1.4: Basic Quantum Models

The first step in solving a quantum mechanical problem is to write a Hamiltonian describing the particles and interactions that give rise to the molecular properties of interest. Formulating a Hamiltonian in a simple but effective manner is a valuable skill and can be built on an understanding of simpler model problems. This section summarizes the time-independent Hamiltonians and wavefunctions that emerge for common models for quantum mechanical objects. These models form the starting point for describing the motion of electrons and the translational, rotational, and vibrational motions of molecules and are the basis for models we will explore further. Understanding them is key in developing intuition about more complex problems.

Waves and free particles

Waves form the basis for our quantum mechanical description of matter. Waves describe the oscillatory amplitude of matter and fields in time and space, taking a number of forms. The simplest form we will use is plane waves, which can be written as \[\psi(\mathbf{r}, t)=\mathbf{A} \exp [\mathrm{i} \mathbf{k} \cdot \mathbf{r}-\mathrm{i} \omega t]\]

The angular frequency \(\omega\) describes the oscillations in time and is related to the number of cycles per second through \(v=\omega / 2 \pi\). The wave amplitude also varies in space as determined by the wavevector \(\mathbf{k}\), where the number of cycles per unit distance (wavelength) is \(\lambda=\omega / k\). Thus, the wave propagates in time and space along a direction \(\mathbf{k}\) with a vector amplitude \(\mathbf{A}\) and a phase velocity \(v_{\phi}=v \lambda\)

For a free particle of mass \(m\) in one dimension, the Hamiltonian only reflects the kinetic energy of the particle energy \[\hat{H}=\hat{T}=\frac{\hat{p}_{x}^{2}}{2 m}=-\frac{\hbar^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}}\]

Solving the time-independent Schrödinger equation leads to plane-wave eigenstates of the form: \[\psi_{k}(x)=\frac{1}{\sqrt{2 \pi \hbar}} \exp \left(\frac{\pm i p x}{\hbar}\right)=\frac{1}{\sqrt{2 \pi}} \exp (\pm i k x)  \] 

where \(k=p / \hbar\). Inserting this expression into the TISE, \(\hat{H} \Psi=E \Psi\), we find that \(k\), the wavenumber, is given by \[k=\sqrt{\frac{2 m E}{\hbar^{2}}}\]

The energy eigenvalues are given by \[E_{k}=\frac{\hbar^{2} k^{2}}{2 m}\]

We added a subscript \(k\) to the expressions for the eigenfunctions and eigenenergies to emphasize that \(k\) is used as a label or index for an eigenstate in the position basis in addition to being a real number.

Free particle plane waves \(\psi_{k}(x)\) are the eigenstates describing a free particle or a particle under the influence of a constant potential, but they form a complete and continuous basis set with which to describe the spatial characteristics of any wavefunction. Note that the eigenfunctions, eq. (1.4.3), are oscillatory over all space. Thus, describing a plane wave allows one to exactly specify the wavevector or momentum of the particle, but one cannot localize it to any point in space. In this form, the free particle is not observable because its wavefunction extends infinitely and cannot be normalized. An observation, however, taking an expectation value of a Hermitian operator will collapse this wavefunction to yield an average momentum of the particle with a corresponding uncertainty relationship to its position.

Particle-in-a-Box

The minimal model for translational motion of a particle that is confined in space is given by the particle-in-a-box. For the case of a particle confined in one dimension in a box of length \(L\) with impenetrable walls, we define the Hamiltonian as \[\begin{align} \hat{H} &=\frac{\hat{p}^{2}}{2 m}+V(x) \\ V(x) &= \begin{cases}0 & 0<x<L_{x} \\ \infty & \text { otherwise }\end{cases} \end{align} \]

The boundary conditions require that the particle cannot have any probability of being within the wall, so the wavefunction should vanish at \(x=0\) and \(L_{x}\), as with standing waves. Solutions to the Schrödinger equation with these boundary conditions lead to normalized eigenfunctions of the form \[\psi_n=\sqrt{\frac{2}{L}} \sin \frac{n \pi x}{L} \qquad \quad  n=1,2,3 \ldots \]

Here values of \(n\) are the integer quantum numbers that describe the harmonics of the fundamental frequency \(\pi / L\) whose oscillations will fit into the box while obeying the boundary conditions. We see that any state of the particle-in-a-box can be expressed in a Fourier series. On inserting eq. (1.4.8) into the time-independent Schrödinger equation (TISE) we find the energy eigenvalues \[E_{n}=\frac{n^{2} \pi^{2} \hbar^{2}}{2 m L^{2}} \]

Recognizing that the wavevector of these waves is \(k_{n}=(n \pi / L)\), we see that the energies correspond to free-particle energies \(E_{n}=\hbar^{2} k_{n}^{2} / 2 m\), although now with limits on what values \(k\) can assume. Note that the spacing between adjacent energy levels grows as \(n(n+1)\).

