All elementary particles have a property called spin, which is interpreted as the intrinsic angular momentum of the particle. In classical mechanics, each angular momentum is associated with rotational energy. Is rotational energy also associated with the spin of elementary particles? If so, how big is it?
The concept of spin was proposed in 1925 by the young physicists Samuel Goudsmit (1902-1978) and George Uhlenbeck (1900-1988) to explain the splitting of electron orbits in an atom when a magnetic field is applied. It was later discovered that not only electrons but all elementary particles possess a spin. However, interpreting spin within the framework of classical mechanics leads to contradictions that Goudsmith and Uhlenbeck were already aware of. The two therefore wanted to withdraw the manuscript they had already submitted before publication. However, their academic teacher Paul Ehrenfest is said to have persuaded them to set aside their concerns by saying, "You are young enough to afford to be stupid. Today, the concept of spin is a generally accepted and proven component of particle physics, without the well-known contradictions having been satisfactorily resolved.
One of these well-known difficulties is encountered when trying to determine the rotational energy associated with the spin of an electron: Let us assume that electrons are rotating homogeneous spheres. The mass of an electron can be determined experimentally and is me = 9,1 ⋅ 10-31 kg. A theoretical radius for the electron, the so-called ‘classical electron radius’ re = 2,8 ⋅ 10-15 m, can be derived from electrodynamics. Experiments to determine the electron radius have led to the conclusion that the radius of the electron must be even smaller, smaller than 10-22 m, if the idea of a spatially extended particle makes any sense at all. According to classical mechanics, the moment of inertia of a homogeneous sphere is Jsphere = 2/5 ⋅ m ⋅ r2 and the rotational energy Erot = 1/2 ⋅ L2 / J. If we now use the spin h/4π and the moment of inertia J of a rotating sphere with radius re and mass me for L, we obtain the rotational energy Erot = 1/2 ⋅ L2 / (2/5 ⋅ me ⋅ re2) = 2 ⋅ 10−9 J. If we compare this result with the rest energy E = me ⋅ c2 = 8,1 ⋅ 10-14 J, we see that the rotational energy must be several orders of magnitude greater than the energy equivalent of its mass at rest. With the experimentally determined upper limit for the electron radius of 10-22 m known today, the discrepancy is even more striking.
A further consideration leads to the conclusion that the electron would have to rotate so fast that its circumferential speed would be a multiple of the speed of light, which is incompatible with the theory of relativity: In classical mechanics, the circumferential speed vu at the equator of a rotating sphere is given by vu = r ⋅ ω with ω = L / Jsphere. Again using the above values for the electron, we obtain vu ≈ 1011 m/s. This value is much higher than the speed of light of 3 ⋅108 m/s.
The model concept that an electron resembles a rotating sphere with mass therefore leads to blatant contradictions. It is therefore usually postulated that the spin cannot be interpreted classically, but is an effect of relativistic quantum mechanics. This is accompanied by the rarely explicitly stated consequence that no rotational energy is associated with the spin of elementary particles. However, this seems unphysical to us, as otherwise any rotation would also result in rotational energy. Is there therefore another way to resolve the contradictions in the interpretation of spin?
Comments powered by CComment