Question

A bedrock topic in quantum mechanics is the uncertainty principle. It is discussed mostly for massive objects in Chapter \(4,\) but the idea also applies to light: Increasing certainty in knowledge of photon position implies increasing uncertainty in knowledge of its momentum, and vice versa. A single-slit pattern that is developed (like the double-slit pattern of Section 3.6 ) one photon at a time provides a good example. Depicted in the accompanying figure, the pattern shows that photons emerging from a narrow slit are spread allover; a photon's \(x\) -component of momentum can be any value over a broad range and is thus uncertain. On the other hand, the \(x\) -coordinate of position of an emerging photon covers a fairly small range, for \(w\) is small. Using the single-slit diffraction formula \(n \lambda=w \sin \theta,\) show that the range of likely values of \(p_{x}\), which is roughly \(p \sin \theta\), is inversely proportional to the range \(w\) of likely position values. Thus, an inherent wave nature implies that the precisions with which the particle properties of position and momentum can be known are inversely proportional.

Short answer

In Conclusion, the Heisenberg's uncertainty principle states that the position and momentum of a photon (or any other particle) cannot both be precisely determined at the same instant. It suggests an inherent inverse relationship - the more precisely one property is known, the less precisely the other can be known. In this case, the range of likely values of the \(x\)-component of momentum (\(p \sin \theta\)) is found to be inversely proportional to the range of likely position values (\(w\)), which is shown using the single-slit diffraction formula.

Step-by-step solution

Analyze the Single-Slit Diffraction Formula

With respect to single-slit diffraction, the formula \(n \lambda=w \sin \theta\) is relevant. Here, \(n\) is the order of the bright fringe, \(\lambda\) is the wavelength of light, \(w\) is the width of the slit, and \(\theta\) is the diffraction angle.

Interpret the Photon's Momentum

The photon's momentum (\(p\)) in the \(x\) direction post diffraction can be expressed as \(p_x = p \sin \theta\). This suggests a direct link between the photon's diffraction angle and its momentum.

Apply the General Perception of Uncertainty

The photon's \(x\) -component of momentum can be any value over a broad range and is thus uncertain. Meanwhile, the \(x\) -coordinate of position of an emerging photon covers a fairly small range when \(w\) is small. Hence, there is an inherent inverse relationship: the more certain the position, the less certain the momentum and vice versa.

Relate Momentum and Position

According to the single-slit diffraction formula, the diffraction angle is directly related to the width \(w\) of the slit. Also, the \(x\)-component of the photon's momentum post diffraction is directly proportional to \(\sin \theta\). When the slit width \(w\) becomes smaller, indicating higher precision in position, the corresponding range of diffraction angle increases, leading to a larger range of possible momentum values. Hence, the precision in momentum decreases. Thus, the range of likely values of \(px\), which is roughly \(p \sin \theta\), is inversely proportional to the range \(w\) of likely position values.

Key concepts

Photon Momentum

Momentum, a fundamental concept in physics, is not just applicable to massive objects but also to light. Interestingly, photons, the particles of light, have momentum even though they lack mass. A photon's momentum is linked to its wave-like nature and is directly related to its frequency.

In the context of single-slit diffraction, when photons pass through a slit, they emerge spread out rather than in a straight line. The photon's momentum in the horizontal direction, denoted as the x-component ( p_x), can be expressed as: p_x = p \sin \theta. Here, \(\theta\) is the angle of diffraction, showing the relationship between how light bends when it passes through a narrow slit and its momentum.

This direct connection between the photon's diffraction angle and momentum reinforces the wave-particle duality of light, emphasizing that light behaves as both a wave and a particle concurrently. As we explore more, this duality will further demonstrate the inherent uncertainties between determining a photon's exact position and momentum.

Single-Slit Diffraction

Single-slit diffraction is an intriguing phenomenon that showcases the wave nature of photons. When a beam of light passed through a narrow slit, it doesn't just move straight on, but instead, it spreads out. This spreading can be explained through the formula: n \lambda = w \sin \theta.

Breaking this down:

• \(n\) is the order of the fringe, marking the intensity position on the display surface.
• \(\lambda\) is the wavelength of the light.
• \(w\) represents the width of the slit.
• \(\theta\) is the diffraction angle.

This formula shows the vital relationship between the slit’s width and the furthest angle that the light spreads out. The narrower the slit (w gets smaller), the more the light spreads, showing increased diffraction. This behavior illustrates critical wave properties, making it clear how the wave nature of light becomes evident in such setups.

Position and Momentum Relationship

The relationship between a photon's position and momentum is a core concept in quantum mechanics and closely ties into the Heisenberg Uncertainty Principle. The principle suggests that one cannot measure both the momentum and position of a photon with absolute precision at the same time.

In the context of the single-slit diffraction:

• The photon's position, particularly its x-coordinate, is fairly definite when it passes through a narrow slit, marked by the width \(w\).
• At the same time, its momentum in the x-direction, \(p_x\), becomes more uncertain, as it can take on a range of values depending on \(\sin \theta\).

As a result, the smaller the width \(w\) indicating higher position precision, the greater the range of possible momentum values \(p_x = p \sin \theta\). This showcases an inverse relationship: increasing precision in one aspect leads to increasing uncertainty in another.

Understanding this relationship in single-slit diffraction is more than just a theoretical exercise. It illustrates a fundamental principle of nature: that greater certainty in one property naturally leads to increased uncertainty in its complementary property. This level of interaction between position and momentum is a hallmark of quantum mechanics, providing profound insights into the limits of measurement.