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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

Expert verified
(a) Speed: \( 1.104 \times 10^{-35} \text{ m/s} \). (b) Time: \( 2.30 \times 10^{27} \) years. No diffraction effects noticed.

Step by step solution

01

Identify the Problem and Given Values

We need to find the speed at which you have a wavelength of 1.0 m. The mass provided is 60.0 kg. We will use the de Broglie wavelength formula, which is given by \( \lambda = \frac{h}{mv} \), where \( h \) is Planck's constant \( 6.626 \times 10^{-34} \text{ m}^2 \text{ kg}/\text{s} \), \( \lambda \) is the wavelength (1.0 m), \( m \) is the mass (60.0 kg), and \( v \) is the speed which needs to be found.
02

Rearrange the de Broglie Equation

Rearrange the de Broglie wavelength formula to solve for velocity \( v \): \[ v = \frac{h}{m\lambda} \]Substitute the values: \( h = 6.626 \times 10^{-34} \text{ m}^2 \text{ kg}/\text{s} \), \( m = 60.0 \text{ kg} \), \( \lambda = 1.0 \text{ m} \).
03

Calculate the Speed

Substitute the values into the equation:\[ v = \frac{6.626 \times 10^{-34}}{60.0 \times 1.0} \]\[ v = 1.104 \times 10^{-35} \text{ m/s} \]This is the speed at which you would have a wavelength of 1.0 m.
04

Calculate the Time to Move 0.80 m

To find the time \( t \) it would take to move 0.80 m at this speed, use the equation \( t = \frac{d}{v} \):\[ t = \frac{0.80}{1.104 \times 10^{-35}} \]\[ t = 7.25 \times 10^{34} \text{ s} \].
05

Convert Time to Years

Convert the time from seconds to years using the conversion factor: 1 year = 31,536,000 seconds.\[ t = \frac{7.25 \times 10^{34}}{31,536,000} \]\[ t \approx 2.30 \times 10^{27} \text{ years} \].
06

Conclusion on Diffraction Effects

It would take approximately \( 2.30 \times 10^{27} \) years to travel 0.80 m at this speed, which is impractically long. Therefore, you would not notice diffraction effects at your regular walking speed; the speed calculated is not attainable.

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Key Concepts

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

Diffraction
Diffraction is a fascinating phenomenon where waves spread out as they pass through a small opening or around the edges of an obstacle. It can be observed in all types of waves, including sound, light, and even particles in the quantum realm. In the context of the de Broglie wavelength, diffraction becomes significant when the wavelength of a moving object is comparable to or larger than the size of an opening. This principle is why waves, having a wavelength of about 1.0 m, experience notable diffraction when passing through a doorway.

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
Planck's constant, denoted as \( h \), is a fundamental constant in physics, pivotal in quantum mechanics. It relates the energy of a photon to its frequency and is crucial when discussing the wave-particle duality of matter. Specifically, in de Broglie's equation, \( h \) links a particle's momentum to its wavelength, making it possible to calculate a particle's wave properties.

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
Finding the velocity at which a large object like a human would exhibit a significant de Broglie wavelength involves using the de Broglie wavelength formula: \( \lambda = \frac{h}{mv} \). For a given wavelength and known mass, this equation can be rearranged to find velocity: \( v = \frac{h}{m\lambda} \).

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 \( \lambda = 1.0 \text{ m} \)
This results in a calculated velocity of \( 1.104 \times 10^{-35} \text{ m/s} \), an exceptionally small and slow speed, highlighting the impracticality of such a scenario in the real world.
Quantum Mechanics
Quantum mechanics is the branch of physics that deals with phenomena at the smallest scales, such as molecules, atoms, and subatomic particles. It introduces concepts that are counter-intuitive to classical physics, such as wave-particle duality, where particles like electrons exhibit both wave- and particle-like properties.

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.

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Most popular questions from this chapter

(a) A nonrelativistic free particle with mass \(m\) has kinetic energy \(K\). Derive an expression for the de Broglie wavelength of the particle in terms of \(m\) and \(K\). (b) What is the de Broglie wavelength of an 800-eV electron?

A pesky 1.5-mg mosquito is annoying you as you attempt to study physics in your room, which is 5.0 m wide and 2.5 m high. You decide to swat the bothersome insect as it flies toward you, but you need to estimate its speed to make a successful hit. (a) What is the maximum uncertainty in the horizontal position of the mosquito? (b) What limit does the Heisenberg uncertainty principle place on your ability to know the horizontal velocity of this mosquito? Is this limitation a serious impediment to your attempt to swat it?

(a) An electron moves with a speed of 4.70 \(\times\) 10\(^6\) m/s. What is its de Broglie wavelength? (b) A proton moves with the same speed. Determine its de Broglie wavelength.

To investigate the structure of extremely small objects, such as viruses, the wavelength of the probing wave should be about one-tenth the size of the object for sharp images. But as the wavelength gets shorter, the energy of a photon of light gets greater and could damage or destroy the object being studied. One alternative is to use electron matter waves instead of light. Viruses vary considerably in size, but 50 nm is not unusual. Suppose you want to study such a virus, using a wave of wavelength 5.00 nm. (a) If you use light of this wavelength, what would be the energy (in eV) of a single photon? (b) If you use an electron of this wavelength, what would be its kinetic energy (in eV)? Is it now clear why matter waves (such as in the electron microscope) are often preferable to electromagnetic waves for studying microscopic objects?

The negative muon has a charge equal to that of an electron but a mass that is 207 times as great. Consider a hydrogenlike atom consisting of a proton and a muon. (a) What is the reduced mass of the atom? (b) What is the ground-level energy (in electron volts)? (c) What is the wavelength of the radiation emitted in the transition from the \(n\) = 2 level to the \(n\) = 1 level?

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