In class today, my professor was discussing how to structure a class. The course primarily uses Java and I have more Java experience than the teacher (he comes from a C++ b
I think your professor's being overly stubborn (probably deliberately, to push you to a fuller understanding). Really the benefits of immutability are not so much what the complier can do with optimisations, but really that it's much easier for us humans to read and understand. A variable that is guaranteed to be set when the object is created and is guaranteed not to change afterwards, is much easier to grok and reason with than one which is this value now but might be set to some other value later.
This is especially true with threading, in that you don't need to worry about processor caches and monitors and all that boilerplate that comes with avoiding concurrent modifications, when the language guarantees that no such modification can possibly occur.
And once you express the benefits of immutability as "the code is easier to follow", it feels a bit sillier to ask for empirical measurements of productivity increases vis-a-vis "easier-to-followness".
On the other hand, the compiler and Hotspot can probably perform certain optimisations based on knowing that a value can never change - like you I have a feeling that this would take place and is a good things but I'm not sure of the details. It's a lot more likely that there will be empirical data for the types of optimisation that can occur, and how much faster the resulting code is.