Chapter 39: Problem 75
If your wavelength were 1.0 m, you would undergo considerable diffraction in moving through a doorway. (a) What must your speed be for you to have this wavelength? (Assume that your mass is 60.0 kg.) (b) At the speed calculated in part (a), how many years would it take you to move 0.80 m (one step)? Will you notice diffraction effects as you walk through doorways?
Short Answer
Step by step solution
Identify the Problem and Given Values
Rearrange the de Broglie Equation
Calculate the Speed
Calculate the Time to Move 0.80 m
Convert Time to Years
Conclusion on Diffraction Effects
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Diffraction
In classical mechanics, we don't frequently encounter diffraction effects for large objects because their de Broglie wavelengths are extremely small compared to visible dimensions, such as doorways. But in quantum mechanics, even particles like electrons display diffraction patterns through slits comparable in size to their wavelength.
Understanding diffraction helps to bridge the gap between classical and quantum mechanics, allowing us to predict when wave-like behavior will be noticeable.
Planck's Constant
The value of Planck's constant is approximately \( 6.626 \times 10^{-34} \, \text{m}^2 \text{ kg/s} \), a very small number because it applies to the quantum realm, where the effects are not readily visible at macroscopic scales. Large objects, like a person weighing 60 kg, have a negligible de Broglie wavelength due to the small value of \( h \), making wave-like properties imperceptible in everyday life.
Velocity Calculation
In the provided exercise, with \( \lambda \) as 1.0 m and mass \( m \) as 60.0 kg, we can substitute into the formula:
- Planck’s constant \( h = 6.626 \times 10^{-34} \, \text{m}^2 \text{ kg/s} \)
- Mass \( m = 60.0 \text{ kg} \)
- Wavelength \( \lambda = 1.0 \text{ m} \)
Quantum Mechanics
In quantum mechanics, the de Broglie hypothesis is fundamental. It suggests that all matter possesses a wave component, as revealed by the equation \( \lambda = \frac{h}{mv} \). Although this wave aspect is negligible for large, everyday objects, it becomes significant when dealing with very small particles.
Understanding quantum mechanics not only helps explain the behavior of particles at microscopic scales but also forms the basis for modern technologies such as semiconductors, lasers, and even contributes to the emerging field of quantum computing.