One Path or Many

Chem 3240 · Lecture 3.1b · from Newton to Feynman

Davit Potoyan

Newton and Hamilton: one physics, two languages

Newton: forces change the motion

m\,\frac{d^2x}{dt^2} = -\frac{dV}{dx}

Hamilton: the energy drives it

H(x,p) = \frac{p^2}{2m} + V(x)

\frac{dx}{dt} = \frac{\partial H}{\partial p} \qquad\qquad \frac{dp}{dt} = -\frac{\partial H}{\partial x}

The first says p = m\,dx/dt, the second says dp/dt = -dV/dx. Together: Newton’s law, the same physics.

Phase space: a state is a point

• A classical state is position and momentum together: one point (x, p)
• Newton follows the mass; Hamilton follows the point
• The point never leaves its curve H = E

Energy conservation, pictured

Any quantity A(x, p) changes as

\frac{dA}{dt} = \{A, H\}

\{A,B\} = \frac{\partial A}{\partial x}\frac{\partial B}{\partial p} - \frac{\partial A}{\partial p}\frac{\partial B}{\partial x}

• \{H, H\} = 0: the energy never changes
• \{x, p\} = 1 becomes [\hat{x}, \hat{p}] = i\hbar

Least action: nature picks one path

• Try every path from A to B (K: kinetic energy)

S = \int_{t_A}^{t_B} (K - V)\,dt

\delta S = 0

• Only one path makes S stationary, and it obeys Newton’s law

One path or many? Feynman drills holes

• Two holes: the electron takes both paths, the double slit
• More holes, more screens, then no screens at all: every path from A to B

Every history at once

• The particle does not choose: every history contributes
• Each carries a phase clock, turned by its action S/\hbar
• Real histories or bookkeeping? The equations do not say; interpretations disagree (Everett: many worlds)

Example: what does a detour cost?

Going from A to B in time T through a hole at height y halfway: how much extra action does the detour cost?

Sideways speed 2y/T on both legs, for a total time T:

\Delta S = \frac{1}{2}m\left(\frac{2y}{T}\right)^{2} T = \frac{2m\,y^2}{T}

Smallest at y = 0: the straight line wins, as Newton says.

• Near y = 0 the cost grows only as y^2: small detours are almost free

Every path adds an arrow

\psi(B) \;\propto \sum_{\text{paths}} e^{\,iS/\hbar}

• Each path: an arrow turned by S/\hbar
• Near the straight path the arrows line up; far ones curl up and cancel
• Free particle: S/\hbar is the phase (px - Et)/\hbar of the wave in Lecture 3.1

Turn down \hbar: one path returns

• Paths stay in step while \Delta S \lesssim \pi\hbar: a band of width \sim\sqrt{\hbar T/m}
• Small \hbar or large m: the band shrinks onto the classical path
• Classical mechanics is the \hbar \to 0 limit

Example: how wide is the band?

Estimate \sqrt{\hbar T/m} for a 0.1 kg ball in flight for 1 s, and for an electron crossing a molecule in 10^{-15} s.

\text{ball:}\quad \sqrt{\frac{(1.05\times10^{-34})(1)}{0.1}} \approx 3\times10^{-17}\ \text{m}

\text{electron:}\quad \sqrt{\frac{(1.05\times10^{-34})(10^{-15})}{9.1\times10^{-31}}} \approx 3\times10^{-10}\ \text{m}

The ball’s band is far smaller than a proton; the electron’s is the size of an atom.

• This is why electrons need quantum mechanics and baseballs do not

Superposition makes the blur

\psi(x) = \sum_k c_k\,e^{ikx}

• One wave: a definite p, found anywhere
• Many waves: localized, but p is spread
• Quantum states are superpositions, so they are blurry

A point becomes a blur

• Classical: a point (x, p) riding its energy curve
• Quantum: a blob of area \sim\hbar riding the same curve
• Its center follows Newton; its width is quantum

Takeaway

Newton, Hamilton and least action all describe one classical path: a point in phase space sliding along its energy contour. Quantum mechanics keeps every path, each with a phase e^{iS/\hbar}. Where S \gg \hbar only the paths near the classical one survive; where S \sim \hbar many do, and the state is a blurry superposition.