First of all, don’t be afraid of the title, scientists love to use fancy words to describe even a simple topic and why wouldn’t they, it makes physics a lot more appealing.

Well, Quantum Harmonic Oscillator is one such topic. Fancy name with fancy application.

The goal of this article is to give a solid foundation in this topic. I will try my best to make you understand this concept and give you the essence of it rather than only spitting out some random equations and their derivations like most textbooks and online sites do, after-all, this the motto of our website.

So, let’s start….

To understand the Quantum Harmonic Oscillator, firstly we have to have an idea on the classical harmonic oscillator and the basic meaning of Harmonic motion.

Harmonic Motion and Classical Harmonic Oscillator

The harmonic motion refers to the periodic motion or periodic vibration. To understand it better, just imagine a string of a guitar is vibrating, then those motions are symmetrical about a region of equilibrium and distance of each trough and crest from the median line is same when a continuous force of same amount is applied. This type of motions may have a single frequency or amplitude or may be composed of two or more such simple type of motions.

Classical Harmonic Oscillator refers to a system which if displaced from the equilibrium, experiences a force named as restoring force. This force is proportional to the displace x. The whole system can be imagined like an object with mass m is attached with the spring, and when the spring vibrates it created a graph with uniform tough and crest.

If that force is the only force acting on the system then the system is called as Simple Harmonic Oscillator and the motion it shows us is called Simple Harmonic motion (SFM) and the equation of force in that case is

F = -kx  (-∞ ≤ x ≥ ∞ )              (1)

Where, F and X are vectors and k (spring constant) is a positive constant

Here in the above equation, the negative sign refers to the opposite direction of the motion as compared to the direction of the force applied.

Before starting our discussion about the quantum harmonic oscillator, we need to know a couple of more formulae and concepts…

The time period (period after which the motion repeats itself) for the simple harmonic motion is

         T = 2π√(m/k)               (2)

Where, m is the mass of the object undergoing the SHM and k is the spring constant

One thing which we should keep in mind that SHM is not damped motion. A damped motion is such kind of motion which dies after a period of time. In the reality most of the motions and oscillations are damped unless and until they are given some sought of force or energy. Discussing further about the formulation and derivations on the damping motion is beyond the scope of this article.

Another formula related to classical harmonic oscillator is the solution of the differential equation of (1),

                                             x(t) = A cos(ωt + φ)        (3)

where, A is amplitude of the wave, ω is the angular frequency and φ is the phase.

Here, ω = √(k/m) and thus eq. (2) can be written as T= 2π/ ω

Also, to get into the Schrodinger’s equation and related it to harmonic oscillation, we need to get the expression for potential, so let’s get it,

V(x) = -F dx = kx dx = kx²/2 + c    (4)

Where, c is the constant of integration

Note here the use of negative during integration. Here, its use to denote the opposite direction of potential energy stored as compared to the direction of force applied.

Equation 4 is also named as Harmonic oscillator potential.

The value of c is normally dropped in the further calculation for the sake of simplicity and also because it doesn’t change the final value drastically.

Quantum Harmonic Oscillator

Here we go, finally in the fun part….

As you know, Quantum Harmonic Oscillator (QHO) is a very vast topic. In fact, it’s one of the few topics in QM for which an analytic and exact solution is known to us. That makes this topic one of the key systems to study QM.

QHO is the quantum mechanical version of classical harmonic oscillator and that’s why I preferred to give a brief intro to the topic.

So, lets start from forming Schrodinger’s equation,

So, by putting equation 4 in Schrodinger’s equation, we get,

              - (ħ²/2m)(d² 𝜓ₙ/dx²) + kx²/2 𝜓ₙ = E 𝜓ₙ                 (5)

This equation will be used in my future articles on this topic.

Since I am intended to write this article as an introductory article to QHO, so I will not go further deep into the formulation on this topic, rather I like to give you a brief overview of it. In future posts, I will tackle each and every necessary topic under QHO.

Now, here I want to introduce you the applications of Quantum Harmonic Oscillator theory:

1) It can be used to describe the vibration spectra of diatomic molecules and complex modes of vibration in larger molecules

2) It is used in the theory of vibrational heat capacity

3) I can be used to understand the motion of atoms in solid lattice and explains who phonons arise from them. In the periodic and elastic arrangement of atoms or molecules of condensed matter (solids and some special liquids), if any collective excitation happens then that called as a Phonon. Using QHO, we can find various properties regarding that, for example, we can find the exact amount of energy, which if supplied to the QHO lattice then it can push it to the next energy level.

4) It can also, be useful to find the expression for thermodynamic vibrational energy.

In my further articles, I will try to tackle each and every necessary topic which are needed to understand QHO. In part 2 of QHO series, I will explain Schrödinger's equation with dimensionless terms. On the basis of that article, I will try to explain the Asymptotic Solution of QHO. Once they are uploaded, the links will be provided here.

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                                                                 -Ratnadeep Das Choudhury
                                                  Founder and writer of The Dynamic Frequency