Consider a two-dimensional harmonic oscillator with displacements in the \(x\) - and \(y\) -directions, the force constants being the same for each direction (the two bending modes of \(\mathrm{CO}_{2}\) is an example ). Show that the state resulting from the excitation of the oscillator to its first excited state can be regarded as possessing one unit of angular momentum about the \(z\) -axis. Hint. Show that \(\psi(x) \psi(y) \propto \mathrm{e}^{\mathrm{jp}}\)

The problem is to show that the wave function of a two-dimensional harmonic oscillator in its first excited state can be considered as having one unit of angular momentum about the z-axis. To show this, we need to find out the wave function of the oscillator in its first excited state, and then show that \(\psi(x) \psi(y)\) is proportional to \(\mathrm{e}^{\mathrm{jp}}\).

The wave function for the first excited state of one-dimensional harmonic oscillator is \(\psi(x)=Ae^{-\frac{x^2}{2}}x\), where \(A\) is the normalization constant. When extended to two dimensions, the wave function becomes \(\psi(x,y)=Ae^{-\frac{x^2+y^2}{2}}(x+iy) = Ae^{-\frac{x^2+y^2}{2}}z\).

Next, we need to derive the required expression \(\psi(x)\psi(y)\). Substituting the expression for \(\psi(x,y)\) we just obtained, \(\psi(x)\psi(y) = A^2e^{-x^2-y^2}z^2\). Now we need to show this is proportional to \(\mathrm{e}^{\mathrm{jp}}\). In quantum mechanics, the term \(e^{jp}\) represents the action of a rotation operator on a state. For a state with one unit of angular momentum, the action of this operator introduces a phase shift of \(\pi/2\). So, our task is to show that \(\psi(x)\psi(y)\) gives a phase shift of \(\pi/2\). This can be seen if we express \(z = re^{i\theta}\) in polar coordinates. When multiplied by \(z\), the exponential becomes \(e^{2i\theta}\), indicating a phase shift of \(\pi/2\) for each \(\pi/2\) increment in \(\theta\). Resulting in \(\psi(x)\psi(y) = A^2e^{-r^2}r^2e^{2i\theta}\).

As we can see from the above derivation, the expression for \(\psi(x)\psi(y)\) contains the desired factor of \(e^{2i\theta}\), demonstrating the angular momentum of one unit about the z-axis. Hence, the state resulting from the excitation of the oscillator to its first excited state can indeed be regarded as possessing one unit of angular momentum about the z-axis. The proportionality is therefore confirmed.