3.1 Introduction to the Quantum Harmonic Oscillator

The square well has introduced to the key concepts of quantum mechanical systems. However, it is not the most realistic model in the world, although it has some uses. In this chapter, we will study the quantum analogue of the simple harmonic oscillator from classical mechanics, which is a very useful quantum mechanical model.

3.1.1 The Importance of the Harmonic Oscillator

It is not an understatement to say that the simple harmonic oscillator is quite possibly the most important model in classical mechanics. Why is this?

The prototypical example of a system exhibiting simple harmonic motion is a mass on a spring, for which the force acting on the mass is given by Hooke’s law $F=-kx$ . This leads to sinusoidal motion with frequency $\omega=\sqrt{\frac{k}{m}}$ .

The reason why Hooke’s law works so well in so many scenarios is that it is always a good approximation around a stable equilibrium point. To see why this is, we take the Taylor expansion of a potential energy function (it does not matter what shape it is) about a minimum located at $x_{0}$ ,

V(x)=V(x_{0})+\frac{\operatorname{\mathrm{d}}\!V(x_{0})}{\operatorname{\mathrm% {d}}\!x}(x-x_{0})+\frac{1}{2}\frac{\operatorname{\mathrm{d}}\!^{2}V(x_{0})}{% \operatorname{\mathrm{d}}\!x^{2}}(x-x_{0})^{2}+\dots.

(3.1)

The term with the first derivative vanishes since $x_{0}$ is a minimum, so for $x$ close to $x_{0}$ , the potential is approximately given by

V(x)\approx V(x_{0})+\frac{1}{2}\frac{\operatorname{\mathrm{d}}\!^{2}V(x_{0})}% {\operatorname{\mathrm{d}}\!x^{2}}(x-x_{0})^{2}.

(3.2)

This gives rise to a force with the same form as Hooke’s law:

F=-\frac{\operatorname{\mathrm{d}}\!V(x)}{\operatorname{\mathrm{d}}\!x}=-\frac% {\operatorname{\mathrm{d}}\!^{2}V(x_{0})}{\operatorname{\mathrm{d}}\!x^{2}}(x-% x_{0}).

(3.3)

If we apply this to situations in quantum mechanics, we could approximate the behaviour of, for example, vibrations in diatomic molecules that consist of two bound atoms such as hydrogen chloride. These interatomic forces can be modelled classically using the Lennard-Jones potential, but at the equilibrium point we can approximate the potential as parabolic. This means that the quantum harmonic oscillator will be a good approximation of the ground state and the first few excited states of this system!

3.1.2 Setting up the Quantum Harmonic Oscillator

For the classical harmonic oscillator, the potential energy is

V(x)=\frac{1}{2}kx^{2}=\frac{1}{2}m\omega^{2}x^{2}.

(3.4)

For the quantum harmonic oscillator, we will write the potential energy with the natural frequency $\omega$ because it is more helpful than referring to an abstract “spring constant” $k$ .

In quantum mechanics, our position variable $x$ gets promoted to an operator $\hat{x}$ , so the potential energy operator for the quantum harmonic oscillator would be

\hat{V}(x)=\frac{1}{2}m\omega^{2}\hat{x}^{2}.

(3.5)

The Hamiltonian operator for the system is then

\hat{H}=\frac{\hat{p}^{2}}{2m}+\frac{1}{2}m\omega^{2}\hat{x}^{2},

(3.6)

which, when written in the position basis, becomes

\hat{H}=-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial x^{2}}+\frac{1}{2}m% \omega^{2}x^{2}.

(3.7)

The time-dependent Schrodinger equation is therefore

-\frac{\hbar^{2}}{2m}\frac{\partial^{2}\psi(x,t)}{\partial x^{2}}+\frac{1}{2}m% \omega^{2}x^{2}\psi(x,t)=i\hbar\frac{\partial\psi(x,t)}{\partial t}.

(3.8)

Just like in the last chapter, we can separate the time-dependence out to get the time-dependent Schrodinger equation for the QHO:

-\frac{\hbar^{2}}{2m}\frac{\operatorname{\mathrm{d}}\!^{2}u(x)}{\operatorname{% \mathrm{d}}\!x^{2}}+\frac{1}{2}m\omega^{2}x^{2}u(x)=Eu(x),

(3.9)

where the full wavefunction is then given by $\psi(x,t)=u(x)e^{-\frac{iEt}{\hbar}}$ .