4.2 Degeneracy

4.2.1 Energy Eigenstates of the 3D Isotropic QHO

As derived in the last section, the energy eigenvalues of the 3D isotropic quantum harmonic oscillator are

E_{n_{x}n_{y}n_{z}}=\hbar\omega\left(n_{x}+n_{y}+n_{z}+\frac{3}{2}\right),

(4.23)

and the spatial parts of the eigenfunctions are

	$\displaystyle u_{n_{x}n_{y}n_{z}}(x,y,z)$	$\displaystyle=u_{n_{x}}(x)u_{n_{y}}(y)u_{n_{z}}(z)$		(4.24)
		$\displaystyle=N_{n_{x}}N_{n_{y}}N_{n_{z}}h_{n_{x}}\left(\frac{x}{x_{0}}\right)% h_{n_{y}}\left(\frac{y}{x_{0}}\right)h_{n_{z}}\left(\frac{z}{x_{0}}\right)e^{-% \frac{m\omega}{2\hbar}(x^{2}+y^{2}+z^{2})},$		(4.25)

where we recall the natural length scale $x_{0}=\sqrt{\frac{\hbar}{m\omega}}$ . The full time-dependent eigenfunctions will then simply be

	$\displaystyle\psi_{n_{x}n_{y}n_{z}}(x,y,z,t)$	$\displaystyle=u_{n_{x}n_{y}n_{z}}(x,y,z)e^{-\frac{iE_{n_{x}n_{y}n_{z}}t}{\hbar}}$		(4.26)
		$\displaystyle=u_{n_{x}}(x)e^{-\frac{iE_{n_{x}}}{\hbar}}u_{n_{y}}(y)e^{-\frac{% iE_{n_{y}}}{\hbar}}u_{n_{z}}(z)e^{-\frac{iE_{n_{z}}}{\hbar}},$		(4.27)

where in the last line we have shown explicitly that the full wavefunction really is the product of the three individual wavefunctions for each direction using $E_{n_{x}n_{y}n_{z}}=E_{n_{x}}+E_{n_{y}}+E_{n_{z}}$ .

The ground state energy for the 3D QHO is found when $n_{x}=n_{y}=n_{z}=0$ , which gives

E_{000}=\hbar\omega\left(0+0+0+\frac{3}{2}\right)=\frac{3}{2}\hbar\omega.

(4.28)

This is three times as much energy as for a 1D oscillator. Since the quantum numbers $n_{x}$ , $n_{y}$ , and $n_{z}$ can vary independently, the spacing between energy levels is still $\hbar\omega$ just like in the one-dimensional case.

The ground state wavefunction looks like

\psi_{000}(x,y,z,t)=N_{000}e^{-\frac{m\omega}{2\hbar}(x^{2}+y^{2}+z^{2})}e^{-% \frac{3}{2}i\omega t}.

(4.29)

In dirac notation, we denote the eigenstates by $\lvert n_{x}n_{y}n_{z}\rangle$ , so the ground state would be denoted $\lvert 000\rangle$ .

The first excited state is the state with the second lowest energy, which would be $E=\frac{5}{2}\hbar\omega$ for the 3D isotropic oscillator. However, from equation 4.22, we can see that there are several states that have this energy! They are:

E_{100}=E_{010}=E_{001}=\frac{5}{2}\hbar\omega.

(4.30)

This was not possible in for the one-dimensional oscillator, but becomes possible now because of the extra degrees of freedom and because of the symmetry of the situation.

The three eigenstates that have this eigenvalue have three distinct wavefunctions, given by

$\displaystyle\psi_{100}(x,y,z,t)$	$\displaystyle=N_{100}xe^{-\frac{m\omega}{2\hbar}(x^{2}+y^{2}+z^{2})}e^{-\frac{% 5}{2}i\omega t}$	(4.31)
$\displaystyle\psi_{010}(x,y,z,t)$	$\displaystyle=N_{010}ye^{-\frac{m\omega}{2\hbar}(x^{2}+y^{2}+z^{2})}e^{-\frac{% 5}{2}i\omega t}$	(4.32)
$\displaystyle\psi_{001}(x,y,z,t)$	$\displaystyle=N_{001}ze^{-\frac{m\omega}{2\hbar}(x^{2}+y^{2}+z^{2})}e^{-\frac{% 5}{2}i\omega t}$	(4.33)

These wavefunctions, which correspond to the states $\lvert 100\rangle$ , $\lvert 010\rangle$ , and $\lvert 001\rangle$ , are still orthonormal and so they represent physically distinct states, yet they have the same energy. These are known as degenerate states.

Definition 4.1.

Degenerate states are states that share the same eigenvalue of a Hermitian operator.

The number of states with the same eigenvalue is called the degeneracy of the eigenvalue. For example, the first excited state of the 3D isotropic quantum harmonic oscillator can correspond to three different eigenstates, so we say that the first excited state has a degeneracy of 3.