Answer :
In this problem, we are determining which objects would exhibit quantum mechanical behavior based on their De Broglie wavelengths and comparing those wavelengths to the characteristic sizes of the objects.
Let's consider each object one-by-one:
1. Ball:
- Mass: [tex]\(125.9 \)[/tex] g
- Diameter: [tex]\(5.5 \)[/tex] cm = [tex]\(0.055 \)[/tex] m
- Speed: [tex]\(25.3 \)[/tex] m/s
The De Broglie wavelength, [tex]\(\lambda\)[/tex], for the ball is [tex]\(20.80\)[/tex] m.
Since the De Broglie wavelength ([tex]\(20.80\)[/tex] m) is much larger than the diameter of the ball ([tex]\(0.055\)[/tex] m), the ball will exhibit quantum mechanical behavior. Therefore, the ball is classified as quantum.
2. Buckyball:
- Mass: [tex]\(1.2 \times 10^{-21} \)[/tex] g
- Diameter: [tex]\(0.7 \)[/tex] nm = [tex]\(0.7 \times 10^{-9} \)[/tex] m
- Speed: [tex]\(18.0 \)[/tex] m/s
The De Broglie wavelength, [tex]\(\lambda\)[/tex], for the buckyball is [tex]\(3.07 \times 10^{24}\)[/tex] m.
Since the De Broglie wavelength ([tex]\(3.07 \times 10^{24}\)[/tex] m) is vastly larger than the diameter of the buckyball ([tex]\(0.7 \times 10^{-9}\)[/tex] m), the buckyball will exhibit quantum mechanical behavior. Therefore, the buckyball is classified as quantum.
3. Airplane:
- Mass: [tex]\(2.34 \times 10^4 \)[/tex] kg
- Length: [tex]\(17.0 \)[/tex] m
- Speed: [tex]\(1600 \)[/tex] km/h = [tex]\(444.44 \)[/tex] m/s
The De Broglie wavelength, [tex]\(\lambda\)[/tex], for the airplane is [tex]\(6.37 \times 10^{-6}\)[/tex] m.
Since the De Broglie wavelength ([tex]\(6.37 \times 10^{-6}\)[/tex] m) is much smaller than the characteristic length of the airplane ([tex]\(17.0 \)[/tex] m), the airplane will behave classically. Therefore, the airplane is classified as classical.
4. Eyelash mite:
- Mass: [tex]\(12.2 \)[/tex] µg = [tex]\(12.2 \times 10^{-6} \)[/tex] g
- Diameter: [tex]\(235 \)[/tex] µm = [tex]\(235 \times 10^{-6} \)[/tex] m
- Speed: [tex]\(47 \)[/tex] µm/s = [tex]\(47 \times 10^{-6} \)[/tex] m/s
The De Broglie wavelength, [tex]\(\lambda\)[/tex], for the mite is [tex]\(1.16 \times 10^{20}\)[/tex] m.
Since the De Broglie wavelength ([tex]\(1.16 \times 10^{20}\)[/tex] m) is vastly larger than the diameter of the mite ([tex]\(235 \times 10^{-6}\)[/tex] m), the mite will exhibit quantum mechanical behavior. Therefore, the mite is classified as quantum.
To summarize:
- Ball: Quantum
- Buckyball: Quantum
- Airplane: Classical
- Eyelash mite: Quantum
Let's consider each object one-by-one:
1. Ball:
- Mass: [tex]\(125.9 \)[/tex] g
- Diameter: [tex]\(5.5 \)[/tex] cm = [tex]\(0.055 \)[/tex] m
- Speed: [tex]\(25.3 \)[/tex] m/s
The De Broglie wavelength, [tex]\(\lambda\)[/tex], for the ball is [tex]\(20.80\)[/tex] m.
Since the De Broglie wavelength ([tex]\(20.80\)[/tex] m) is much larger than the diameter of the ball ([tex]\(0.055\)[/tex] m), the ball will exhibit quantum mechanical behavior. Therefore, the ball is classified as quantum.
2. Buckyball:
- Mass: [tex]\(1.2 \times 10^{-21} \)[/tex] g
- Diameter: [tex]\(0.7 \)[/tex] nm = [tex]\(0.7 \times 10^{-9} \)[/tex] m
- Speed: [tex]\(18.0 \)[/tex] m/s
The De Broglie wavelength, [tex]\(\lambda\)[/tex], for the buckyball is [tex]\(3.07 \times 10^{24}\)[/tex] m.
Since the De Broglie wavelength ([tex]\(3.07 \times 10^{24}\)[/tex] m) is vastly larger than the diameter of the buckyball ([tex]\(0.7 \times 10^{-9}\)[/tex] m), the buckyball will exhibit quantum mechanical behavior. Therefore, the buckyball is classified as quantum.
3. Airplane:
- Mass: [tex]\(2.34 \times 10^4 \)[/tex] kg
- Length: [tex]\(17.0 \)[/tex] m
- Speed: [tex]\(1600 \)[/tex] km/h = [tex]\(444.44 \)[/tex] m/s
The De Broglie wavelength, [tex]\(\lambda\)[/tex], for the airplane is [tex]\(6.37 \times 10^{-6}\)[/tex] m.
Since the De Broglie wavelength ([tex]\(6.37 \times 10^{-6}\)[/tex] m) is much smaller than the characteristic length of the airplane ([tex]\(17.0 \)[/tex] m), the airplane will behave classically. Therefore, the airplane is classified as classical.
4. Eyelash mite:
- Mass: [tex]\(12.2 \)[/tex] µg = [tex]\(12.2 \times 10^{-6} \)[/tex] g
- Diameter: [tex]\(235 \)[/tex] µm = [tex]\(235 \times 10^{-6} \)[/tex] m
- Speed: [tex]\(47 \)[/tex] µm/s = [tex]\(47 \times 10^{-6} \)[/tex] m/s
The De Broglie wavelength, [tex]\(\lambda\)[/tex], for the mite is [tex]\(1.16 \times 10^{20}\)[/tex] m.
Since the De Broglie wavelength ([tex]\(1.16 \times 10^{20}\)[/tex] m) is vastly larger than the diameter of the mite ([tex]\(235 \times 10^{-6}\)[/tex] m), the mite will exhibit quantum mechanical behavior. Therefore, the mite is classified as quantum.
To summarize:
- Ball: Quantum
- Buckyball: Quantum
- Airplane: Classical
- Eyelash mite: Quantum
