In this video, Dr. Jacob Hudis explains the connection between the Hermitian matrix operators used in Heisenberg’s matrix mechanics and the differential Hermitian operators found in the Schrödinger equation. The video highlights how 𝑖 times the derivative is a Hermitian operator related to momentum, while x represents the Hermitian operator associated with position. These operators have key eigenfunctions, such as the Dirac delta function and the complex exponential plane wave, both of which are essential in quantum mechanics. Dr. Hudis illustrates how the wave nature and matrix nature of quantum mechanics share the same mathematical foundation, ultimately pointing to Hamiltonians built from non-commuting operators in different bases.

Other useful videos to look at:
1) Wavefunction Mathematics and The Continuous Basis of Position and Momentum (QM math 3B)
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2) Quantum Mechanics Math 3A Fourier Series and Transformations
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