A macroscopic uncertainty principle in quantum mechanics: we cannot measure angular momentum precisely?

In our previous blog post we analyzed Anthony J. Leggett's claim that a circulating superconducting current in a loop cannot decay to a lower angular momentum state because there is a huge potential wall which prevents this transition.

The angular momentum of a complex system does not need to be an integer multiple of ħ

For a single microscopic particle, quantum mechanics dictates that its "orbital" angular momentum to the z direction in a stationary state has to be a multiple of ħ. Does the same hold for a macroscopic system?

That does not hold for a complex microscopic system, if we allow the system to change its state. The system does not need to return to the original state after one rotation, because the system may have a different state during the 2nd rotation, yet another state during the 3rd rotation, and so on. The angular momentum can be any fraction of ħ.

What if we demand that the state of the system is the same after one rotation? It is not clear what "the same state" exactly means for a complex system.

A rotating donut: can we change its angular momentum by a fraction of ħ?

We assume that we have a macroscopic rotating donut in otherwise empty space. The temperature of the donut is exactly 0 kelvin.

                                                         ~~~~~

        radius 1 meter

                _____

              /           \_____ mirror       ~~~~~

              \______/  1 meter

                 ---->  rotation

                                                         ~~~~~

                                                    1,000-meter-wide

                                                    laser beam

                                                    λ = 1 meter

The donut has an attached mirror whose size is one meter.

We have a laser beam which is 1,000 meters wide and whose wavelength is 1 meter. The beam is defocused by rougly 0.001 radians because the beam is 1,000 meters wide.

Can the laser reduce or increase the angular momentum of the donut by a fraction of ħ?

We have

       λ = h / p,

where p is the momentum of the photon. Since λ = 1 meter, the numerical value of p is the same as the Planck constant h has.

We assume that the donut rotates very slowly. When the laser is switched on, the mirror is at a small angle relative to the laser. The mirror will scatter incoming photons to various directions, but most of the photons get deflected very little. We can put a photographic plate far away from the mirror and measure where each photon hits the plate.

If a photon is deflected by, say 0.01 radians, then it probably was reflected or scattered by the mirror. It gave some 0.01 h = 0.06 ħ of angular momentum to the donut.

We can give a fraction of ħ as new angular momentum to the donut.

Adding angular momentum to a system in fractions of ħ

Assume that we have a (macroscopic) system whose angular momentum is quantized to be an integer times ħ. Using the method sketched in the previous section we add angular momentum to it in fractions of ħ. What happens?

If we just send a single photon at a time, then the system can only scatter the photon in the way where the angular momentum of the system stays as an integer times ħ.

If the system is macroscopic and the laser beam is macroscopic, then the system must behave classically, because of the principle of the classical limit. There cannot be any problem in absorbing angular momentum from the laser beam, even though in terms of individual photons it comes as fractions of ħ.

An uncertainty principle for a macroscopic system

Maybe there cannot exist a macroscopic system whose angular momentum is quantized as an integer times ħ?

Uncertainty principle for a macroscopic system. We cannot measure the angular momentum of a macroscopic system at the precision of ħ.

What chance we might have for measuring a macroscopic system at the precision of ħ? We would need to measure the state of a huge number of particles. The measurement would add a lot of energy and angular momentum into the system.

In quantum mechanics we do not like hidden variables. If we cannot measure something, even in principle, then it does not possess a sharp value at all.

General uncertainty principle for a macroscopic system. It is not possible to measure any property of a macroscopic system at the precision allowed for a microscopic system.

Can we find a counter-example for the general principle?

Conclusions

Our macroscopic uncertainty principle may solve some other paradoxes, in addition to the quantization of angular momentum.

Let us have a macroscopic pulse of laser light. How many photons are in it? We have earlier written that since the pulse can excite oscillators of various frequencies, it does not make sense to claim that the pulse contains some precise integer number N of photons of a precise wavelength λ. The set of photons in the pulse is inherently fuzzy: we can decide at which frequencies we harvest energy from the pulse. The pulse, in this sense, is strictly a classical entity. It cannot be quantized. "Quantization" only happens at the moment when we use quantum harmonic oscillators to harvest energy from the pulse.

If the donut in our example would have its angular momentum quantized in units of ħ or in some fractions of ħ, then that would severely restrict how individual photons can scatter from the mirror. But in the classical limit, a macroscopic laser beam must scatter from the mirror in the classical way. Does this show that the angular momentum of the macroscopic donut cannot be quantized?

We will investigate this more. The fuzzy line between the quantum world and the classical world may be clarified.