The quantum particle

The Quantum Particle and The Two Slit Interference Experiment

A classical particle is an object that has mass, and with the property of having a definite location and momentum at any instant of time. In this post, we will consider the quantum version of the classical particle, which we will refer to as the quantum particle or simply particle. In order for us to discover how the quantum description of a particle arises, we will revisit the well-known two slit experiment involving electrons. This will eventually lead us into the construction of the particle Hilbert space.

Consider a collimated weakly interacting (for all practical purposes non-interacting) electron beam incident upon a pair of slits. On the opposite of the slits is a film that serves as a screen on which we observe the arrival of an electron. Also on the upper slit there is a detector that tells which slit the electron has passed through on its way to the screen. If the detector lights up, the electron went through the upper slit; otherwise, through the lower slit. We are all aware of the result of the experiment: In the presence of the detector, no interference appears on the screen; otherwise, in the absence of the detector, interference appears. Let us go beyond appearances and quantify this result.

Let us suppose that the screen is dressed with an array of detectors with some width $\delta x$, and that the $j$-th detector is centered at the point $x_j$. The $x_j$’s then give the location where an electron is detected. First, let us consider when the detector is present. Let there be a total of $N$-electrons that arrived at the screen after the experiment. Let us denote the number of electrons that arrived at the detector at $x_j$ through the upper slit by $N_u(x_j)$, and through the lower slit by $N_l(x_j)$; moreover, let $N_t(x_j)$ be the total number of electrons that arrived at $x_j$. When the detector is present, we find that the total number of arrivals at $x_j$ is just $N_t(x_j)=N_u(x_j)+N_l(x_j)$.

Now when the detector is absent so that there is no way for us to determine which slit an electron went through, an uncanny behavior is observed. The distribution of arriving electrons no longer resemble that of the former configuration. That means we can no longer describe the number of arrivals at a certain location as the sum of the electrons passing through the upper and lower slits. That is we can no longer describe that a certain number $N_u(x_j)$ passed through the upper slit or a certain number $N_l(x_j)$ through the lower slit. To describe what we observe is to go beyond counting. Nevertheless, we can still count the number of electrons that arrived at the location $x_j$, and say it is given by $N_t(x_j)$.

To describe the distribution on the screen we go through the following manipulation—almost a magical one—on the number $N_t(x_j)$. Since $N_t(x_j)$ is positive, we can always find a complex number $\eta(x_j)$ such that $N_t(x_j)=\bar{\eta}(x_j)\eta(x_j)$. Now it turns out that the only way to describe the result of the experiment is to associate a complex valued function to each of the upper and lower slits, $\eta_u(x_j)$ and $\eta_l(x_j)$, respectively; and that the complex valued fuction $\eta(x_j)$ is a linear sum of the two, $\eta(x_j)=\eta_u(x_j)+\eta_l(x_j)$. On writing $\eta_u(x_j)=|\eta_u(x_j)| \exp\left(i \phi_u(x_j)\right)$ and $\eta_u(x_j)=|\eta_l(x_j)| \exp\left(i \phi_l(x_j)\right)$, the total number of electrons that arrived at $x_j$ assumes the form
\begin{equation}
N_t(x_j)=\left|\eta_u(x_j)\right|^2+\left|\eta_l(x_j)\right|^2 + 2 \sqrt{\left|\eta_u(x_j)\eta_l(x_j)\right|} \cos\left(\phi_u(x_j)-\phi_l(x_j)\right).
\end{equation}The first two terms are positive and the third term oscillates between $\pm 2 \sqrt{\left|\eta_u(x_j)\eta_l(x_j)\right|}$ for all $j$’s. Clearly it is the third term that is responsible for the observed interference pattern on the screen.

This tells us that in order to describe the result of the experiment, we have to introduce complex quantities out of the measurable quantities. In this case, knowledge of the complex quantities can determine the distribution of the results of the experiments. Note that the result of the first experiment can be brought into the framework of the second experiment by identifying $N_u(x_j)=\left|\eta_u(x_j)\right|^2$ and $N_l(x_j)=\left|\eta_l(x_j)\right|^2$, with the third term missing. It can be interpreted that the act of extracting information on which slit an electron has passed through has caused the third term to disappear. How the measurement process reduces the third term away is the central problem of quantum measurement theory which we will consider in detail later.

A characteristic feature of the two experiments is that where a single electron arrives on the screen is completely random. We can imagine several similar experiments done simultaneously. Necessary care is made such that the conditions of these experiments are the same; in particular, their initial conditions are the same. If we observe where the first electron of each experimental set-up ends up on the screen, we find that they do not land on the same point. The same thing with the second, the third, and so on. More importantly, we cannot, on the basis of the complete knowledge of the initial conditions of the experimental set-up, predict where a single electron will land on the screen.

