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Free Particle Propagator from the Path Integral Formulation

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Free Particle Kernel from the Path Integral Formulation
In my post on the deriving the non-relativistic path integral, I motivated our discussion by the question of finding the transition probability amplitude of a particle existing in a position eigenstate, $|x_a\rangle$, to a new position eigenstate, $|x_b\rangle$. The power of the path integral, however, comes not in finding the transition probability amplitude, but in finding the Green's function (or "kernel") that describes the propagation of some arbitrary quantum mechanical system under some arbitrary Hamiltonian to a different state (by state, I am actually referring to the notion of a "state" in quantum mechanics). I must reiterate the last sentence because it is important - once we find the propagator, all we need to do is specify the initial state (the initial conditions). The boundary conditions are already taken care of - they are tucked away in the Hermitian nature of the operators that we deal …

Derivation of Electronic Transition Rate

We have a particle that is in a vacuum, meaning that the Fock state representing the photon is in its ground state. In making our calculations, we assume that the photonic ground state is the asymptotic state and that the interaction occurs for only a finite amount of time (this is the basis of perturbative quantum field theory). We represent the combined state of the photon and the electron's initial state as a direct product, such that $$|\Psi \rangle = |p_n^{(i)} \rangle \bigotimes |0\rangle = |p_n^{(i)} \rangle |0\rangle$$ Similarly, we represent the final combined state in its asymptotic limit as $$|\Psi \rangle = |p_n^{(f)} \rangle \bigotimes |0\rangle = |p_n^{(f)} \rangle |0\rangle$$ Great, Zetilli represents the quantized free electromagnetic field as a discrete sum of momenta over a large volume (the difference between Zetilli and many other quantum texts being that they let the spectrum be discrete [confining the photons to a finite volume], whereas books on QFT an…

Derivation of the Non-Relativistic Path Integral

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The path integral formulation of both relativistic and non-relativistic quantum mechanics is arguably one of the most beautiful and far-reaching ideas of physics post-1950's. It is, however, one of the least intuitive. By introducing the path integral in the non-relativistic case, where canonical quantization is done on position and momentum, we introduce the idea that a particle moving between any two points takes an infinite number of paths during its journey. In the relativistic case, where canonical quantization is done on fields, we introduce the idea that fields themselves occupy an infinite number of classical field configurations in a field's journey between any two states of the field (say, the ground state configuration to some excitation). The derivation for both is surprisingly similar, as simply switching $\hat{x}$ for $\hat{\phi}$ (where, in the most simple case, $\hat{\phi}$ is a scalar field) and $\hat{p}$ for $\partial_t \hat{\phi}$ supplemented with the cor…

Emmy Noether's Theorem and Energy/Momentum Conservation

Perhaps one of the most important consequences of classical field theory is a definitive answer to the question "why is momentum and energy conserved"? Indeed, even during your first days of being a wee little physics student, you took energy and momentum conservation as a given - an axiom of the theories underlying physics. Momentum and energy conservation, however, come from a much more deep connection between symmetries and physical laws. In this article, we will try to give the reader an understanding of where these symmetries come from and how they birth energy and momentum conservation. Let us first introduce the definition of the Lagrangian density $$S=\int d^4 x L(\phi, \partial_\mu \phi)$$ where $S$ is the action and the argument of the integral is the "Lagrangian density", which depends upon the scalar field, $\phi$, and its spatio-temporal derivatives, $\partial_\mu \phi$. The integration goes over time and three spatial dimensions. Minimizing the act…

How Do Magnets Work? Spin, Paramegnetism, Ferromagnetism and the Ising Model

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Magnets are everywhere - they play a crucial role in correcting the trajectories of charged particles in particle accelerators, they act as very great emergency breaking systems, they encode your credit card information whenever you swipe to make a purchase, and you may even have a few sitting on your refrigerator. Unfortunately, however, even many undergraduate physics majors never get to fully appreciate the origin of this strange phenomena that we call "permanent magnetism". Indeed, not every metal can become a permanent magnet. As it turns out, magnetic materials tend to require that the individual atoms that make up the metal be sufficiently close to each other to experience an "interaction potential" whose origin is fundamentally quantum mechanical. I will attempt to make this discussion tailored toward the intermediate undergraduate physics major or the even the advanced high school student. Why so? Well, the full beauty of permanent magnetism is best d…

An Explanation of Paramagnetism from the Framework of Statistical Mechanics

Here, we discuss a beautifully straight-forward way of calculating the effects of an incident magnetic field on a substance composed of atoms with a net spin of $s=\frac{1}{2}$ from the framework of statistical mechanics. As a side note on notation, most derivations will use the notation for the magnetic field being as being $\mathbf{H}$; however, I am not fond of this misuse of notation, for the "true" magnetic field is $\mathbf{B}$, $\mathbf{H}$ being merely a measure of the magnetic field caused by free currents, called the auxiliary field (which does not have the same units of $\mathbf{B}$). Now, consider a system composed of a of weakly interacting atoms spin-1/2 atoms (i.e. the spin-spin interactions of each atom can be neglected) at some temperature T. Then we can consider the probability of a single atom occupying a particular energy state to be described by the Boltzmann distribution: $$p(E_r,T)=\frac{\exp(-\beta E_r)}{\sum_r \exp(-\beta E_r)}$$ which we can …

The Heisenberg Uncertainty Principle

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