Wave Functions and Hilbert Spaces in Quantum Mechanics
Introduction
In classical mechanics, the main concern is finding the position function $\mathbf{r}(t)$ that satisfies Newton’s second law $\mathbf{F} = m \mathbf{a}$ under given conditions. For example, the time-dependent position of an object launched with initial velocity $\mathbf{v}_{0} = (v_{0}\cos\theta, v_{0}\sin\theta)$ in two-dimensional space can be found by solving the following system of equations, and this is called projectile motion.
$$ \begin{align*} m \dfrac{d^{2}\mathbf{r}}{dt^{2}} &= -mg\hat{\mathbf{y}} \\ \mathbf{v}(0) &= (v_{0}\cos\theta, v_{0}\sin\theta) \end{align*} $$
Solving the above equation, we find that the function representing the object’s position is as follows.
$$ \mathbf{r}(t) = -\dfrac{1}{2}gt^{2}\hat{\mathbf{y}} + t \mathbf{v}_{0} $$
Once we know the position, we can find the velocity $\mathbf{v} = \dfrac{d\mathbf{r}}{dt}$, and once we know the velocity, we can find the momentum $\mathbf{p} = m\mathbf{v}$ and the kinetic energy $T = \dfrac{1}{2}mv^{2}$. That is, the reason finding $\mathbf{r}(t)$ is important is that it lets us know the physical information about the object.
Similarly, what we are interested in in quantum mechanics is the wave function of a particle. Whereas classical mechanics solves Newton’s second law to analyze the motion of an object, quantum mechanics solves the Schrödinger equation. This is because this function contains the physical information about the object.
Definition
The solution of the following Schrödinger equation is called the wave function.
$$ \begin{align*} \i\hbar\frac{ \partial \psi}{ \partial t} &= \left(-\frac{\hbar^{2}}{2m}\frac{ \partial ^{2} }{\partial x^{2} }+V\right)\psi & (\text{1-dim}) \\[1em] \i\hbar\frac{ \partial \psi}{ \partial t} &= \left(-\frac{\hbar^{2}}{2m}\nabla^{2}+V\right)\psi & (\text{3-dim}) \end{align*} $$
Here $\hbar$ is the reduced Planck constant, $V$ is the potential, and $\nabla^{2}$ is the Laplacian.
Explanation
The notations mainly used for the wave function are as follows.
$$ \Psi(x, t),\quad \psi(x, t),\quad \phi(x, t),\quad u(x) $$
At Freshrimp Sushi House, the wave function with respect to position and time is denoted by $\psi (x,t)$, and the wave function that is independent of time and depends on position is denoted by $u(x)$. In the case where there is no potential, that is, for a free particle, the wave function is as follows.
$$ \begin{align*} \psi(x, t) &= e^{\i (kx - \omega t)} = e^{\i (px - Et)/\hbar} & (\text{1-dim}) \\ \psi(\mathbf{r}, t) &= e^{\i (\mathbf{k}\cdot \mathbf{r} - \omega t)} = e^{\i (\mathbf{p}\cdot \mathbf{r} - Et)/\hbar} & (\text{3-dim}) \end{align*} $$
Setting $V = 0$ in the Schrödinger equation above and substituting $\psi$, one can easily verify that the equation holds.
$$ {-} \dfrac{\hbar^{2}}{2m}\dfrac{\partial^{2}}{\partial x^{2}}\psi = - \dfrac{\hbar^{2}}{2m} \dfrac{(\i p)^{2}}{\hbar^{2}} \psi = \dfrac{p^{2}}{2m}\psi = E\psi $$
$$ \i \hbar \dfrac{\partial \psi}{\partial t} = \i \hbar \left( -\i E/\hbar \right)\psi = E\psi $$
Note, however, that this plane wave solution is not square-integrable and thus cannot be normalized, so by itself it is a formal solution that does not represent a physical state. The state of an actual free particle is constructed by superposing such plane waves into a wave packet.
Interpretation
We said above that the wave function contains physical information about the object; more specifically, following Max Born’s interpretation, $\left| \psi(x, t) \right|^{2}$ is treated as the probability density function for the particle to exist at some point $x$ at time $t$. Therefore, the following equation means the probability that the particle exists in the interval $[a, b]$ at time $t$.
$$ \int _{a} ^b |\psi (x,t)|^2dx \\[1em] = \text{The probability that a particle exists in the interval } [a,b] \text{ at time } t $$
Hilbert Space
The Hilbert space referred to in quantum mechanics is, mathematically, the same as the following Lebesgue space $L^{2}$. This is just one of many Hilbert spaces, but since quantum mechanics is not concerned with any Hilbert space other than this one, they are treated as effectively the same.
$$ L^{2} = \left\{ \psi : \int \left| \psi \right|^{2} dx < \infty \right\} $$
This means that the square integral of the wave function must not diverge. A Hilbert space is originally a space equipped with an inner product, and the inner product of two wave functions is defined as follows.
$$ \braket{\psi | \phi} := \int \psi^{\ast}(x, t) \phi(x, t) dx $$
Strictly speaking, a vector space in which an inner product is defined and which is complete with respect to the norm induced by that inner product is called a Hilbert space, but an undergraduate physics student need not know the mathematical definition of all this. It is enough to understand a Hilbert space as “the set of square-integrable functions.”
A wave function, that is, a physical state, is among the elements of $L^{2}$ one that is normalized ($\left\| \psi \right\| = 1$), and as long as $\psi, \phi \in L^{2}$, the above inner product is guaranteed to be finite.
Cauchy–Schwarz Inequality
Also, since there is an inner product, the Cauchy–Schwarz inequality holds. For two wave functions $\psi$, $\phi$,
$$ \left| \braket{\psi | \phi} \right| \leq \sqrt{\braket{\psi | \psi}}\sqrt{\braket{\phi | \phi}} $$
or it is sometimes expressed as follows.
$$ \left| \braket{\psi | \phi} \right|^{2} \le \braket{\psi | \psi} \braket{\phi | \phi} $$