Ch 38: Quantization

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 38: Quantization

Problem 19a

Chapter 38, Problem 19a

What is the de Broglie wavelength of a 200 g baseball with a speed of 30 m/s?

Verified step by step guidance

Step 1: Recall the formula for the de Broglie wavelength, which is given by \( \lambda = \frac{h}{p} \), where \( \lambda \) is the wavelength, \( h \) is Planck's constant \( (6.626 \times 10^{-34} \, \text{J·s}) \), and \( p \) is the momentum of the object.

Step 2: Calculate the momentum \( p \) of the baseball using the formula \( p = m \cdot v \), where \( m \) is the mass of the baseball (200 g converted to kilograms, \( 0.2 \; \text{kg} \)) and \( v \) is its velocity (30 m/s).

Step 3: Substitute the values of \( m \) and \( v \) into the momentum formula to find \( p \). This gives \( p = 0.2 \cdot 30 \; \text{kg·m/s} \).

Step 4: Substitute the calculated momentum \( p \) into the de Broglie wavelength formula \( \lambda = \frac{h}{p} \). Use \( h = 6.626 \times 10^{-34} \; \text{J·s} \).

Step 5: Simplify the expression to find the de Broglie wavelength \( \lambda \). Note that the result will be extremely small due to the large mass and velocity of the baseball.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

de Broglie Wavelength

The de Broglie wavelength is a concept in quantum mechanics that describes the wave-like behavior of particles. It is given by the formula λ = h/p, where λ is the wavelength, h is Planck's constant (6.626 x 10^-34 Js), and p is the momentum of the particle. This concept illustrates that all matter exhibits both particle and wave characteristics, especially at the quantum scale.

Momentum

Momentum is a physical quantity defined as the product of an object's mass and its velocity, expressed as p = mv. In this context, the momentum of the baseball can be calculated using its mass (200 g or 0.2 kg) and its speed (30 m/s). Understanding momentum is crucial for calculating the de Broglie wavelength, as it directly influences the wave properties of the object.

Planck's Constant

Planck's constant is a fundamental constant in quantum mechanics, denoted as h, with a value of approximately 6.626 x 10^-34 Js. It relates the energy of a photon to its frequency and plays a critical role in the formulation of the de Broglie wavelength. This constant is essential for understanding the scale at which quantum effects become significant, particularly for small particles.

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