Hi DiscoStu,
If you have sown the seen evenly at the correct rate, I feel sure that over the few days, you will notice it becoming thicker and thicker as more seeds germinate as they are unlikely to germinate all at once.
Initially each seed will produce just 1or 2 fine blades and I can see why you think it's looking sparse, so I'd give it a while longer to see how it developes and over-seed it later if neccessary.
Wait for the grass to reach at least 2" before making your first cut and then set your mower on a high setting.
As it is cut more and more, each individual grass plant will produce more leaves at the base to hopefully form a good thick sword of turf. Good Luck. Tbird+