91Ó°ÊÓ

High-speed electrons are used to probe the interior structure of the atomic nucleus. For such electrons the expression \(\lambda=h / p\) still holds, but we must use the relativistic expression for momentum, \(p=m v / \sqrt{1-v^{2} / c^{2}}\) (a) Show that the speed of an electron that has de Broglie wavelength \(\lambda\) is $$v=\frac{c}{\sqrt{1+(m c \lambda / h)^{2}}}$$ (b) The quantity \(h / m c\) equals \(2.426 \times 10^{-12} \mathrm{m}\) . (As we saw in Section \(38.3,\) this same quantity appears in Eq. \((38.7),\) the expression for Compton sattering of photons by electrons.) If \(\lambda\) is small compared to \(h / m c,\) the denominator in the expression found in part (a) is close to unity and the speed \(v\) is very close to \(c .\) In this case it is convenient to write \(v=(1-\Delta) c\) and express the speed of the electron in terms of \(\Delta\) rather than \(v .\) Find an expression for \(\Delta\) valid when \(\lambda \ll h / m c\) [Hint: Use the binomial expansion \((1+z)^{n}=1+n z+n(n-1) z^{2} / 2+\cdots\) valid for the case \(|z|<1.1(\mathrm{c})\) How fast must an electron move for its de Broglie wavelength to be \(1.00 \times 10^{-15} \mathrm{m},\) comparable to the size of a proton? Express your answer in the form \(v=(1-\Delta) c,\) and state the value of \(\Delta .\)

Step by step solution

01

Understanding the de Broglie wavelength equation

We know that the de Broglie wavelength \( \lambda \) is given by \( \lambda = \frac{h}{p} \), where \( h \) is Planck's constant and \( p \) is the momentum.

02

Expression for relativistic momentum

The relativistic expression for momentum is \( p = \frac{m v}{\sqrt{1 - v^2/c^2}} \), where \( m \) is mass, \( v \) is velocity, and \( c \) is the speed of light.

03

Relating equations to find velocity

Substitute the expression for momentum \( p \) in the de Broglie wavelength equation: \( \lambda = \frac{h}{\frac{m v}{\sqrt{1 - v^2/c^2}}} \). This simplifies to \( \lambda = \frac{h \sqrt{1 - v^2/c^2}}{m v} \).

04

Solving for velocity

Rearrange the equation to find \( v \): \( \lambda m v = h \sqrt{1 - v^2/c^2} \). Square both sides and solve for \( v \) to get \[ v = \frac{c}{\sqrt{1 + \left(\frac{mc\lambda}{h}\right)^2}} \].

05

Approximation for small \( \lambda \)

When \( \lambda \ll h/mc \), set \( z = \left(\frac{mc\lambda}{h}\right)^2 \) in the expression for \( v \). Use the binomial expansion for \( \sqrt{1 + z} \) to approximate \( v \approx c (1 - \frac{1}{2}z) \), leading to \( v = (1 - \Delta)c \) with \( \Delta = \frac{(mc\lambda)^2}{2h^2} \).

06

Calculating specific case for \( \lambda = 1.00 \times 10^{-15} \mathrm{m} \)

Use the approximation to calculate \( \Delta \). Given \( h/mc = 2.426 \times 10^{-12} \mathrm{m} \), then \[ \Delta = \frac{1}{2} \left(\frac{\lambda mc}{h}\right)^2 = \frac{1}{2} \left(\frac{1.00 \times 10^{-15} \times mc}{h}\right)^2 \]. Solving gives \( \Delta \approx 2.12 \times 10^{-7} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

de Broglie Wavelength

The de Broglie wavelength is a fundamental concept in quantum mechanics, linking wave-like properties and momentum. According to de Broglie's hypothesis, every particle has a wavelength, given by \( \lambda = \frac{h}{p} \), where:

• \( \lambda \): de Broglie wavelength
• \( h \): Planck's constant
• \( p \): momentum of the particle

This relationship means that particles exhibit both wave-like and particle-like behavior, a duality central to quantum theory.
This principle helps in understanding how electrons act at atomic and subatomic levels. For electrons, this concept is particularly relevant as their wavelengths can be comparable to atomic scales, revealing insights into structures like atomic nuclei.

Binomial Expansion

Binomial expansion is a powerful mathematical method used to approximate expressions under certain conditions. In the context of relativistic momentum and de Broglie wavelength, it helps simplify calculations when variables are small. The expansion formula is:\[ (1+z)^{n}=1+n z+\frac{n(n-1) z^{2}}{2}+ \cdots \]This formula is valid when \(|z| \) is less than 1. For small values \( \text{like} \; z \approx 0 \), using just the first two terms often suffices, allowing easy calculations in problems involving tiny quantities.
In exercises like electron de Broglie wavelength calculations, approximations simplify into straightforward expressions, enhancing understanding of electron speed relative to light.

Speed of Light

The speed of light, denoted by \( c \), is a constant vital in physics, defined as approximately 299,792,458 meters per second. It not only set limits on how fast objects can travel in the universe but also features prominently in relativity and quantum mechanics.
In relativistic momentum equations such as \( v=\frac{c}{\sqrt{1+(mc\lambda/h)^2}} \), \( c \) benchmarks the speed of particles like electrons. Movies and myths often romanticize light speed, but scientifically, it frames how we understand electromagnetic waves and fundamental particle behavior.

Planck's Constant

Planck's constant \( h \) is a pivotal figure in quantum mechanics, introduced by Max Planck. A miniscule number, about \( 6.626 \times 10^{-34} \; \text{Js} \), it represents the elemental quantum of action, interlinking energy and frequency of photons via \( E = hf \).
This constant impacts de Broglie's equation \( \lambda = \frac{h}{p} \), indicating how electrons' wave properties describe quantum behavior. In physics, understanding \( h \) enables comprehension of how energy quantization leads to discrete systems, from atomic to cosmic scales, anchoring foundational theories like quantum entanglement.