Answer :
To determine whether each object would require quantum mechanics (showing wave-particle duality) or can be adequately described by classical mechanics, we use the de Broglie wavelength. The de Broglie wavelength [tex]\(\lambda\)[/tex] is given by:
[tex]\[ \lambda = \frac{h}{mv} \][/tex]
where:
- [tex]\(h\)[/tex] is the Planck constant,
- [tex]\(m\)[/tex] is the mass, and
- [tex]\(v\)[/tex] is the velocity of the object.
In this alternate universe, the Planck constant [tex]\(h = 6.62607 \times 10^6 \, J \cdot s\)[/tex].
An arbitrary threshold for deciding quantum vs classical behavior is set at [tex]\(1 \times 10^{-9} \, m\)[/tex]. If the de Broglie wavelength of an object is greater than [tex]\(1 \times 10^{-9} \, m\)[/tex], the object exhibits quantum behavior. If it is less than this threshold, the object can be described by classical mechanics.
Let's examine each object:
1. Virus
- Mass: [tex]\(1.1 \times 10^{-17} \, kg\)[/tex]
- Velocity: [tex]\(1.30 \times 10^{-6} \, m/s\)[/tex]
Calculating the de Broglie wavelength:
[tex]\[ \lambda_{\text{virus}} = \frac{6.62607 \times 10^6}{(1.1 \times 10^{-17}) \times (1.30 \times 10^{-6})} \approx 4.63 \times 10^{29} \, m \][/tex]
Since [tex]\(4.63 \times 10^{29} \, m\)[/tex] is much greater than [tex]\(1 \times 10^{-9} \, m\)[/tex], the virus shows quantum properties.
2. Car
- Mass: [tex]\(1100 \, kg\)[/tex]
- Velocity: [tex]\(66.0 \, km/h = 18.33 \, m/s\)[/tex]
Calculating the de Broglie wavelength:
[tex]\[ \lambda_{\text{car}} = \frac{6.62607 \times 10^6}{(1100) \times (18.33)} \approx 328.57 \, m \][/tex]
Since [tex]\(328.57 \, m\)[/tex] is greater than [tex]\(1 \times 10^{-9} \, m\)[/tex], the car surprisingly shows quantum properties.
3. Grain of Sand
- Mass: [tex]\(205 \times 10^{-6} \, kg\)[/tex]
- Velocity: [tex]\(1.00 \times 10^{-3} \, m/s\)[/tex]
Calculating the de Broglie wavelength:
[tex]\[ \lambda_{\text{sand}} = \frac{6.62607 \times 10^6}{(205 \times 10^{-6}) \times (1.00 \times 10^{-3})} \approx 3.23 \times 10^{13} \, m \][/tex]
Since [tex]\(3.23 \times 10^{13} \, m\)[/tex] is much greater than [tex]\(1 \times 10^{-9} \, m\)[/tex], the grain of sand shows quantum properties.
4. Atom
- Mass: [tex]\(1.0 \times 10^{-26} \, kg\)[/tex]
- Velocity: [tex]\(366 \, m/s\)[/tex]
Calculating the de Broglie wavelength:
[tex]\[ \lambda_{\text{atom}} = \frac{6.62607 \times 10^6}{(1.0 \times 10^{-26}) \times (366)} \approx 1.81 \times 10^{30} \, m \][/tex]
Since [tex]\(1.81 \times 10^{30} \, m\)[/tex] is much greater than [tex]\(1 \times 10^{-9} \, m\)[/tex], the atom shows quantum properties.
Summary:
| object | quantum or classical? |
|--------|-----------------------|
| A virus with a mass of [tex]\(1.1 \times 10^{-17} \, kg\)[/tex], moving at [tex]\(1.30 \, \mu m/s\)[/tex]. | quantum |
| A car with a mass of [tex]\(1100 \, kg\)[/tex], moving at [tex]\(66.0 \, km/h\)[/tex]. | quantum |
| A grain of sand with a mass of [tex]\(205 \, mg\)[/tex], moving at [tex]\(1.00 \, mm/s\)[/tex]. | quantum |
| An atom with a mass of [tex]\(1.0 \times 10^{-26} \, kg\)[/tex], moving at [tex]\(366 \, m/s\)[/tex]. | quantum |
In this alternate universe, all the given objects exhibit quantum properties due to the extraordinarily high value of the Planck constant.
