5.17: Density Operator Approach to the Double-Slit Experiment

A sharply focused particle beam (photons, electrons, molecules, etc.) is incident on a screen with two slits. According to quantum mechanics the individual particles are represented by a coherent superposition of being simultaneously at both slits.

\[ | \Psi \rangle = \frac{1}{ \sqrt{2}} \left[ |1 \rangle + |2 \rangle \right] \nonumber \]

In the interest of mathematical simplicity, 1 and 2 label slits that are infinitesimally narrow in the x-direction and infinitely long in the y-direction. The density operator for this state is

\[ \hat{ \rho} = | \Psi \rangle \langle \Psi | = \frac{1}{2} \left[ |1 \rangle + |2 \rangle \right] \left[ \langle 1 | + \langle 2 | \right] = \frac{1}{2} \left[ |1 \rangle \langle 1 | + |1 \rangle \langle 2 | + |2 \rangle \langle 1 | + |2 \rangle \langle 2 | \right] \nonumber \]

The expectation value for the arrival of a particle at position x on the detection screen is

\[ \langle x | \hat{ \rho} | x \rangle = \frac{1}{2} \left[ \langle x |1 \rangle \langle 1 | x \rangle + \langle x | 1 \rangle \langle 2 | x \rangle + \langle x | 2 \rangle \langle 1 | x \rangle + \langle x | 2 \rangle \langle 2 | x \rangle \right] \nonumber \]

Rearrangement yields,

\[ \langle x | \hat{ \rho} | x \rangle = \frac{1}{2} \left[ | \langle x | 1 \rangle |^2 + | \rangle x |2 \rangle |^2 + \langle x | 1 \rangle \langle x | 2 \rangle* + \langle x | 2 \rangle \langle x | 1 \rangle* \right] \nonumber \]

The probability amplitudes in this equation represent the phase of a particle on arrival at position x from slits 1 and 2. For example, using Euler’s equation we calculate the phase of a particle arriving at x from slit 1 as follows,

\[ \langle x | 1 \rangle = \frac{1}{ \sqrt{2 \pi r}} exp \left( i 2 \pi \frac{ \delta x_1}{ \lambda} \right) \nonumber \]

where δx₁ is the distance from slit 1 to position x on the detection screen and λ is the de Broglie wavelength of the particle.

Using this form for the probability amplitudes we can write the expectation value in terms of the distances to x from slits 1 and 2.

\[ \langle x | \hat{ \rho} | x \rangle = \frac{1}{4 \pi} \left[ 2 + exp \left( i 2 \pi \frac{( \delta x_1 - \delta x_2)}{ \lambda} \right) + exp \left( -i2 \pi \frac{( \delta x_1 - \delta x_2)}{ \lambda} \right) \right] = \frac{1}{2 \pi} \left[ 1 + \cos \left( \frac{( \delta x_1 - \delta x_2)}{ \lambda} \right) \right] \nonumber \]

Clearly \( \frac{( \delta x_1 - \delta x_2)}{ \lambda}\) will vary continuously along the x-axis of the detector from large negative values at one end to large positive values at the other end leading to minima and maxima in the cosine term and therefore \( \langle x | \hat{ \rho} | x \rangle\), thereby yielding the well-known interference fringes associated with the double-slit experiment. Naturally a more realistic slit geometry will lead to a mathematically more complicated expression for the expectation value.

If one takes a classical view of the double-slit experiment that assumes the particle goes through one slit or the other, and has a 50% chance of going through either slit, the coherent superposition,

\[ \hat{ \rho} = \frac{1}{2} |1 \rangle \langle 1 | + \frac{1}{2} |2 \rangle \langle 2 | \nonumber \]

The expectation value for the arrival of the particle at x on the detection screen is now,

\[ \langle x | \hat{ \rho}_{cl} | x \rangle = \frac{1}{2} \langle x | 1 \rangle \langle 1 | x \rangle + \frac{1}{2} \langle x | 2 \rangle \langle 2 | x \rangle = \frac{1}{2} | \langle x | 1 \rangle |^2 + \frac{1}{2} | \langle x | 2 \rangle |^2 = \text{ constant} \nonumber \]

which has a constant value with no oscillations in arrival probability as a function of x. In other words, no interference fringes.