39.6 The Heisenberg Uncertainty Principle

If matter has wave-like aspects and a de Broglie wavelength, then the expression $Δ f Δ t \geq 1$ must somehow apply to matter. How? And what are the implications?

Consider a particle with velocity $v_{x}$ as it travels along the $x$ -axis with de Broglie wavelength $λ = h / p_{x}$ . Figure 39.11 showed a history graph $(ψ versus t)$ of a wave packet that might represent the particle as it passes a point on the $x$ -axis. It will be more useful to have a snapshot graph $(ψ versus x)$ of the wave packet traveling along the $x$ -axis.

The time interval $Δ t$ is the duration of the wave packet as the particle passes a point in space. During this interval, the packet moves forward

Δ x = v_{x} Δ t = \frac{p_{x}}{m} Δ t

(39.24)

where $p_{x} = m v_{x}$ is the $x$ -component of the particle’s momentum. The quantity $Δ x$ , shown in FIGURE 39.16, is the length or spatial extent of the wave packet. Conversely, we can write the wave packet’s duration in terms of its length as

Δ t = \frac{m}{p_{x}} Δ x

(39.25)

The figure illustrates a snapshot graph of a wave packet.

FIGURE 39.16 Full Alternative Text

You will recall that any wave with sinusoidal oscillations must satisfy the wave condition $λ f = v$ . For a material particle, where $λ$ is the de Broglie wavelength, the frequency $f$ is

f = \frac{v}{λ} = \frac{p_{x} / m}{h / p_{x}} = \frac{p_{x}^{2}}{h m}

If the momentum $p_{x}$ should vary by the small amount $Δ p_{x}$ , the frequency will vary by the small amount $Δ f$ . Assuming that $Δ f ≪ f$ and $Δ p_{x} ≪ p_{x}$ (reasonable assumptions), we can treat $Δ f$ and $Δ p_{x}$ as if they were differentials $d f$ and $d p_{x}$ . Taking the derivative, we find

Δ f = \frac{2 p_{x} Δ p_{x}}{h m}

(39.26)

Multiplying together these expressions for $Δ t$ and $Δ f$ , we find that

Δ f Δ t = \frac{2 p_{x} Δ p_{x}}{h m} \frac{m Δ x}{p_{x}} = \frac{2}{h} Δ x Δ p_{x}

(39.27)

Because $Δ f Δ t \geq 1$ for any wave, one last rearrangement of Equation 39.27 shows that a matter wave must obey the condition

Δ x Δ p_{x} \geq \frac{h}{2} (Heisenberg uncertainty principle)

(39.28)

This statement about the relationship between the position and momentum of a particle was proposed by Werner Heisenberg, creator of one of the first successful quantum theories. Physicists often just call it the uncertainty principle.

NOTE In statements of the uncertainty principle, the right side is sometimes $h / 2$ , as we have it, but other times it is just $h$ or contains various factors of $π$ . The specific number is not especially important because it depends on exactly how $Δ x$ and $Δ p$ are defined. The important idea is that the product of $Δ x$ and $Δ p_{x}$ for a particle cannot be significantly less than Planck’s constant h. A similar relationship for $Δ y Δ p_{y}$ applies along the $y$ -axis.