If a CPU has two core, it means that the CPU contains two fully functional processors. 64bit instructions are regular instructions that are fed into one of the core of your CPU. More specifically, 64bit instructions can access more and wider registers (64bits). They are not split into two 32bit instructions that are dispatched amongst the cores. Each of yhe cores is capable of executing the 64bit instructions fed into it.
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The primary difference between 32- and 64-bit modes is the use of 32-bit pointers on the former, and longer ones on the latter. A consequence is the ability to use more RAM in 64-bit mode, where, for example, most 32-bit Windows versions use no more than 3GB.
In 64-bit mode, there is native support for arithmetic on 64-bit integers, while in 32-bit mode, 64-bit integer arithmetic is carried out by combinations of 32-bit operations.
Certain CPUs in the past split 128-bit operands into 2 64-bit chunks, to be processed by separate units on a single core. Those which supported 64-bit mode didn't change this behavior between 32- and 64-bit modes.