# Quantum Books on Harmonic Oscillators

1. Aug 14, 2015

### snatchingthepi

I was thinking about harmonic oscillators last night when explaining the non-zero ground state energy of the quantum version to a non-scientific friend and the conversation pushed my curiosity a little so I'm wondering if anyone happens to know about any books solely about harmonic oscillators. Ideally I'd be looking for something that covers the subject in extreme detail and covers both classical and quantum versions. Thanks all.

2. Aug 14, 2015

### DEvens

I don't think you can usefully fill a book solely with material about SHO. You can work out the classical one in great detail in 5 pages. Then you can work out the quantum one also in great detail in 10 pages. Then you are pretty much done as far as solely about the SHO.

If you want to start doing variations and applications of those variation you can get a lot more material. For example, SHO and the black body problem. Or SHO as a model for the behaviour of gas molecules. Or SHO as a model for the behaviour of atoms in a crystal. Or small perturbations to the SHO as a way to study interactions. Or one I did as an exercise during my PhD, SHO on the null plane to understand the null gauge often used in string theory. And so on. But this is far beyond solely the SHO. You might want to pick your topic and branch out.

3. Aug 14, 2015

### snatchingthepi

Well I was thinking about oscillators in general and not just the simple harmonic oscillator. I'd be looking for something like a compendium or "handbook to" harmonic oscillators as a whole at an advanced level. But your observations are useful. Maybe I should look at specific applications or variations instead of looking for a complete collection. I'm just wondering if such a thing is out there.

4. Aug 15, 2015

### vanhees71

Indeed, you might not find a book exclusively treating only the harmonic oscillator, but almost any QT textbook will cover this topic. It's the one simple model that can be solved completely and analytically for all kinds of problems, and it's a very fruitful endeaver to do this. The best thing is, you can do it yourself, and it's big fun: I'd suggest the following sequence of study:

Harmonic oscillator in 1 dimension
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(1) starting from the Heisenberg algebra for position and momentum, define the "ladder operators" for the energy eigen values and derive the complete spectrum and energy eigenbasis in representation free form. Look at this in the Schrödinger and the Heisenberg picture of time evolution.

(2) Work out the wave functions in terms of the position and momentum representations. You'll learn a lot about Hermite polynomials and techniques to handle orthogonal polynomials in general (Rodrigues formula, integral formulae etc.)

(3) Try to get the propagator in the position representation in two different ways: (a) use the just elaborated energy eigen functions and derive the corresponding series. With help of the integral representation of the eigenfunction you can even do the sum analytically, but that's tricky and you should look for it in textbooks and online material; (b) use the Heisenberg picture and solve the operator equations of motion for the position operators and then evaluate the propagator as
$$U(t,x;t',x')=\langle x,t|x',t' \rangle.$$

(4) Derive coherent and squeezed states as the eigen functions of the annihilation operator.

Harmonic oscillator in 2 and 3 dimensions
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(1) Show that the harmonic oscillator in 2 and 3 dimensions in cartesian coordinates is simply given by the three non-interacting 1d harmonic oscillators; i.e., the harmonic oscillator has a pretty large symmetry group which goes beyond the space-time symmetries (dynamical symmetry).
(2) Using the annihilation and creation operators to show that the symmetry groups are SU(2) and SU(3) respectively (for an n-dimensional oscillator it's SU(n)). This gives a great way to derive all the irreducible representations of these groups (see Sakurai, modern quantum mechanics).
(3) Solve the energy eigenvalue problem by separation of the time-independent Schrödinger equation in polar and spherical coordinates (for the 2D and 3D case, respectively).

5. Aug 17, 2015

### SredniVashtar

So, you need a book on vibrations.
French's "Vibration and Waves" is in my opinion the best so far. The material is very clearly and logically laid out, and it is also short. It's only about classical oscillators though.
Another great classic, who might at first seem convoluted in its exposition is Crawford's fourth volume of the Berkeley Physics course. Still mostly classical, with a hint of quantum physics in the appendices.
But the book you might be looking for is "Waves and Oscillations - A Prelude to Quantum Mechanics", by Walter Fox Smith. It uses the bra and ket formalism for classical oscillators, thus making it easier to draw parallels between the two worlds.

6. Aug 24, 2015

### snatchingthepi

Thanks to all of you.