How many wave functions can be represented as a superposition in a simple harmonic oscillator?

Question

I'm teaching myself about QM, but there are something really puzzling me about the simple harmonic oscillator: $$H=\frac{p^2}{2m}+\frac{m\omega^2x^2}{2}.$$

I've learned how to use ladder operators to obtain the eigenvalues of this oscillator. Also, I'm able to write out the eigenstates in the form of position space wave functions: $$\langle x'|n\rangle=\left(\frac{1}{\pi^{1/4}\sqrt{2^n n!}}\right)\left(\frac{1}{x^{n+1/2}_0}\right)\left(x'-x^2_0\frac{d}{dx'}\right)^n\exp\left[-\frac{1}{2}\left(\frac{x'}{x_0}\right)^2\right],$$ where $$x_0\equiv \sqrt{\frac{\hbar}{m\omega}}.$$ So the initial wave function must be a superposition of these eigenfunctions.

However, given an arbitrary normalized wave function $\langle x'|\alpha\rangle$ which is not necessarily a proper superposition, I can use $|\alpha\rangle$ as an initial state and make it evolve according to the Schrödinger equation: $$\langle x'|i\hbar\frac{\partial}{\partial t}|\alpha;t\rangle = \langle x'|H|\alpha;t\rangle,$$ which seems to make sense.

So my question is:

1. Can any normalized wave function be represented as a superposition of the eigenfunctions?
2. If not, what would happen if I set the initial state to a wave function that is not a superposition of the eigenfunctions?

Also, there is another question which might be related:

3. The numbers of eigenfunctions for $x$ and $p$ are obviously uncountably infinite. But how could it be that this number is countably infinite for $H$?

Answer 1

1. The states $\{| n \rangle \}$ form a complete basis, so at any time you can expand any state $|\alpha(t)\rangle $ as a linear combination of $| n \rangle$,

$$ | \alpha(t) \rangle = \sum_n c_n(t)| n \rangle \tag{1} $$

Now is a matter of finding the coefficients $c_n$. To do that note that

\begin{eqnarray} i\hbar \frac{{\rm d}}{{\rm d}t}| \alpha(t) \rangle &=& i\hbar\sum_n \dot{c}_n(t) | n \rangle \\ H| \alpha(t) \rangle &=& \sum_n c_n(t) H | n \rangle = \sum_n c_n(t) E_n | n \rangle \\ \Rightarrow i\hbar\sum_n \dot{c}_n(t) | n \rangle &=& \sum_n c_n(t) E_n | n \rangle \end{eqnarray}

with $E_n = \hbar \omega(n + 1/2)$. Multiplying both sides by a state $| m\rangle$, and recalling that $\langle m | n \rangle = \delta_{nm}$

$$ i\hbar \dot{c}_m(t) = c_n(t)E_n $$

whose solution is

$$ c_n(t) = c_n(0)e^{-iE_n t/\hbar} \tag{2} $$

The coefficients $c_n(0)$ are easily obtained from Eq. (1):

$$ \langle m | \alpha(0)\rangle = \sum_n c_n(0)\langle m | n\rangle = c_m(0) $$

That is

$$ c_n(0) = \langle n | \alpha(0)\rangle \tag{3} $$

so, the evolution of any state $|\alpha(t)\rangle$ can be written as Eq. (1), where the coefficients $c_n(t)$ evolve according to Eq. (2) with initial conditions given by Eq. (3)

2. Since the set $\{ | n \rangle \}$ you can always write any state as a linear combination of eigenstates of $H$

3. Please follow this link, but intuitively speaking, the potential $V(x) = m\omega^2 x^2/2$ has infinitely many bound states, this means that no matter how large the energy of a particle is, you can always contain it with $V$. In this case the states are also countable, indeed, you can label them with a single integer $n$

Answer 2

I'll address 3. The word "basis" is often defined to only allow linear combinations of finitely many basis elements. This is the definition in the famous theorem stating all bases have the same cardinality. Hilbert spaces are unusual in that any finite-norm linear combination of orthogonal elements will define a unique element of the space; we say it's metric-complete, which doesn't apply to arbitrary inner product spaces. (This is somewhat analogous to the fact that, although $\mathbb{R}$ contains the limits of all its Cauchy sequences, general Cauchy sequences in $\mathbb{Q}$ converge to arbitrary real numbers.) A "basis" of a Hilbert space is defined more generally. As a result, a Hilbert space's bases can vary in cardinality.