J.R. S. answered 05/31/18
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The copper and the zinc will undergo oxidation/reduction where reduction will take place at the Cu electrode and oxidation at the Zn electrode. This is based on the fact that reduction potential for Cu2+ is greater than that of Zn2+. Thus,
Cu2+ + 2e- ==> Cu(s) at the Cu electrode and
Zn(s) ==> Zn2+ + 2e- at the Zn electrode
Without a salt bridge, and with both electrodes in the same bath, the current will not last long because there will be no neutralization of charge build up. So, while a voltage will pass, it will be of very short duration. Also, I wouldn't agree that you have only a half cell. You still have 2 half cells, but they aren't separated.