For an $n$-fold degenerate energy level, there are $n$ independent wave functions $\psi{1}, \psi{2}, \ldots, \psi_{n}$, each having the same energy eigenvalue $w$ :
\(
\begin{equation}
\hat{H} \psi{1}=w \psi{1}, \quad \hat{H} \psi{2}=w \psi{2}, \quad \ldots, \quad \hat{H} \psi{n}=w \psi{n} \tag{3.76}
\end{equation}
\)
We wish to prove the following important theorem: Every linear combination
\(
\begin{equation}
\phi \equiv c{1} \psi{1}+c{2} \psi{2}+\cdots+c{n} \psi{n} \tag{3.77}
\end{equation}
\)
of the wave functions of a degenerate level with energy eigenvalue $w$ is an eigenfunction of the Hamiltonian operator with eigenvalue $w$. [A linear combination of the functions $\psi{1}, \psi{2}, \ldots, \psi_{n}$ is defined as a function of the form (3.77) where the $c$ 's are constants, some of which might be zero.] To prove this theorem, we must show that $\hat{H} \phi=w \phi$ or
\(
\begin{equation}
\hat{H}\left(c{1} \psi{1}+c{2} \psi{2}+\cdots+c{n} \psi{n}\right)=w\left(c{1} \psi{1}+c{2} \psi{2}+\cdots+c{n} \psi{n}\right) \tag{3.78}
\end{equation}
\)
Since $\hat{H}$ is a linear operator, we can apply Eq. (3.9) $n-1$ times to the left side of (3.78) to get
\(
\hat{H}\left(c{1} \psi{1}+c{2} \psi{2}+\cdots+c{n} \psi{n}\right)=\hat{H}\left(c{1} \psi{1}\right)+\hat{H}\left(c{2} \psi{2}\right)+\cdots+\hat{H}\left(c{n} \psi{n}\right)
\)
Use of Eqs. (3.10) and (3.76) gives
\(
\begin{aligned}
\hat{H}\left(c{1} \psi{1}+c{2} \psi{2}+\cdots+c{n} \psi{n}\right) & =c{1} \hat{H} \psi{1}+c{2} \hat{H} \psi{2}+\cdots+c{n} \hat{H} \psi{n} \
& =c{1} w \psi{1}+c{2} w \psi{2}+\cdots+c{n} w \psi{n} \
\hat{H}\left(c{1} \psi{1}+c{2} \psi{2}+\cdots+c{n} \psi{n}\right) & =w\left(c{1} \psi{1}+c{2} \psi{2}+\cdots+c{n} \psi{n}\right)
\end{aligned}
\)
which completes the proof.
For example, the stationary-state wave functions $\psi{211}, \psi{121}$, and $\psi{112}$ for the particle in a cubic box are degenerate, and the linear combination $c{1} \psi{211}+c{2} \psi{121}+c{3} \psi{112}$ is an eigenfunction of the particle-in-a-cubic-box Hamiltonian with eigenvalue $6 h^{2} / 8 m a^{2}$, the same eigenvalue as for each of $\psi{211}, \psi{121}$, and $\psi{112}$.
Note that the linear combination $c{1} \psi{1}+c{2} \psi{2}$ is not an eigenfunction of $\hat{H}$ if $\psi{1}$ and $\psi{2}$ correspond to different energy eigenvalues $\left(\hat{H} \psi{1}=E{1} \psi{1}\right.$ and $\hat{H} \psi{2}=E{2} \psi{2}$ with $E{1} \neq E{2}$ ).
Since any linear combination of the wave functions corresponding to a degenerate energy level is an eigenfunction of $\hat{H}$ with the same eigenvalue, we can construct an infinite number of different wave functions for any degenerate energy level. Actually, we are only interested in eigenfunctions that are linearly independent. The $n$ functions $f{1}, \ldots, f{n}$ are said to be linearly independent if the equation $c{1} f{1}+\cdots+c{n} f{n}=0$ can only be satisfied with all the constants $c{1}, \ldots, c{n}$ equal to zero. This means that no member of a set of linearly independent functions can be expressed as a linear combination of the remaining members. For example, the functions $f{1}=3 x, f{2}=5 x^{2}-x$, $f{3}=x^{2}$ are not linearly independent, since $f{2}=5 f{3}-\frac{1}{3} f{1}$. The functions $g{1}=1$, $g{2}=x, g_{3}=x^{2}$ are linearly independent, since none of them can be written as a linear combination of the other two.
The degree of degeneracy of an energy level is equal to the number of linearly independent wave functions corresponding to that value of the energy. The one-dimensional free-particle wave functions (2.30) are linear combinations of two linearly independent functions that are each an eigenfunction with the same energy eigenvalue $E$. Thus each such energy eigenvalue (except $E=0$ ) is doubly degenerate (meaning that the degree of degeneracy is two).