Quantum reactions do have lag times, for CO2 they are measured in tens of milliseconds.
As for the calculations, nothing special, I will show my work and you can disprove it, if you can!
The IPCC cites Baede, et al (2001) as the more comprehensive description
of the key concepts of climate science.
http://www.climatechange2013.org/images/report/WG1AR5_Chapter01_FINAL.pdf
Within Baede we find, the core of the IPCC's AGW concept.
http://www.grida.no/climate/ipcc_tar/wg1/pdf/tar-01.pdf
Two parts, the direct response warming from doubling the CO2 level,
and then the
belief that the warming from the CO2 will be amplified by feedbacks to add
additional warming between .3 and 3.3 °C.
Their confidence is expressed in the massive range of the prediction.
So back to the calculation.
The IPCC says doubling the CO2 level instantaneously, would cause warming of 1.2°C.
(they did not mention a lag time for direct response!)
1.2°C/ln(560ppm/280ppm)=1.73,
lets validate that number, 1.73 X ln(560/280)=1.1994...°C
Well that's within the ±10%.
So, to find out how much warming would have been caused by moving the CO2 level from
280 ppm to to 400 ppm, we do what I showed before,
1.73 X ln(400/280)=.61 °C
Please point out any mistakes I may have made!
P.S. and show your work also!