I have never seen a question asked about quant, not been asked a question about it! But, I will give it a go.
Imagine you are on a train of length '2x' with a light beam source directly in the middle, so the distance to either end of the carriage is 'x' and at either end of the carriage there are light detectors, when hit, will instantaneously open the carriage door.
Now quantitatively, to the person inside the train (stationary relative to the train), the train is moving at a relative velocity of 0 -- hence you can figure out the time taken to hit either of the carriage doors as being distance/speed = x/c
Now, to the person outside, the train is moving at say 0.5c
So, you can figure out how far the train will have moved before the light beam hits the rear door
Distance travelled by the beam = speed x time
lets called the distance x'
x' = c x t
now x - x' corresponds to the amount which the train has shifted in the same time period
x - x' = c/2 x t
This tells you that the train has travelled half as far as the light beam, therefore we can say that the light beam has travelled (2/3)x and the train has travelled (1/3)x when the light has hit the rear door.
Now, after time 't', if the train has moved (1/3)x and the light has moved (2/3)x, then going towards the front of the train, it would only be (2/3)x from its point of origin and is now (2/3)x away from the door (keeping in mind the relative speed of light to the observer is always 'c' and train is moving at 'c/2')
Ok, it is late, but that is my attempt at explaining it somewhat quantitatively.
I hope it makes more sense to you then it did to me whilst I was typing it
