Jack the cat has a mass of 6.35kg, and runs with a speed of 13. What is his De Broglie wavelength? (Assume a classical momentum.)
The correct answer and explanation is:
To find Jack the cat’s De Broglie wavelength, we use the De Broglie equation: λ=hp\lambda = \frac{h}{p}
Where:
- λ\lambda is the De Broglie wavelength (in meters),
- hh is Planck’s constant (6.626×10−34 J\cdotps)(6.626 \times 10^{-34} \, \text{J·s}),
- pp is the momentum, calculated as p=mvp = mv,
- m=6.35 kgm = 6.35 \, \text{kg} (Jack’s mass),
- v=13 m/sv = 13 \, \text{m/s} (Jack’s speed).
Step 1: Calculate the momentum
p=mv=6.35×13=82.55 kg\cdotpm/sp = mv = 6.35 \times 13 = 82.55 \, \text{kg·m/s}
Step 2: Calculate the wavelength
λ=6.626×10−3482.55≈8.03×10−36 meters\lambda = \frac{6.626 \times 10^{-34}}{82.55} \approx 8.03 \times 10^{-36} \, \text{meters}
✅ Final Answer:
λ≈8.03×10−36 m\boxed{\lambda \approx 8.03 \times 10^{-36} \, \text{m}}
✍️ Explanation (300 words):
The De Broglie wavelength is a concept from quantum mechanics that describes the wave-like nature of particles. According to Louis de Broglie’s hypothesis, any object with momentum has an associated wavelength, given by λ=hp\lambda = \frac{h}{p}, where hh is Planck’s constant, and pp is the momentum of the object.
In this case, Jack the cat is treated like a classical particle with mass 6.35 kg moving at 13 m/s. His momentum is the product of mass and velocity: p=6.35×13=82.55 kg\cdotpm/sp = 6.35 \times 13 = 82.55 \, \text{kg·m/s}. Plugging this into the De Broglie equation, we find his wavelength is approximately 8.03×10−36 m8.03 \times 10^{-36} \, \text{m}, which is unimaginably small.
This result highlights a key idea in quantum mechanics: while all matter exhibits wave-like behavior, the effect is only significant at extremely small scales—like atoms or subatomic particles. For large objects like cats, cars, or even grains of sand, the De Broglie wavelength is so tiny that any wave-like behavior is undetectable and irrelevant in practice.
Hence, while Jack technically has a De Broglie wavelength, it doesn’t affect his behavior in any noticeable way. This exercise is valuable primarily to show the universal nature of quantum principles, even though classical mechanics remains sufficient for understanding macroscopic objects.