Problem 46 - Essential Calculus Early Transcendentals

Question

In the theory of relativity, the energy of a particle is $$E=\sqrt{m_{0}^{2} c^{4}+h^{2} c^{2} / \lambda^{2}}$$ where \(m_{0}\) is the rest mass of the particle, \(\lambda\) is its wave length, and \(h\) is Planck's constant. Sketch the graph of \(E\) as a function of \(\lambda .\) What does the graph say about the energy?

Step-by-Step Solution

Answer

The graph shows energy asymptotically approaches rest mass energy at large \( \lambda \) and increases sharply at small \( \lambda \).

Step 1: Understand the relation

The given formula is \( E = \sqrt{m_0^2 c^4 + \frac{h^2 c^2}{\lambda^2}} \). This is a function of wavelength \( \lambda \), which depicts how energy \( E \) changes with respect to changes in wavelength \( \lambda \).

Step 2: Identify Asymptotic Behavior

As \( \lambda \rightarrow \infty \), the term \( \frac{h^2 c^2}{\lambda^2} \rightarrow 0 \). Thus, \( E \rightarrow \sqrt{m_0^2 c^4} = m_0 c^2 \), indicating energy approaches a constant value equal to the rest mass energy.

Step 3: Analyze for Small \(\lambda\)

As \( \lambda \rightarrow 0 \), the term \( \frac{h^2 c^2}{\lambda^2} \) grows large. Therefore, \( E \rightarrow \frac{hc}{\lambda} \), showing that energy increases sharply as wavelength decreases.

Step 4: Sketch the Graph

Start the curve from the asymptotic limit at \( m_0 c^2 \) for large \( \lambda \) and let the curve sharply increase as \( \lambda \) decreases. This shows a steep upward trend as energy increases without bound as \( \lambda \to 0 \).

Step 5: Interpret the Graph

The graph indicates that for very large wavelengths, energy approaches a constant equal to the rest mass energy, \( m_0 c^2 \). However, as the wavelength decreases, energy significantly increases, indicating a high-energy regime as wavelength approaches zero.

Key Concepts

Wavelength

Wavelength is a fundamental concept in the study of wave phenomena, including light and other types of radiation. It is defined as the distance between consecutive peaks of a wave. Represented by the symbol \( \lambda \), it is crucial in understanding different properties of waves, such as energy and frequency.
Here are few key things to remember about wavelength:

• It determines the type of wave. For instance, longer wavelengths can correspond to radio waves, while shorter ones relate to gamma rays.
• Inversely related to frequency, which means shorter wavelengths have higher frequencies and vice versa.
• Plays an important role in energy, as described by the equation \( E = \sqrt{m_{0}^{2} c^{4}+\frac{h^{2} c^{2}}{\lambda^{2}}} \), linking it directly to energy.

Understanding wavelength is important because it helps in explaining how particles and waves interact with one another across different fields of physics.

Energy as a function of wavelength

Energy, in the context of relativity, can be intricately affected by the wavelength of a particle. The given equation \( E = \sqrt{m_{0}^{2} c^{4}+\frac{h^{2} c^{2}}{\lambda^{2}}} \) provides an insight into how energy \( E \) changes concerning the wavelength \( \lambda \).
Let's break it down:

• The squaring operation under the square root ensures that energy is always a positive value, reflecting physical realities.
• As wavelength \( \lambda \) changes, it affects the second term \( \frac{h^{2} c^{2}}{\lambda^{2}} \) in the equation, which directly affects the value of energy \( E \).
• This equation shows that energy has a complex, inverse-square relationship with wavelength. As the denominator (wavelength squared) decreases, the overall energy contribution from this term increases.
• This relationship is essential in understanding phenomena in both macroscopic and quantum scales, indicating how energy propagates in waves.

The concept also illustrates how energy and wavelength are deeply intertwined, forming the foundational basis for understanding aspects such as photon energy in electromagnetic radiation.

Asymptotic behavior of energy

Asymptotic behavior refers to how functions behave as they approach certain limits, either approaching infinity or zero. In the given equation for energy, the asymptotic behavior is crucial in explaining the limits of energy as a function of wavelength \( \lambda \).
When considering large values of \( \lambda \) (wavelength approaching infinity):

• The term \( \frac{h^{2} c^{2}}{\lambda^{2}} \) trends towards zero, simplifying the energy equation to \( E \approx m_{0}c^{2} \), which is the rest energy of a particle. This indicates that at large wavelengths, energy becomes primarily defined by its rest mass.

When dealing with small \( \lambda \) (wavelength approaching zero):

• The term \( \frac{h^{2} c^{2}}{\lambda^{2}} \) grows without bound, indicating that energy \( E \) increases significantly. This suggests an intense rise in energy, corresponding to scenarios such as particles moving close to light speed or demonstrating high-frequency characteristics.

Understanding these asymptotic behaviors is vital as they help plot how energy behaves at the extremities of the wavelength, providing a more comprehensive understanding of energy variation and the underlying physics.