MFQM-2 From Position Space to Momentum Space. Scaled Fourier Transform. John Harland.
Channel: Math 4 Wisdom
Date: 10/03/2024

John Harland continues presenting his mathematical framework for quantum mechanics (MFQM). He reviews the basic set up from the first video.  https://www.youtube.com/watch?v=yKR8mRRF6mo  Then at 28:10 he starts to talk about a custom made "scaled Fourier transform" that takes us from position space to momentum space. In subsequent videos, this will allow us to present the Schroedinger equation in momentum space and calculate the dispersal of a wave packet.  

John has a PhD in mathematics (functional analysis) and a passion for independent research into the relationship between quantum physics and classical physics. Momentum space is key to his approach.

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0:00 Introduction
1:50 The Basic Setup. Hilbert space. Future topic: Quantum Boolean algebra. Observables are self-adjoint operators.
4:10 Schroedinger equation. Self-adjoint operators. Unitary group of operators applied to the initial state. Spectral theorem breaks up wave function in terms of eigenfunctions of the Hamiltonian. Explicit formula for time evolution.
5:20 Question about measurements. Eigenfunctions of self-adjoint operators (like position) may not be the same eigenfunctions for the Hamiltonian.
Symmetries of the Hamiltonian simplify spectral decomposition.
8:25 Born Rule gives the average value of the observable.
9:45 Time variation of an observable A is a commutator 1/i h-bar [A,H]. If the observable commutes with the Hamiltonian, then it is conserved. This is a version of Noether's theorem, a future topic. The role of symmetry.
14:30 Uncertainty principle in terms of the commutator of an operator. Noncommutativity as the basis for Heisenberg's uncertainty principle.
15:20 "Multiplication by position" as an abstract operator. Born rule gives the mean value of position. Born rule relates states to measurements in the laboratory.
17:40 Mathematician's view in terms of self-adjoint operators, unitary dynamics, inner products. 
18:25 Orthogonal polynomial point of view.
21:30 Consider velocity.
23:40 Schroedinger's equation says that the Hamiltonian is giving the time evolution of the wave function. Explicitly given by unitary dynamics.
24:15 Work out the average velocity in terms of the commutator. This gives the operator (h/i d/dx) associated with momentum.
26:40 The connection between position and momentum is on the level of averages. This is based on the Born rule.
28:10 Starting point is our definition of momentum as an operator. Schroedinger equation. Can we rewrite differentiation by x, transform it, so that it looks like multiplication by momentum? 
33:10 The relationship between momentum as an average and an observable vs. momentum as an operator in the quantum world.
37:45 Fourier transform takes us from differentiation to multiplication. Products become convolutions.
39:40 Mathematician's Fourier transform is unitary on Hilbert space, preserves inner products.
41:40 Fourier transform of a derivative. Deriving this using integration by parts.
45:10 We want to intertwine h/i d/dx with multiplication by the parameter. So we need to scale.
47:30 Define a new Fourier transform, which is a scaled Fourier transform.
51:00 Define phi_t(zeta) to be this scaled Fourier transform of the wave function.
52:00 Schroedinger equation. Take scaled Fourier transform of both sides. 
55:30 Defining the convolution.
56:45 Argument that zeta can be interpreted as momentum. Showing that they have the same averages. Zeta parametrizes momentum.
59:00 In free space, the potential disappears and Schroedinger's equation becomes very simple in momentum space. Can use the inverse Fourier transform to go back to position space.
1:00:10 Considering what would be simpler in position space. 
1:00:55 Time evolution is simple if you have eigenfunctions of the Hamiltionian. That is an alternatew way to proceed. But John prefers to think in terms of momentum space because he likes to connect the Hamiltonian to projection in momentum space.
1:02:00 Next time will argue that zeta is momentum and will solve Schroedinger equation in free space and transform back to position space and calculate the dispersal of a wave packet.
1:02:40 Reviewing what we did today. Introduced a scaled Fourier transform.

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