BER for AWGN Channel
1.414 is the square root of 2. If the same signal is sent over the 2 independent channels, then your received Noise PSD is the same, but your signal is now twice as powered. You add the signals, but the noise is not additive because it is IID / uncorrelated. Twice as much power is a 3dB Gain. I'm a little hazy on BER curves, but if that Q(sqrt(g)) is the likelyhood function, then instead of g you have 2*g, and Sqrt(2*g) = Sqrt(2)*sqrt(g)

Yeah. The argument under the square root is the received Signal to noise ratio in Eb/No (energy-per-bit/Noise-Power-in-1Hz-bandwidth). One receiver and one channel is R = S + N. In this example R = S1 + S2 + N. If S1 and S2 are equal then you have 2S + N, thus twice the SNR.