LOG#203. Action gym.

A challenge post! 😉

Quantum physics becomes important when the magnitude called action is of order of Planck constant (an action itself!). Action is quantized. It is a much more essential quantization than that of energy or angular momentum (action itself the latter!). Whenever S\sim 10^{-34}J\cdot s you have quantum effects.

Exercise 1. Compute the action from an antenna with power 1kW and frequency f=1MHz. Is it quantum?

Exercise 2. A pocket clock. It uses a device of size 10^{-4}m and mass 10^{-4}kg to get times with precision of 1s. Compute its action. Is it quantum?

Exercise 3. Compute the action for a single atomic nucleus. Typical energies are about 1 MeV and distances about 1 fm (10^{-15}m). Binding energy per nucleon is about 10^{-12}J, mass of the nucleus can be taken as the proton mass. Compute the action. Is it quantum?

Exercise 4. Compute the action for an electron in the 1s shell of the hydrogen atom. Is it quantum?

Exercise 5. Compute the so-called Fermi energy in the case of potassium (metal). The atomic weight is about 39 g/mol, and its atom density 0.86 g per cubic centimeter. Suppose a single electron per atom. Compute its action. Is it quantum?

Remark:

    \[\left[\hbar\right]=ML^2T^{-1}=\mbox{Energy}\cdot \mbox{Time}=\mbox{Momentum x Length}=\mbox{angular momentum x angle}\]

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