: compute the gap V0 - E. A small gap suggests a longer penetration depth and more tunneling.
2. Check barrier width: compare the width to the penetration depth. If the barrier is only a few penetration depths wide, transmission may be measurable; if it is many penetration depths wide, transmission becomes tiny.
3. Consider the particle mass: electrons tunnel much more readily than heavier particles through the same barrier shape.
4. Translate to an experimental signal: ask what “tiny” means in context (e.g., a detectable current in a circuit, or a decay rate in a nucleus).

4) Common misconceptions: tunneling is not literal “energy borrowing”

Misconception: “The particle borrows energy to climb over the barrier”

A popular story says the particle “borrows energy” for a short time and then “pays it back.” This can be misleading if taken as literal bookkeeping. In tunneling through a static barrier, the particle’s total energy is not temporarily increased in a measurable, classical sense. The barrier region does not correspond to the particle having a well-defined negative kinetic energy that must be compensated by a loan.

A better statement is: the quantum state in the barrier is a non-oscillatory (decaying) solution consistent with the same total energy E. Transmission occurs because the state is not confined to a single classical trajectory; it is determined by boundary matching and the resulting probability flow.

Misconception: “The particle loses energy while tunneling”

For a time-independent barrier, the transmitted component emerges with the same energy as the incident component (though with reduced amplitude, meaning reduced probability/current). What changes is the likelihood of transmission, not the energy of those particles that do transmit.

Misconception: “Tunneling means the particle is ‘inside the barrier’ like a classical object”

Inside the barrier, the state’s amplitude is nonzero, so there is a nonzero probability density. But interpreting that as a classical path through the barrier is not required. The most operational interpretation is: if you place a detector sensitive to presence in the barrier region (and accept that this changes the situation), you can sometimes register the particle there. Without such intervention, the barrier region is part of the quantum evolution that connects incoming and outgoing probability flow.

5) Case examples

Example A: Alpha decay as barrier escape

In alpha decay, an alpha particle (a helium nucleus) is bound inside a larger nucleus by the strong nuclear force, but it faces a repulsive barrier due to electric (Coulomb) repulsion when trying to leave. Classically, if the alpha particle’s energy is below the top of this Coulomb barrier, it would be trapped forever.

Quantum mechanically, the alpha particle’s state has a small but nonzero probability to tunnel through the barrier and appear outside the nucleus. Once outside, it can move away classically (the repulsion accelerates it outward).

Step-by-step physical picture:

1. Inside the nucleus: the alpha particle is confined in a region where it can exist as a bound configuration.
2. At the nuclear edge: it encounters the Coulomb barrier that is too high to cross classically.
3. Within the barrier: the amplitude decays with radius; most of the probability reflects back inward.
4. Beyond the barrier: a tiny transmitted amplitude corresponds to a small probability per unit time of escape.
5. Observed half-life: the decay rate is extremely sensitive to barrier width/height and the alpha energy, explaining why half-lives vary enormously across isotopes.

Design insight: small changes in alpha energy (or effective barrier shape) can change decay rates by huge factors because the tunneling probability depends exponentially on barrier parameters.

Example B: Scanning tunneling microscopy (STM) and tunneling current

An STM uses a sharp conductive tip brought extremely close to a conductive or semiconductive surface—so close that electrons can tunnel across the vacuum gap between tip and sample. Classically, a vacuum gap is an insulator: electrons should not cross it without enough energy to overcome the barrier. Quantum mechanically, electrons have a nonzero tunneling probability across a thin gap.

Operationally, the STM measures a tunneling current, which depends very strongly on the tip–sample separation.

Step-by-step mechanism:

1. Set a small bias voltage: this creates an energy difference that favors electron flow from one side to the other.
2. Bring the tip close: the vacuum gap acts as a barrier with a width approximately equal to the separation distance.
3. Electrons tunnel: the wave-like electron states decay into the vacuum but can reach the other side if the gap is thin enough.
4. Measure current: the tunneling probability translates into a measurable current; because of exponential sensitivity, moving the tip by a fraction of a nanometer can change the current dramatically.
5. Map the surface: by scanning and maintaining constant current (or constant height), the instrument infers surface topography and electronic structure with very high spatial resolution.

Probability-flow viewpoint: the measured current is literally a controlled probability flow per unit time from tip to sample, regulated by barrier width (gap distance) and barrier height (related to work function and bias conditions).

6) Design-style prompts: how parameter changes affect tunneling rates

Use the prompts below as “engineering questions” about tunneling. For each, predict whether the tunneling rate (or transmission probability / tunneling current) increases or decreases, and explain using penetration depth and exponential sensitivity.

Barrier geometry

• Width: If you increase the barrier width by 20%, what happens to transmission? What if you halve the width?
• Height: If the barrier height increases while the particle energy stays fixed, how does the penetration depth change?
• Shape: Compare a rectangular barrier to a smoothly varying barrier with the same maximum height and similar area. Which tends to tunnel more, and why might the “effective width” matter?

Particle properties

• Mass: Compare electron tunneling to proton tunneling through the same barrier width/height. Which is more suppressed, and how would that show up in an experiment?
• Energy tuning: If you can increase the particle energy so that E approaches V0 from below, how does transmission change? What happens once E exceeds V0?

STM-style control questions

• Tip–sample distance: If the tip retracts by 0.1 nm, does the current drop a little or a lot? How would you use feedback to keep current constant while scanning?
• Work function changes: If the surface has a region with a higher effective barrier height, what happens to the tunneling current at fixed distance and bias?

Nuclear tunneling prompts (alpha decay)

• Energy shift: If an isotope emits alpha particles with slightly higher energy, how does that affect half-life qualitatively?
• Barrier width: If the effective barrier becomes wider (e.g., due to a change in nuclear charge distribution), what happens to decay rate?