Lower yourself towards the BH with your back towards it. You'll observe the entire universe blue-shifting and getting squished into a structure in front of you. At the same time, all things happening in the universe will appear in fast motion and so by the time you reach the horizon you would...
Well, we break up h. So,
h = m c \lambda_{C} and,
h = m v \lambda_{B}.
Here, v is the velocity of the particle with mass m and lambda_{C} and lambda_{B} are its Compton and D'Broglie wavelengths respectively. Also, v_{g} = v.
Cancelling off m and juggling the rhs and lhs terms...
v_g and v_p explanations suck in textbooks, especially the long detour they take. Its pretty simple v_g/c = c/v_p means that the dimensions used can be written in unit terms, ie, light-seconds covered per second for v_g (the particle's velocity) and seconds spent per light-second for v_p (by...