The Path Integral and the Sum Over Histories
A position-space propagator can be expressed as a path integral with phase factor exp(iS/ℏ), where S is the action and ℏ is the reduced Planck constant. Under suitable semiclassical conditions, stationary-action paths give leading contributions. The same amplitude framework describes interference and tunnelling.
What a learner can do afterwards
- Identifies the path-integral phase factor and integration domain, and distinguishes amplitudes from probabilities
- Explains stationary-action contributions in a semiclassical approximation and identifies limits of a single-path account
- Explains coherent double-slit interference and tunnelling without assigning damping to individual real-action phase factors
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For one nonrelativistic particle, K(x_f,t_f;x_i,t_i)=<x_f|U(t_f,t_i)|x_i> is a position-space propagator. Formally, K=∫D[x] exp(iS[x]/ℏ), with endpoint positions and times fixed. S[x]=∫L dt is the action and ℏ=h/(2π) is the reduced Planck constant. Here i²=-1. For real S, exp(iS/ℏ) has modulus one.
D[x] denotes integration over admissible histories, including normalization. A construction such as time slicing defines the formal expression. Include nonclassical histories as well as classical ones. Add complex amplitudes before taking a squared modulus; individual paths do not carry separate probabilities. Applying K to an initial wavefunction gives the final wavefunction, whose squared modulus is position probability density.
For neighboring histories, the phase difference is ΔS/ℏ. Rapid phase variation makes nonstationary contributions largely cancel. At a stationary-action path, δS=0 for endpoint-preserving variations, so the first-order phase change vanishes. Nearby contributions can reinforce. Stationarity gives the classical equations of motion; it does not require zero action or the shortest geometric route.
This stationary-phase argument gives an approximation under suitable conditions, not exact deletion of other paths. Large absolute action alone is insufficient; relevant phase variation matters. Several stationary paths can contribute, and their amplitudes must be combined. Turning points and caustics can need different local approximations. These limitations do not make the full path integral undefined.
For coherent slit contributions A1 and A2, the screen amplitude is A1+A2 and intensity is proportional to |A1+A2|². Equal contributions with the same phase reinforce. Equal contributions a and -a cancel exactly. Opposite phase alone is insufficient: 2a and -a leave a. Each slit contribution represents a family of histories, not one measured particle route.
The same propagator describes a finite static barrier. Where E<V, no real classical trajectory at energy E crosses the forbidden region, but the quantum transmitted amplitude can be nonzero. For a thick smooth barrier, a semiclassical calculation gives exponential suppression of transmission. That suppression comes from evaluating the amplitude, not damping each individual real-action phase factor. Total energy is conserved.
In a simple alpha-decay model, λ=fT. Here T is barrier transmission probability, λ is decay rate, and f includes attempt frequency and alpha-formation probability. The half-life is t_half=ln(2)/λ. With f fixed, tripling T triples λ and divides the half-life by three. Transmission alone does not determine a half-life. This barrier-penetration model does not require borrowed energy.
Sum amplitudes with phase exp(iS/ℏ). Stationary action guides a semiclassical approximation; interference and tunnelling require the amplitude sum.
Original lesson, questions, worked solutions and flashcards written for this dataset.
Original readings and flashcards written for this dataset. Linked videos: Young's double slit introduction and Young's double slit equation, by Khan Academy.
More to read
Add amplitudes before finding probabilities
A path integral constructs an amplitude between specified endpoints. The factor exp(iS/ℏ) has modulus one when S is real. Its phase depends on the action. The integration measure supplies the rest of the construction, including normalization. A phase factor is not a probability for a particle to take one route.
For a double slit, group histories into the two slit alternatives. Call their contributions A1 and A2. The combined amplitude is A1+A2. Squaring its modulus gives |A1|²+|A2|²+2 Re(A1* A2), where * denotes complex conjugation and Re means real part. The last term is the interference term.
For equal magnitudes and opposite phases, the total amplitude vanishes. Unequal magnitudes leave a remainder. Adding separate probabilities would discard the interference term and would miss this distinction.
The propagator acts on the initial wavefunction to produce the final wavefunction. The squared modulus of that final wavefunction gives position probability density. This is different from interpreting |K|² as an ordinary normalized probability distribution for an exactly localized starting state.
Individual detections and an interference pattern
A screen can record separate photon or electron arrivals while the accumulated distribution has interference fringes. Sending particles one at a time does not by itself remove the fringes. The distribution depends on the prepared state and the experimental conditions.
Path-integral histories are alternatives used to calculate an amplitude. Including them does not establish that a detected particle followed one particular unobserved classical route, or that a detector observed every included history.
Interference depends on the relation between alternatives. A stable relative phase can preserve an interference term. Random relative phases can make that term average to zero across trials. A distinguishable which-slit record can also remove interference from the unconditioned screen distribution.
Thus two open slits do not by themselves guarantee visible fringes. Preparation coherence and the availability of distinguishing information matter. Individual detection events and the distribution of many events answer different experimental questions.
Transmission through a forbidden classical region
Consider a particle of fixed energy E meeting a finite static barrier. Where the potential V exceeds E, a real classical trajectory at that energy cannot cross. The quantum wavefunction can remain nonzero in the barrier and beyond it. Matching the wavefunction across the barrier can give transmission while conserving total energy.
For a rectangular barrier of width w and height V>E, define κ=sqrt(2m(V-E))/ℏ. In the thick-barrier regime, the transmission probability has an exponential factor exp(-2κw), multiplied by a prefactor. This is a regime-dependent approximation. It does not say that transmission is exponential directly in mass or barrier height for every potential.
The path-integral and wavefunction calculations describe the same transmission. Each real-action real-time phase factor still has modulus one. The suppression belongs to the evaluated result, not to a rule that reduces the magnitude of each barrier-crossing history.
In a simple alpha-decay rate model, transmission is multiplied by factors for attempts and alpha formation. Comparing half-lives therefore requires more than a transmission probability alone.
Uncertainty and a classical approximation
When the relevant state variances exist, position and momentum standard deviations satisfy ΔxΔp≥ℏ/2. These spreads describe outcome distributions across repeated preparations. They are not only instrument errors. A smaller position spread raises the lower bound on the momentum spread, but the actual product need not equal the bound.
This relation supplies no permission to borrow energy during tunnelling. A time-independent Hamiltonian conserves energy. Barrier transmission can occur within that constraint.
The classical approximation needs separate reasoning. In a path integral, phase changes are action differences divided by ℏ. Rapidly varying contributions can largely cancel, while neighborhoods of stationary-action histories can contribute at leading order. Several such histories may remain relevant.
Adding the same constant to every history's action multiplies the propagator by a common phase. It changes neither action differences nor stationarity. Large absolute action alone therefore cannot prove that one classical trajectory gives the whole prediction. Turning points and caustics also require care with the local approximation.
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