[tex]\[ \lambda = \frac{h}{mv} \][/tex]
where:
- [tex]\(h\)[/tex] is the Planck constant,
- [tex]\(m\)[/tex] is the mass, and
- [tex]\(v\)[/tex] is the velocity of the object.
In this alternate universe, the Planck constant [tex]\(h = 6.62607 \times 10^6 \, J \cdot s\)[/tex].
An arbitrary threshold for deciding quantum vs classical behavior is set at [tex]\(1 \times 10^{-9} \, m\)[/tex]. If the de Broglie wavelength of an object is greater than [tex]\(1 \times 10^{-9} \, m\)[/tex], the object exhibits quantum behavior. If it is less than this threshold, the object can be described by classical mechanics.
Let's examine each object:
1. Virus
- Mass: [tex]\(1.1 \times 10^{-17} \, kg\)[/tex]
- Velocity: [tex]\(1.30 \times 10^{-6} \, m/s\)[/tex]
Calculating the de Broglie wavelength:
[tex]\[ \lambda_{\text{virus}} = \frac{6.62607 \times 10^6}{(1.1 \times 10^{-17}) \times (1.30 \times 10^{-6})} \approx 4.63 \times 10^{29} \, m \][/tex]
Since [tex]\(4.63 \times 10^{29} \, m\)[/tex] is much greater than [tex]\(1 \times 10^{-9} \, m\)[/tex], the virus shows quantum properties.
2. Car
- Mass: [tex]\(1100 \, kg\)[/tex]
- Velocity: [tex]\(66.0 \, km/h = 18.33 \, m/s\)[/tex]
Calculating the de Broglie wavelength:
[tex]\[ \lambda_{\text{car}} = \frac{6.62607 \times 10^6}{(1100) \times (18.33)} \approx 328.57 \, m \][/tex]
Since [tex]\(328.57 \, m\)[/tex] is greater than [tex]\(1 \times 10^{-9} \, m\)[/tex], the car surprisingly shows quantum properties.
3. Grain of Sand
- Mass: [tex]\(205 \times 10^{-6} \, kg\)[/tex]
- Velocity: [tex]\(1.00 \times 10^{-3} \, m/s\)[/tex]
Calculating the de Broglie wavelength:
[tex]\[ \lambda_{\text{sand}} = \frac{6.62607 \times 10^6}{(205 \times 10^{-6}) \times (1.00 \times 10^{-3})} \approx 3.23 \times 10^{13} \, m \][/tex]
Since [tex]\(3.23 \times 10^{13} \, m\)[/tex] is much greater than [tex]\(1 \times 10^{-9} \, m\)[/tex], the grain of sand shows quantum properties.
4. Atom
- Mass: [tex]\(1.0 \times 10^{-26} \, kg\)[/tex]
- Velocity: [tex]\(366 \, m/s\)[/tex]
Calculating the de Broglie wavelength:
[tex]\[ \lambda_{\text{atom}} = \frac{6.62607 \times 10^6}{(1.0 \times 10^{-26}) \times (366)} \approx 1.81 \times 10^{30} \, m \][/tex]
Since [tex]\(1.81 \times 10^{30} \, m\)[/tex] is much greater than [tex]\(1 \times 10^{-9} \, m\)[/tex], the atom shows quantum properties.
Summary:
| object | quantum or classical? |
|--------|-----------------------|
| A virus with a mass of [tex]\(1.1 \times 10^{-17} \, kg\)[/tex], moving at [tex]\(1.30 \, \mu m/s\)[/tex]. | quantum |
| A car with a mass of [tex]\(1100 \, kg\)[/tex], moving at [tex]\(66.0 \, km/h\)[/tex]. | quantum |
| A grain of sand with a mass of [tex]\(205 \, mg\)[/tex], moving at [tex]\(1.00 \, mm/s\)[/tex]. | quantum |
| An atom with a mass of [tex]\(1.0 \times 10^{-26} \, kg\)[/tex], moving at [tex]\(366 \, m/s\)[/tex]. | quantum |
In this alternate universe, all the given objects exhibit quantum properties due to the extraordinarily high value of the Planck constant.
