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Wave Functions and Uncertainty

A Wave Analysis of the Double-Slit Experiment

Two waves traveling from the slits have displacements where a is the amplitude of each wave, k is the wave number, and r 1 and r 2 are the distances from the two slits. The net displacement is The function A ( x ) is called the amplitude function. The intensity is proportional to A 2 :

Connecting the Wave and Photon Views

• (^) The intensity of the light wave is correlated with the probability of detecting photons. That is, photons are more likely to be detected at those points where the wave intensity is high and less likely to be detected at those points where the wave intensity is low.
• (^) Imagine a narrow strip of width δx located at x with area A=H δx. The energy falling on this strip in each second is E(in δx at x) = I(x) H δx
• (^) From Photon perspective, E(in δx at x)=N (in δx at x) hf
• (^) The probability that a photon lands in the narrow strip is
• (^) The probability of detecting a photon at a particular point is directly proportional to the square of the light-wave amplitude function at that point:     I ^ x ^ x hfN

H

N

N in xat x Pr ob in xat x tot tot

Probability Density

• (^) The same way we define mass density, We can define the probability density P ( x ) such that
• (^) In one dimension, probability density has SI units of m–1. Thus the probability density multiplied by a length yields a dimensionless probability.
• (^) P ( x ) itself is not a probability. You must multiply the probability density by a length to find an actual probability.
• (^) The photon probability density is directly proportional to the square of the light-wave amplitude: - (^) P(x) is independent of δx

Normalization

• (^) A photon or electron has to land somewhere on the detector after passing through an experimental apparatus.
• (^) Consequently, the probability that it will be detected at some position is 100%.
• (^) The statement that the photon or electron has to land somewhere on the x -axis is expressed mathematically as
• (^) Any wave function must satisfy this normalization condition.

Examples

• (^) Consider the electron wave function
  • (^) Determine the normalization constant c.
  • (^) Draw graph of |ψ(x)|^2 over the interval -2 cm ≤ x ≤ 2 cm.
  • (^) If 104 electrons are detected, how many will be in the interval 0.00 cm ≤ x ≤ 0.50 cm?   

0 x 1 m c 1 x x 1 cm x 2

The Heisenberg Uncertainty Principle

• (^) If matter has wave-like aspects and a de Broglie wavelength, then the expression ΔfΔt ≥ 1 must somehow apply to matter.
• Consider a particle with velocity vx as it travels along x-axis with de Broglie wavelength λ=h/px.
• (^) The quantity Δ x is the length or spatial extent of a wave packet.
• (^) Δpx is a small range of momenta corresponding to the small range of frequencies within the wave packet.
• (^) Any matter wave must obey the condition
• (^) This is a statement about the relationship between the position and momentum of a particle.

The Heisenberg Uncertainty Principle

• (^) If we want to know where a particle is located, we

measure its position x with uncertainty Δ x.

• (^) If we want to know how fast the particle is going, we need

to measure its velocity v

x

or, equivalently, its momentum

p

x

. This measurement also has some uncertainty Δ p

x

• You cannot measure both x and p x

simultaneously with

arbitrarily good precision.

• (^) Any measurements you make are limited by the condition

that Δ x Δ p

x

≥ h /.

• (^) Our knowledge about a particle is inherently

uncertain.