Schrodinger equation

Bra-ket state vectors in Hilbert space (linear algebra, matrices a bit foreign in the rest of physics syllabus)

Feynman sum of all paths functional calculus (calculus not quite in the rest of the syllabus)

To better align with other parts of the syllabus, and build on what students have already learnt in the rest of physics, the Schrodinger equation seems the most appropriate starting point

Motivate this equation using concept of mechanical energy

Interpret solutions as waves, build upon wave concepts

If can introduce Fourier transform in topic of waves, then some concepts like Heisenberg uncertainty principle can be understood on that basis

Complication of complex numbers being involved, but not getting students to really grapple with the technicalities of solving

Can mention Feynman sum of paths formalism, but this might be a bit too much for many students, path integrals, functional integration, and also Lagrangian and Hamiltonian methods (worth a mention in framework of classical mechanics)

Matrices and state vectors are powerful and more modern, simpler... maybe better to reserve it for university courses

Spin-statistics theorem is great, but maybe too advanced, though would be a good example to highlight the deep interconnections across areas of physics (some idea of angular momentum would be important pre-requisite though)

Should anyway introduce Maxwell distribution of classical particles before fermion/boson distributions