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I was reading an article online pertaining to quantum mechanics and I stumbled across these few sentences.

A look at the corresponding energy regimes shows (Beck and Eccles 1992) that quantum processes are distinguishable from thermal processes for energies higher than $10^{-2}$ eV (at room temperature). Assuming a typical length scale for biological microsites of the order of several nanometers, an effective mass below 10 electron masses is sufficient to ensure that quantum processes prevail over thermal processes.

I would like to know what they mean when they say "is sufficient to ensure that quantum processes prevail over thermal processes". Or possibly just what the difference is between a quantum process and a thermal process.

Original Text (Section 4.4 Beck and Eccles: Quantum Mechanics at the Synaptic Cleft)

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up vote 1 down vote accepted

The quantum uncertainty in position of particles (micro sites) of mass $m$ moving or vibrating at thermal speeds characteristic for a temperature $T$, is given by the thermal De Broglie wavelength $\sqrt{\frac{2 \pi \hbar^2}{m \ kT}}$. If this uncertainty in particle (micro site) position is larger than or comparable to its size, the inter-particle (inter site) interactions must be described quantum mechanically.

For the example quoted, $m$ is 10 electron masses and $T$ is 300 K (room temperature), it follows that the De Broglie wavelength is a few nanometer. Hence, micro sites weighing 10 electron masses can be treated classically (using Newton's laws rather than quantum mechanics) provided they are significantly larger than a few nanometers. If these micro sites are of size a few nanometer (or smaller) quantum uncertainty kicks in an the full quantum physics machinery needs to be brought in.

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Because of particle-wave duality one can assign a wavelength (the so called de-Broglie wavelength) to a particle (or compound object). De-Broglie found that this wavelength is


where $h$ is Planck's constant and $p$ the particle's momentum (at low velocities $p=mv$). Very handwavingly one can imagine the object as a wave-packet with size roughly of the order of \lambda. In classical physics the extension of the wave-packet would be just a single dot, i.e., the center of mass of the object. Now, what is the approximate size of the wave-packets of these biological microsites? The thermal de Broglie wavelength uses the average thermal energy of a particle to calculate the wave-packet's extension. For an ideal gas one would obtain

$\Lambda=\sqrt{\frac{2\pi\hbar^2}{m k_B T}} \approx 1.3\times 10^{-9} m$

(i.e., of the order of 1 nanometer)


(all units in SI)

  • $\hbar \approx 10^{-34}$
  • $m \approx 10 * m_e = 9.1\times10^{-30}$
  • $k_B = 1.38 \times 10^{-23}$
  • $T \approx 300$

(the question is if the ideal-gas-approximation is justified or not in your case or not)

If the thermal de Broglie wavelength $\Lambda$ is of the order or larger than the spacing between "biological microsites" (which is the case here), then there is significant overlap of the wavefunctions of two neighboring sites, that is, one can't say where one microsite ends and the other begins, it's like a big soup: The system is said to be quantum degenerate (or sometimes even ultracold) and quantum effects are not negligible.

If on the other hand, the extension of the wave-packets was so small that there would be essentially no overlap, one can approximate the system as thermal (and therefore neglect quantum effects). So I think it's not about the length scales per se, but whether there is wavefunction overlap between different objects or not.

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I'm still a little confused, so a thermal process involves processes at scales larger than the De Broglie wavelength and quantum processes are on scales smaller than the De Broglie wavelength? And the De Broglie wavelength is determined by the particles mass, temperature, and Kb (I think boltzmans constant?)? – KDecker Jul 25 '13 at 13:32

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