The double slit, told in full

The previous five sections have piled up two facts that pull against each other: light knocks out electrons one at a time, and electrons diffract like waves.

One experiment forces both into the same picture. We will go through it in four steps, each harder to explain than the last.

Step one: light is a wave (1801)

Thomas Young sent a beam of light through two parallel narrow slits and saw alternating bright and dark bands on a screen behind them.

Work it out: why you never see fringes in daily lifebasic~4 minexpand

Take $\lambda=550$ nm (green), $L=1$ m, $d=0.1$ mm:

$$\Delta y=\frac{5.5\times10^{-7}\times1}{10^{-4}}=5.5\times10^{-3}\ \text{m}=5.5\ \text{mm}\tag{1.6.3}$$

Perfectly visible. But with a slit separation of 1 mm, $\Delta y=0.55$ mm — just about visible. At 1 cm, $55\ \mu\text{m}$, beyond the eye’s resolution.

That is why two ordinary light sources (two lamps, say) never show interference: the separation is far too large and the fringes far too fine. On top of which ordinary sources are incoherent, their phases jumping randomly, so any fringes would be averaged away in time anyway.

Up to 1801 everything is comfortable: light is a wave, done.

Step two: send them one at a time (from 1909)

This is where the trouble starts.

Dim the source until at any moment there is on average less than one photon in the apparatus. Replace the screen with a detector that registers single photons.

• The first few dozen photons: scattered points on the screen with no discernible pattern.
• After a few hundred: some regions start accumulating points while others never receive any.
• After tens of thousands: interference fringes stand out clearly, in exactly the positions they occupy with bright light.

In 1989 Tonomura at Hitachi did the same experiment with electrons, filming the entire build-up one electron at a time. In 2002 a Physics World readers’ poll voted it the most beautiful experiment in the history of physics.

Try it yourself

Step three: what if we peek

The natural next question: put a detector at the slits and see which one it went through.

The result: as soon as you can determine the path, the fringes vanish, and the distribution degenerates into the simple sum of the two slits’ individual contributions. Switch the detector off and the fringes come straight back.

It also does not matter whether you ever read the record — as long as the information is available in principle, the fringes are gone.

Step four: delayed choice (proposed 1978, realised 2007)

Wheeler pushed this to its final step with an almost provocative question:

What if we wait until the photon has already passed the slits before deciding whether to measure which path or measure interference?

On the naive picture the photon must decide at the slits: either behave as a particle and pick one, or behave as a wave and take both. And we make our choice after it has gone through — must it foresee the future?

What to take away

∎Key formulas

• Double-slit fringe spacing

  $$\Delta y=\frac{\lambda L}{d}$$

  Wider slit separation, finer fringes; the first method of measuring wavelength

• Bright-fringe condition

  $$d\sin\theta=m\lambda,\qquad m=0,\pm1,\pm2,\dots$$

  Constructive where the path difference is a whole number of wavelengths

• Coherent two-slit intensity

  $$I=|\psi_1+\psi_2|^2=I_1+I_2+2\sqrt{I_1I_2}\cos\delta$$

  The third term is the interference term; the fringes live entirely in it

• With which-path information

  $$I=|\psi_1|^2+|\psi_2|^2$$

  The interference term is gone, leaving the plain sum of the two slits

?Self-check4 questions

1. 1.

   With the source dimmed so that only one photon is in the apparatus at a time, long accumulation on the screen produces:

   • Interference fringes, in exactly the positions seen with bright light
   • Two bright bands, the shadows of the two slits
   • A uniform scatter of points
   • Nothing at all

   Submit
2. 2.

   About "the fringes vanish when a which-path detector is installed", which statements are correct? (Select all that apply.)

   Select all that apply

   • The fringes vanish as long as the path information is available in principle, whether or not you read it
   • "The probe photon knocked the particle sideways" holds in some setups but is not the essential reason
   • Which-path schemes that transfer no momentum also destroy the fringes
   • It happens because the detector absorbs some photons and the intensity drops

   Submit
3. 3.

   Which class of explanation does Wheeler's delayed-choice experiment rule out?

   • That a particle decides at the slits whether it will "behave as a wave or a particle"
   • That photons have definite energies
   • That interference requires two or more particles
   • That measurement affects the system being measured

   Submit
4. 4.

   A double-slit experiment uses λ = 500 nm, slit separation d = 0.20 mm and screen distance L = 2.0 m. What is the spacing between adjacent bright fringes, in millimetres?

   mm10% relative tolerance

   Submit

What comes next

The double slit forces a conclusion: “where the electron is” has no definite answer before measurement.

So what does “uncertain” actually mean — that we do not know, or that there is nothing to know? The last section of this chapter traces where the concept came from, and what was wrong with the argument that originally sold it.