Mathematical physics is the study of physics using mathematics. It might be argued that all of physics is mathematical physics, but in practice, most physics is done on a more intuitive/approximate or even questionable level. Mathematical physics tries to study physics on a more abstract and rigorous level than typical.
Quantum mechanics can't be understood without a good knowledge of mathematics. It's not surprising then that its developed version under the name of quantum field theory is the most abstract, mathematical-based and backward influential to mathematics area of physical sciences.
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