If the effects described here
If the effects described here are present in catalysis, their empirical confirmation poses many challenges. Cat state correlations are fragile outside an environment especially suited for their preservation. As present day observational techniques whether in vivo or in vitro were not designed bearing this in mind, they probably would have a destructive effect on such correlations. Indirect observations may be possible and realistic simulations may provide further insights. Some simulation results are intriguing, for instance, Romm et al. (2012) speaking of tunneling in glutamate mutase state that “The enzyme uses both classical and tunneling motions for a successful catalysis.” We have not incorporated a contribution of classical (decohered) processes in our analog to keep the analysis simple (otherwise higher dimensional models would be necessary), but they are surely present in real situations. The authors further state: “Consideration of the tunneling contributions of more and more atoms around the active center shows that the motions at the Fostriecin sodium salt ring play a central role during the tunneling enhancement of the hydrogen transfers.” This can be interpreted in various ways. On the one hand it could be another hint of helpful effects (classical or quantum) of crowding (by atoms in this case), or, on the other hand, as indicating that quantum coherence can be enhanced by involving more atoms. This is not explicitly a cat state simulation, but such can certainly be carried out. Another reason unequivocal quantum effects may be hard to observe is due to something one can state as “life thrives on the edge of quantum”. In the cases where biological quantum effects have now been widely acknowledged, namely, photosynthesis, animal navigation, and olfaction, most of the processing can be easily understood through conventional physics, chemistry and thermodynamics. Quantum mechanics becomes important at the end of the process. Conventional processes are robust and quantum processes are efficient but fragile. Life uses robustness as far as that can take it, calling upon quantum processes for short times, assuring coherence, to effectively complete the task. See McFadden and Al-Khalili (2015) for a quantum layperson\'s view of the cited examples. The fragility of quantum states and the “edge of quantum” situation present serious obstacles for experimental verification of the relevant quantum processes, requiring the development of proper experimental procedures bearing all of this in mind, possibly using quantum metrology itself. See Mairhofer et al. (2017) for a recent, if not the first, use of quantum metrology to study biomolecules.
To go beyond an analog to a true quantum model, one has to incorporate a lot more information concerning enzyme dynamics. Two references should be particularly useful in this regard: Somogy et al. (1984) and Igamberdiev (1999)
The delta potential on a line Consider the Schrödinger equationTo the right and left of q=0 the equation describes a free particle and so has sinusoidal waves (pure phase) as energy eigenfunctions. In three dimensions these are called plane waves. Writing Ψ(q, t)=exp(−iEt)ψ(q) then one has to havewith E=k2. We are for now assuming E>0. Continuity at q=0 imposes A+B=C+D. Call this number F. To take care of the delta potential one needs a discontinuity in ψ′(q) at q=0 so that the derivative of this discontinuity gives a delta function that cancels the potential. One must then have:which translates to Consider now the case that a flux of particles comes in from the left, part of which is reflected by the potential and part of which tunnels through. There are no particles coming in from the right. The above wave function is also an eigenstate of the momentum operator with eigenvalue k. As we have set the mass to be 1 the velocity of the particle is k. The incoming flux from the left is then k A 2 seeing as the density of these particles is A 2. The outgoing flux on the right is k C 2. If we assume D=0, that is there are no incoming particles from the right, then the fraction of left incoming particles that tunneled through isand is known as the transmission coefficient. Without loss of generality we now take D=0 and A=1, so F=1+B. The wave function that has tunneled through is then Fe. Condition (4) then reads and we finally find: