Two and three-dimensional harmonic osciilators

In more than one dimension, there are several different types of Hooke's law forces that can arise. Consider a diatomic molecule AB separated by a distance $r$ with an equilbrium bond length $R_{\rm eq}$. If we consider the bond between them to be approximately harmonic, then there is a Hooke's law force between then of the form

$$F = -k(r-R_{\rm eq})$$

which arises from a potential energy

$$V = {1 \over 2}k(r-R_{\rm eq})^2$$

Note, however, that if the diatomic exists in two dimensions, then $r = \vert{\bf r}\vert$, where ${\bf r}$ is the relative vector ${\bf r}= {\bf r}_{\rm A}-{\bf r}_{\rm B} = (x,y)$. and $r = \sqrt{x^2 + y^2}$. The Hooke's law potential is no longer a sum of terms involving only $x$ and only $y$. This is also true in three dimensions where ${\bf r}= (x,y,z)$ and $r = \sqrt{x^2 + y^2 + z^2}$. This is a somewhat complicated case that we will discuss in the next chapter.

For now, let us take $R_{\rm eq}=0$ as an approximation. Then, in two dimensions, the Hooke's law potential becomes a harmonic potential in $x$ and a harmonic potential in $y$:

$$V(x,y) = {1 \over 2}k\left(x^2 + y^2\right)$$

and the classical energy

$${p_x^2 \over 2\mu} + {p_y^2 \over 2\mu} + {1 \over 2}k\left(x^2 + y^2\right) = E$$

where $\mu$ is known as the reduced mass of the diatomic

$$\mu = {m_{\rm A}m_{\rm B} \over m_{\rm A} + m_{\rm B}}$$

Now we see that the classical energy is a sum of terms involving motion and forces in the $x$ direction and motion and forces in the $y$ direction:

$$E = \varepsilon_x + \varepsilon_y$$

$$\varepsilon_x = {p_x^2 \over 2\mu} + {1 \over 2}kx^2$$

$$\varepsilon_y = {p_y^2 \over 2\mu} + {1 \over 2}ky^2$$

As with the particle in a two-dimensional box, we will need two independent integers $n_x$ and $n_y$ to satisfy the boundary conditions along the $x$ and $y$ directions, and the allowed values of $\varepsilon_x$ and $\varepsilon_y$ become

$$\varepsilon_{n_x} = \left(n_x + {1 \over 2}\right)h\nu \;\;\... ...;\;\;\; \varepsilon_{n_y} = \left(n_y + {1 \over 2}\right)h\nu$$

so that the allowed values of the total energy are

$$E_{n_x n_y} = \left(n_x + n_y + 1\right)h\nu$$

Also, as with the particle in a two-dimensional box, the wave functions are products of harmonic oscillator wave functions in the $x$ and $y$ directions. Some examples are

$$\psi_{00}(x,y) = \left({\alpha \over \pi}\right)^{1/2} e^{-\alpha (x^2+y^2)/2}$$

$$\psi_{10}(x,y) = \left({4\alpha^3 \over \pi}\right)^{1/4} \left({\alpha \over \pi}\right)^{1/4} xe^{-\alpha (x^2+y^2)/2}$$

$$\psi_{11}(x,y) = \left({4\alpha^3 \over \pi}\right)^{1/2} xye^{-\alpha (x^2+y^2)/2}$$

The figure below shows 6 such wave functions, $\psi_{00}(x,y)$, $\psi_{10}(x,y)$, $\psi_{20}(x,y)$, $\psi_{03}(x,y)$, $\psi_{11}(x,y)$, $\psi_{21}(x,y)$:

Figure: 6 Wave functions for a two-dimensional harmonic oscillator.

All of this generalizes straightforwardly to three dimensions, again assuming $R_{\rm eq}=0$. In this case

$$V(x,y,z) = {1 \over 2}k\left(x^2 + y^2 + z^2\right)$$

the classical energy is

$${p_x^2 \over 2\mu} + {p_y^2 \over 2\mu}+ {p_z^2 \over 2\mu} + {1 \over 2}k\left(x^2 + y^2 + z^2\right) = E$$

which is a sum of three terms $\varepsilon_x + \varepsilon_y + \varepsilon_z$, and we need three integers $n_x$, $n_y$, and $n_z$. The allowed values of these three energies will be

$$\varepsilon_{n_x} = \left(n_x + {1 \over 2}\right)h\nu \;\;\... ...;\;\;\; \varepsilon_{n_z} = \left(n_z + {1 \over 2}\right)h\nu$$

and the allowed values of the total energy will be

$$E_{n_x n_y n_z} = \left(n_x + n_y + n_z + {3 \over 2}\right)h\nu$$

Similarly, the wave functions will be products of one-dimensional harmonic oscillator functions in the $x$, $y$, and $z$ directions. Thus, the ground state would be

$$\psi_{000}(x,y,z) = \left(\alpha \over \pi\right)^{3/4} e^{-\alpha (x^2 + y^2 + z^2)/2}$$

and other wave functions can be constructed in a similar manner.