Imagine an alternate universe where the value of the Planck constant is ×6.62607·10−12Js.In that universe, which of the following objects would require quantum mechanics to describe, that is, would show both particle and wave properties? Which objects would act like everyday objects, and be adequately described by classical mechanics?object quantum or classical?A mosquito with a mass of 2.2 mg, 6.6 mm long, moving at 2.4 m/s.classicalquantumA ball with a mass of 115. g, 4.4 cm wide, moving at 14.4 m/s.classicalquantumA bacterium with a mass of 6.0 pg, 2.0 µm long, moving at 10.00 µm/s.classicalquantumAn alpha particle with a mass of 6.6 x 10-27 kg, 8.0 x 10-15 m wide, moving at 16. km/s.classicalquantum
Question
Imagine an alternate universe where the value of the Planck constant is ×6.62607·10−12Js.In that universe, which of the following objects would require quantum mechanics to describe, that is, would show both particle and wave properties? Which objects would act like everyday objects, and be adequately described by classical mechanics?object quantum or classical?A mosquito with a mass of 2.2 mg, 6.6 mm long, moving at 2.4 m/s.classicalquantumA ball with a mass of 115. g, 4.4 cm wide, moving at 14.4 m/s.classicalquantumA bacterium with a mass of 6.0 pg, 2.0 µm long, moving at 10.00 µm/s.classicalquantumAn alpha particle with a mass of 6.6 x 10-27 kg, 8.0 x 10-15 m wide, moving at 16. km/s.classicalquantum
Solution
To determine whether an object requires quantum mechanics to describe or can be adequately described by classical mechanics, we can use the de Broglie wavelength formula:
λ = h / (m*v)
where λ is the de Broglie wavelength, h is the Planck constant, m is the mass of the object, and v is the velocity of the object. If the de Broglie wavelength is comparable to the size of the object, then quantum mechanics is needed. If the de Broglie wavelength is much smaller than the size of the object, then classical mechanics can be used.
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Mosquito: Mass (m) = 2.2 mg = 2.2 * 10^-6 kg Velocity (v) = 2.4 m/s Size = 6.6 mm = 6.6 * 10^-3 m λ = h / (m*v) = 6.62607 * 10^-12 Js / (2.2 * 10^-6 kg * 2.4 m/s) = 1.26 * 10^-6 m Since the de Broglie wavelength is much smaller than the size of the mosquito, the mosquito can be described by classical mechanics.
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Ball: Mass (m) = 115 g = 0.115 kg Velocity (v) = 14.4 m/s Size = 4.4 cm = 0.044 m λ = h / (m*v) = 6.62607 * 10^-12 Js / (0.115 kg * 14.4 m/s) = 4.01 * 10^-14 m Since the de Broglie wavelength is much smaller than the size of the ball, the ball can be described by classical mechanics.
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Bacterium: Mass (m) = 6.0 pg = 6.0 * 10^-12 kg Velocity (v) = 10.00 µm/s = 10.00 * 10^-6 m/s Size = 2.0 µm = 2.0 * 10^-6 m λ = h / (m*v) = 6.62607 * 10^-12 Js / (6.0 * 10^-12 kg * 10.00 * 10^-6 m/s) = 0.11 m Since the de Broglie wavelength is much larger than the size of the bacterium, the bacterium requires quantum mechanics to describe.
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Alpha particle: Mass (m) = 6.6 * 10^-27 kg Velocity (v) = 16. km/s = 16. * 10^3 m/s Size = 8.0 * 10^-15 m λ = h / (m*v) = 6.62607 * 10^-12 Js / (6.6 * 10^-27 kg * 16. * 10^3 m/s) = 6.3 * 10^-10 m Since the de Broglie wavelength is much larger than the size of the alpha particle, the alpha particle requires quantum mechanics to describe.
Similar Questions
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quantum mechanics, science dealing with the behaviour of matter and light on the atomic and subatomic scale. It attempts to describe and account for the properties of molecules and atoms and their constituents—electrons, protons, neutrons, and other more esoteric particles such as quarks and gluons.
Which phenomenon BEST supports the theory that matter has a wave nature?*1 pointphoton diffractionelectron diffractionphoton momentumelectron momentum
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