Oak, Osage, locust, hard maple all have a higher modulus of elasticity.
See I know big words too ...
The simple answer is the chart of woods elasticity is what tells you which woods will span farther with thinner boards.
If those vertical lines are small corner braces, then yes, thinner boards will work. Careful how deep you make your dado so you don't cause the backboard to split.
In pine, you can use 3/4" stock to build that shelf. I have seen tons of 'em that size.
As you thin the boards, the fasteners become the challenge. In the harder woods you will have to pre drill. The more visible the pores are on the wood you pick, the higher the tendency to split or run out.
I would suggest half inch as a minimum thickness considering the dado and fastening issues. However, if you must press your luck, 3/8 and bring the shelf supports in about 2 to 3 " from the end. Mathematics say the interior span will support better than the open end so dont be tempted to bring them to quarter/half/quarter position for better support.
