Question

After you told him about de Broglie's hypothesis that particles of momentum \(p\) have wave characteristics with wavelength \(\lambda=h / p\), your 60.0 -kg roommate starts thinking of his fate as a wave and asks you if he could be diffracted when passing through the 90.0 -cm-wide doorway of your dorm room. a) What is the maximum speed at which your roommate can pass through the doorway in order to be significantly diffracted? b) If it takes one step to pass through the doorstep, how long should it take your roommate to make that step (assume the length of his step is \(0.75 \mathrm{~m}\) ) in order for him to be diffracted? c) What is the answer to your roommate's question? Hint: Assume that significant diffraction occurs when the width of the diffraction aperture is less that 10.0 times the wavelength of the wave being diffracted.

Short answer

Answer: No, the student's roommate cannot be diffracted while passing through a doorway.

Step-by-step solution

Calculate the wavelength for de Broglie's hypothesis

To calculate the wavelength, first we need to find the momentum. Momentum (p) is defined as the product of mass (m) and velocity (v). The formula is p=mv. Given the mass of the roommate (m=60.0 kg), we can calculate the wavelength (λ) using de Broglie's hypothesis formula: λ=h/p.

Determine the maximum wavelength for significant diffraction

According to the hint, significant diffraction occurs when the width of the diffraction aperture (in this case, the width of the doorway which is 0.9 m) is less than 10.0 times the wavelength of the wave being diffracted. That is, 0.9 m < 10λ. From this, we can conclude that λ < 0.09 m = 90 cm.

Calculate the maximum velocity for significant diffraction

Using the maximum wavelength calculated in step 2 (λ= 90 cm), we can now solve for the maximum velocity (v). Using de Broglie's hypothesis equation again (λ=h/p) and the formula for momentum (p=mv), we can find the maximum velocity: v = h/(m*λ).

Calculate the time taken for one step to ensure diffraction

Suppose it takes one step to pass through the doorway, we are given that the length of the step is 0.75 m. Now to calculate the time taken, we use the formula: time = distance/velocity. Since we have the maximum velocity from step 3 and the step length of 0.75 m, we can find the time it takes for the student's roommate to make that step in order for him to be diffracted.

Answer the question of whether the roommate could be diffracted or not

After calculating the maximum velocity and time taken for the student's roommate to be diffracted in the previous steps, we can compare the results to practical possibilities (speed and time). If the results are achievable and practical, we can conclude that the roommate could be diffracted. Otherwise, we can conclude that the roommate cannot be diffracted.

Key concepts

Particle-Wave Duality

De Broglie's hypothesis is central to understanding particle-wave duality, a foundational concept in quantum mechanics. This hypothesis suggests that particles, like electrons or in our amusing scenario, your roommate, exhibit both particle-like and wave-like properties. This dual nature comes into play depending on the observational context. Traditionally, we consider objects like baseballs or cars as particles, but quantum mechanics tells us that on a smaller scale, particles also behave like waves, and their position isn't well defined until measured.

Think of it this way, while we don't notice the wave characteristics of a car due to their massive size in comparison to the tiny particles like electrons, theoretically, all matter has a wave side. That means even your roommate walking through a doorway could, on a very, very small scale and under the right conditions, exhibit wave-like behavior such as diffraction, which is the bending of waves around obstacles and the spreading out of waves past narrow openings.

Quantum Mechanics

Quantum mechanics is the branch of physics that deals with particles at the atomic and subatomic levels. It describes their behavior and interactions in ways that classical physics cannot. For instance, classical mechanics would suggest that particles have a definite position and velocity at any given time, but that's not the case in quantum mechanics. Here, particles are described by probabilities, encapsulated in a mathematical function called the wave function.

This wave function can tell us the probability of finding a particle in a particular position but not the precise location until it is observed. Quantum mechanics has strange and counter-intuitive implications, like the uncertainty principle, which states that you cannot know both the position and momentum of a particle with absolute certainty. The behavior of particles elucidated by quantum mechanics is crucial in devices like transistors and lasers, affecting our day-to-day technology.

Diffraction of Particles

In our context, diffraction refers to the phenomenon where waves, including those associated with particles according to de Broglie's hypothesis, spread out as they pass through a narrow opening or move around an obstacle. For significant diffraction to occur, the size of the opening or obstacle should be on the same order as the wavelength of the particle's wave.

For your roommate to experience any noticeable diffraction while passing through the doorway, their associated wavelength would need to be on the order of the doorway's width, which is implausible due to their large mass resulting in an extremely small wavelength. In the world of quantum mechanics, it's usually the subatomic particles that we observe being diffracted, such as electrons passing through a narrow slit in the famous double-slit experiment.

Momentum

Momentum, a fundamental quantity in physics, is the product of an object's mass and velocity. It's a vector quantity, which means it has both magnitude and direction. The higher an object's mass or velocity, the greater its momentum. In quantum mechanics, momentum plays a key role because de Broglie's hypothesis links it to the wave-like nature of particles.

In the exercise, we use the momentum (mass times velocity) of your roommate to determine his associated de Broglie wavelength. However, since he has such a large mass, the velocity required for significant diffraction (as computed in the solution) is exceptionally low, much lower than the average walking speed of a human. It illustrates how in the macroscopic world we inhabit, quantum effects like wave-particle duality rarely manifest in a way that's directly observable in our day-to-day experiences.