Lecture 9 | The Theoretical Minimum

The paragraph discusses the importance of entanglement in quantum mechanics, noting it as the heart of the subject. It emphasizes the shift from studying the foundations to examining examples within the field. The speaker also touches on the basic empirical observations in quantum mechanics, using the example of an apparatus that measures spin, and discusses the concept of a system involving a singlet and a triplet state with a photon.

This section delves into the concept of superposition in quantum mechanics, illustrating how states of particles, including the number of particles, can be superposed. It explains the transition of a system from an unentangled to an entangled state upon photon emission. The paragraph also explores the mathematical interlude involving Fourier analysis and the Dirac Delta function, which are essential tools for understanding particle motion in quantum mechanics.

The paragraph focuses on Fourier analysis, explaining how a wide class of functions can be expressed as sums or integrals of plane waves. It details the process of decomposing a function into plane waves and introduces the concept of wave amplitude. The discussion also covers the Fourier transform theorem, which allows for the reconstruction of a function from its wave components.

This part explores the properties of the Delta function, a key concept in quantum mechanics. It discusses the intuitive definition of the Delta function as an infinite, narrow function with an area of one. The paragraph outlines the properties of the Delta function, including its behavior under integration and its use in representing the operation on a function. It also presents a theorem that provides a representation of the Delta function in terms of Fourier transforms.

The speaker introduces the concept of describing the state of a quantum mechanical particle using an orthonormal basis of states. It contrasts this with classical mechanics, where both position and momentum are known. In quantum mechanics, due to the Heisenberg uncertainty principle, one can only know either position or momentum, but not both simultaneously. The paragraph also discusses the eigenvectors of position and their use in constructing wavefunctions.

This section discusses the duality in quantum mechanics where a particle can be described by either its position or momentum, but not both. It explains the concept of wavefunctions as projections of a state onto a state of definite position and how these wavefunctions can be used to calculate probabilities. The paragraph also touches on the normalization of quantum states and the role of the Dirac Delta function in continuous variable systems.

The paragraph explores the concept of momentum in quantum mechanics, introducing the momentum operator and its action on wavefunctions. It discusses the eigenvectors of the momentum operator and their relationship to the observable values of momentum. The speaker also explains the mathematical form of these eigenvectors and how they relate to the wavelength of a particle, hinting at the de Broglie wavelength concept.

This section focuses on the wavefunctions of particles with definite momentum, noting their oscillatory nature and the fact that they are delocalized. It discusses the concept of a momentum space wavefunction and how it is related to the probability amplitude for finding different momenta. The paragraph also explains how to calculate the momentum space wavefunction using the Fourier transform of the position space wavefunction.

The speaker touches on the uncertainty principle, relating it to Fourier transforms and the behavior of conjugate variables like position and momentum. It also introduces the concept of Hamiltonians in quantum mechanics, which govern the time evolution of state vectors and are related to the classical idea of energy. The paragraph sets the stage for a deeper exploration of quantum mechanical evolution and the Hamiltonian's role.

This part discusses the time-dependent Schrödinger equation and its implications for the evolution of quantum states. It explores simple Hamiltonians and their effects on wavefunctions, illustrating how different Hamiltonians can lead to different behaviors of quantum states over time. The speaker also draws parallels between quantum and classical mechanics, particularly in the context of the Hamiltonian's role in system evolution.

The paragraph concludes with a discussion on the Schrödinger equation for a non-relativistic particle, which is a more complex scenario than the simple Hamiltonian cases previously discussed. It highlights that for such particles, the wavefunction does not maintain its shape and tends to disperse over time, contrasting with the rigid movement seen in the simpler models. The speaker also notes that this equation is a type of wave equation, with the wavefunction representing the quantum state of the particle.