Wave Function and Hilbert Space in Quantum Mechanics
Build-Up
In Classical Mechanics, the main concern is finding the position function $\mathbf{r}(t)$ that satisfies Newton’s Second Law $\mathbf{F} = m \mathbf{a}$ given specific conditions. For instance, the position of an object launched with an initial velocity $\mathbf{v}_{0} = (v_{0}\cos\theta, v_{0}\sin\theta)$ in 2-dimensional space over time 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 reveals that the function representing the object’s position is as follows:
$$ \mathbf{r}(t) = -\dfrac{1}{2}gt^{2}\hat{\mathbf{y}} + t \mathbf{v}_{0} $$
Knowing the position, one can determine the Velocity $\mathbf{v} = \dfrac{d\mathbf{r}}{dt}$. Knowing the velocity allows us to find the Momentum $\mathbf{p} = m\mathbf{v}$ and Kinetic Energy $T = \dfrac{1}{2}mv^{2}$. In other words, finding $\mathbf{r}(t)$ is crucial as it provides physical information about the object.
Similarly, in Quantum Mechanics, we are interested in a particle’s wave function. While the goal in classical mechanics is to solve Newton’s Second Law to analyze the motion of an object, in quantum mechanics, we solve the Schrödinger Equation. This function contains physical information about the object.
Definition
The solution to the Schrödinger Equation, as shown below, 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 Planck constant, $V$ is the Potential, and $\nabla^{2}$ is the Laplacian.
Explanation
Notations commonly used for wave functions include the following:
$$ \Psi(x, t),\quad \psi(x, t),\quad \phi(x, t),\quad u(x) $$
In Sashimi restaurants, the wave function concerning position and time is denoted by $\psi (x,t)$, and the wave function concerning only position, irrespective of time, is denoted by $u(x)$. In the absence of a potential, i.e., for a free particle, the wave function is as follows:
$$ \begin{align*} \psi(x, t) &= e^{\i (kx - \omega t)} = e^{\i (px - Et)/\hbar} & (\text{1-dim}) \\ \psi(\mathbf{r}, t) &= e^{\i (\mathbf{k}\cdot \mathbf{r} - \omega t)} = e^{\i (\mathbf{p}\cdot \mathbf{r} - Et)\hbar} & (\text{3-dim}) \end{align*} $$
Substituting $V = 0$ into the Schrödinger Equation and then substituting $\psi$ confirms the validity of the equation.
$$ {-} \dfrac{\hbar^{2}}{2m}\dfrac{\partial^{2}}{\partial x^{2}}\psi = - \dfrac{\hbar^{2}}{2m} \dfrac{(\i p)^{2}}{\hbar^{2}} \psi = \dfrac{p^{2}}{2m}\psi = E\psi $$
$$ \i \hbar \dfrac{\partial \psi}{\partial } = \i \hbar \dfrac{(\i E)^{2}}{\hbar^{2}} \psi = E\psi $$
Interpretation
It was mentioned earlier that the wave function encapsulates physical information about the object. Specifically, according to Max Born’s Interpretation, $\left| \psi(x, t) \right|^{2}$ is treated as the Probability Density Function for finding the particle at point $x$ when the time is $t$. Therefore, the equation below signifies the probability of finding the particle within 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 collection of all wave functions is called Hilbert Space. Although there is a more rigorous mathematical definition, this basic understanding should suffice for physicists. (A detailed explanation can be found through the link.) In other words, Hilbert Space can be expressed as follows:
$$ \text{Hilbert space} = \left\{ \psi_{p, E}(x, t) = e^{\i (px - Et)/\hbar} : \forall p, E \in \mathbb{R} \right\} $$
In Hilbert Space, an Inner Product is defined between two elements. The Inner Product of two wave functions is defined as follows:
$$ \braket{\psi | \phi} := \int \psi^{*}(x, t) \phi(x, t) dx $$
Cauchy-Schwarz Inequality
With an inner product, the Cauchy-Schwarz Inequality holds. For two wave functions $\psi$ and $\phi$,
$$ \left| \braket{\psi | \phi} \right| \leq \sqrt{\braket{\psi | \psi}}\sqrt{\braket{\phi | \phi}} $$
It can also be expressed as follows:
$$ \left| \braket{\psi | \phi} \right|^{2} \le \braket{\psi | \psi} \braket{\phi | \phi} $$