Having Fun With the Quantum Harmonic Oscillator: Non-Trivial Exercises With Detailed Solutions
Abstract
The harmonic oscillator is one of the most important elementary systems in both classical and quantum physics. The present eBook is a – hopefully successful – attempt to present some of the many important aspects of the one-dimensional quantum harmonic oscillator (QHO), through a series of non-trivial exercises, which are solved in detail. In each solution, all the steps are elaborated and all the equations are analyzed, so that the reader can follow it without difficulty. We work in the Schrödinger picture and, without completely neglecting the analytical treatment, we choose a rather algebraically-oriented approach to the problems, as this provides a deeper and richer insight to the nature of the QHO. The reader is assumed to have a basic knowledge of the postulates and the mathematical formalism of quantum mechanics, including the Dirac notation and the ladder operator method of the QHO.
The purpose of this work is to introduce, in a simple, intuitive way, the coherent and squeezed states of the quantum harmonic oscillator (QHO), through a series of exercises, which are solved in detail.
Starting from the application of a spatial translation to the ground state of the QHO, we introduce the spatial and momentum translations, focusing on their application to the QHO, which leads us to the displacement operator.
Next, we introduce the coherent states and examine their basic aspects.
We then proceed to give a simple and purely intuitive introduction to the squeezed states and we conclude by identifying the coherent states as states of minimum energy expectation value compared to the respective squeezed states.
The reader is assumed to have a basic knowledge of the postulates and the mathematical formalism of quantum mechanics, including the Dirac notation and the ladder operator method of the QHO.
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