Question

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

(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

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.

Rearrange the de Broglie Equation

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

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.

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 $t=\frac{d}{v}$:$$t=\frac{0.80}{1.104\times {10}^{-35}}$$$$t=7.25\times {10}^{34}\text{ s}$$.

Convert Time to Years

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

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.

Key concepts

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.