In an alternate universe where the value of the Planck constant is significantly larger, at [tex]$6.62607 \times 10^9 \, \text{J} \cdot \text{s}$[/tex], we need to determine which objects would require quantum mechanics to be adequately described and which would act like everyday objects, adequately described by classical mechanics.
To decide whether an object behaves quantum mechanically, we consider the de Broglie wavelength, [tex]\(\lambda\)[/tex], given by:
[tex]\[
\lambda = \frac{h}{mv}
\][/tex]
where:
- [tex]\(h\)[/tex] is the Planck constant,
- [tex]\(m\)[/tex] is the object's mass,
- [tex]\(v\)[/tex] is the object's velocity.
If the de Broglie wavelength is comparable to or larger than [tex]\(1 \, \text{nm} = 1 \times 10^{-9} \, \text{m}\)[/tex], the object shows quantum properties. Otherwise, it behaves classically.
### Objects Analysis
1. Virus
- Mass: [tex]\(9.2 \times 10^{-17} \, \text{g} = 9.2 \times 10^{-20} \, \text{kg}\)[/tex]
- Velocity: [tex]\(1.10 \, \mu \text{m/s} = 1.10 \times 10^{-6} \, \text{m/s}\)[/tex]
Using the given properties in the alternate universe, the virus has quantum properties.
2. Airplane
- Mass: [tex]\(2.67 \times 10^4 \, \text{kg}\)[/tex]
- Velocity: [tex]\(1300 \, \text{km/h} = 1300 \times \frac{1000}{3600} \, \text{m/s} = 361.11 \, \text{m/s}\)[/tex]
Using the given properties in the alternate universe, the airplane also demonstrates quantum properties.
3. Ball
- Mass: [tex]\(95 \, \text{g} = 95 \times 10^{-3} \, \text{kg}\)[/tex]
- Velocity: [tex]\(19.7 \, \text{m/s}\)[/tex]
Using the given properties in the alternate universe, the ball displays quantum properties.
4. Mosquito
- Mass: [tex]\(1.0 \, \text{mg} = 1.0 \times 10^{-6} \, \text{kg}\)[/tex]
- Velocity: [tex]\(3.1 \, \text{m/s}\)[/tex]
Using the given properties in the alternate universe, the mosquito shows quantum properties.
### Conclusion:
Given the significantly larger value of the Planck constant in this alternate universe, all the objects exhibit quantum mechanical properties. Hence:
| object | quantum or classical? |
|:------:|:---------------------:|
| Virus | quantum |
| Airplane | quantum |
| Ball | quantum |
| Mosquito | quantum |
All listed objects (virus, airplane, ball, and mosquito) require quantum mechanics to describe their behavior in this scenario.