This model is readily extended to a three-dimensional box by separating the box into \(x, y\), and \(z\) coordinates. Then \[\hat{H}=\hat{H}_{x}+\hat{H}_{y}+\hat{H}_{z}\]

in which each term is specified as eq. (1.4.6). Since \(\hat{H}_{x}, \hat{H}_{y}\), and \(\hat{H}_{z}\) commute, each dimension is separable from the others. Then we find \[\begin{align} \psi(x, y, z) &=\psi_{x} \psi_{y} \psi_{z} \\[3pt]  E_{x, y, z} &=E_{x}+E_{y}+E_{z} \end{align}\]

which follow the definitions given in eq. (1.4.8) and eq. (1.4.9) above. The state of the system is now specified by three quantum numbers with positive integer values: \(n_{x}, n_{y}, n_{z}=1,2,3 \ldots\)

Figure \(\PageIndex{1}\). Particle-in-a-box potential wavefunctions that are plotted superimposed on their corresponding energy levels.

Figure \(\PageIndex{2}\). Harmonic oscillator potential showing wavefunctions that are superimposed on their corresponding energy levels. 

Harmonic Oscillator

The harmonic oscillator Hamiltonian refers to a particle confined to a parabolic, or harmonic, potential. We will use it to represent vibrational motion in molecules, but it also becomes a general framework for understanding all bosons. For a classical particle bound in a one-dimensional potential, the potential near the minimum \(x_{0}\) can be expanded as \[V(x)=V\left(x_{0}\right)+\left(\frac{\partial V}{\partial x}\right)_{x=x_{0}}\left(x-x_{0}\right)+\frac{1}{2}\left(\frac{\partial V^{2}}{\partial x^{2}}\right)_{x=x_{0}}\left(x-x_{0}\right)^{2}+\cdots\]

Setting \(x_{0}\) to 0 , the leading term with a dependence on \(x\) is the second-order (harmonic) term \(V=\) \(-\kappa x^{2} / 2\), where the force constant \(\kappa=-\left(\partial^{2} V / \partial x^{2}\right)_{x=0}\). The classical Hamiltonian for a particle of mass \(m\) confined to this potential is \[H=\frac{p^{2}}{2 m}+\frac{1}{2} \kappa x^{2} \]

Noting that the force constant and frequency of oscillation for a harmonic oscillator are related by \(\kappa=m \omega_{0}^{2}\), we can substitute operators for \(p\) and \(x\) in eq. (1.4.14) to obtain the quantum Hamiltonian \[\hat{H}=-\frac{1}{2} \frac{\hbar^{2}}{m} \frac{\partial^{2}}{\partial x^{2}}+\frac{1}{2} m \omega_{0}^{2} \hat{x}^{2}\]

We will also make use of reduced, or mass-weighted, coordinates defined as \[\begin{align} \begin{aligned}
\underset{^\sim}{p} &=\sqrt{\frac{1}{m \hbar \kern.05em \omega_0 }} \hat{p} \\[4pt]
\underset{^\sim}{x} &=\sqrt{\frac{m \omega_{0}}{\hbar}} \hat{x}
\end{aligned} \end{align}\]

for which the Hamiltonian can be written as \[\hat{H}=\frac{1}{2} \hbar \kern.05em \omega_{0} \left( {\underset{^\sim}{p}}^{2}+{\underset{^\sim}{x}}^{2} \right)\]

In this form, one can readily see that the energy—the expectation value of the Hamiltonian—is equally split between the kinetic and potential energy.

The energy eigenvalues for the harmonic oscillator Hamiltonian are discrete equally spaced in units of the vibrational quantum \(\hbar \kern.05em \omega_{0}\) above the zero-point energy \(\hbar \kern.05em \omega_{0} / 2\). \[E_{n}=\hbar \kern.05em \omega_{0}\left(n+\frac{1}{2}\right) \quad n=0,1,2 \ldots\]

where \(n\) is the vibrational quantum number. The corresponding eigenstates in the position representation are expressed in terms of Hermite polynomials, \(\mathcal{H}_{n}(x)\), \[\psi_{n}(x)=\sqrt{\frac{\alpha \vphantom{\times} }{2^{n} \sqrt{\pi} n !}} e^{-\alpha^{2} x^{2} / 2} \, \mathcal{H}_{n}(\alpha x) \quad \quad  n=0,1,2 \ldots\]

where \(\alpha=\sqrt{m \omega_{0} / \hbar}\) and the Hermite polynomials are obtained from \[\mathcal{H}_{n}(x)=(-1)^{n} e^{x^{2}} \frac{d^{n}}{d x^{n}} e^{-x^{2}}\]

The ground state wavefunction is a Gaussian function: \(\psi_{0}(x)=\left(m \omega_{0} / \pi \hbar\right)^{1 / 4} \exp \left(-m \omega_{0} x^{2} / 2 \hbar\right)\).