However, we find that as the number of electrons arriving gets more and more numerous a pattern on the screen emerges. And this pattern is identical to all experiments involving arbitrarily large number of electrons. While the result of a single measurement is unpredictable and is not determined by the initial configurations of the measuring instruments, the result of a large number of measurement is predictable and is determined by the initial configuration of the system and the measuring apparatus. The existence of an underlying distribution of results imply that, while the occurrence of an event is random, one can assign a definite probability for the occurrence of that event. It is the business of quantum mechanics to providing the recipe in computing for such probabilities.

The Wavefunction

We find above that in order to describe the result of the two-slit experiment we need to introduce a complex-valued quantity. This, in the context of the experiment, is what we call a number amplitude because the modulus of which gives the number of electrons arriving at a given point on the screen. Let $N$ be the total number of electrons that arrived at the screen during the duration of the experiment. Then consider the quantity $N_t(x_j)/N \delta x = \delta P(x_j)/\delta x$, where $\delta P(x_j)=N_t(x_j)/N$ gives the probability of locating a single electron in the immediate vicinity of $x_j$. The quantity $\delta P(x_j)/\delta x$ gives us the probability density.

Now let the number $N$ of electrons gets large indefinitely and let the width of the detector $\delta x$ gets smaller indefinitely as well. Quantum mechanics presupposes that there exists a well-defined limit,
\begin{equation}
\rho(x)=\lim_{\delta x\rightarrow 0,\; N\rightarrow \infty} \frac{N_t(x)}{N\delta x}.
\end{equation}Moreover the quantities $\eta(x_j)/\sqrt{N\delta x}$, $\eta_u(x_j)/\sqrt{N\delta x}$, and $\eta_l(x_j)/\sqrt{N\delta x}$ likewise obtain the limiting values $\psi(x)$, $\psi_u(x)$, and $\psi_l(x)$, respectively, such that $\rho(x)=\bar{\psi}(x) \psi(x)=\left|\psi(x)\right|^2$ and $\psi(x)=\psi_u(x)+\psi_l(x)$.

From the way we have arrived at $\psi(x)$ the quantity $\left|\psi(x)\right|^2 dx$ yields the probability for the electron to arrive at $x$ or for us to find the electron in the neighborhood of $x$. Knowledge of $\psi(x)$ does not allow us to predict where an electron will arrive on the screen; however, it gives complete information on the distribution of outcome of position measurements, which is given by $|\psi(x)|^2$. The complex-valued function $\psi(x)$ is known as the wave-function.

The Superposition Principle

The second experiment teaches us a very important lesson on the quantum description of a quantum particle. Two or more results are mutually exclusive if one result excludes the other. An electron taking the upper slit and the same electron taking the lower slit are mutually exclusive because a detector interrogating the electron can only register one of the two possibilities. If the detector lights up, then the electron went through the upper slit and not through the lower slit, and vice versa. Now if the experimental set up is such that no information can be extracted on which of the mutually exclusive events is the case, then the description of the object requires that a complex-valued quantity be assigned to each mutually exclusive event and that the state of the object is given by a complex-valued quantity which is the linear sum of these complex-valued quantities corresponding to each possibility. This is the well-known phenomenon of self-interference, which is embodied by the quantum superposition principle.

The Hilbert Space of the Quantum Particle

Quantum mechanics postulates that the state of a quantum particle in one dimension is completely described by the wave-function $\psi(x)$. We now turn into constructing the Hilbert space of the quantum particle. We start with the identification that $|\psi(x)|^2$ is the probability density of locating the particle at $x$, so that the probability of finding it in the interval $\Delta$ is given by the integral
\begin{equation}
P(\Delta)=\int_{\Delta} |\psi(x)|^2 \mbox{d}x.
\end{equation}We can discretize the entire line into disjoint intervals, $\{\Delta_k\}$, and obtain the probability of finding the particle in each interval, $\{P(\Delta_k)\}$. The law of probability demands that the sum of all probabilities must be unity, so that $\sum_k P(\Delta_k)=1$. This translates into what is known as the normalizability condition
\begin{equation}
\int_{\mathbb{R}}|\psi(x)|^2\, \mbox{d}x=1 .
\end{equation}If it happens that the wave-function is not normalized, i.e. $\int_{\mathbb{R}}|\psi(x)|^2\, \mbox{d}x\neq 1$, it can be normalized with the replacement $\psi(x)\rightarrow \psi(x)/\int_{\mathbb{R}}|\psi(x)|^2\mbox{d}x$. Quantum mechanics does not distinguished these two functions: They represent the same quantum state of the particle.

The foregoing dictates that any complex valued function $\psi(x)$ is a legitimate state of a quantum particle provided it is normalizable; in particular, it satisfies what is known as the square integrability condition
\begin{equation}
\int_{\mathbb{R}}\left|\psi(x)\right|^2 \, \mbox{d}x<\infty . \end{equation}Let us denote the set of complex valued, square-integrable functions in the real line by $\mathcal{S}^2(\mathbb{R})$. This set is a complex vector space under the usual rules of pointwise addition of functions, and complex multiplication of functions with complex numbers. This follows from the well-known inequality \begin{equation} \int_{\mathbb{R}}\left|\alpha\psi(x)+\beta\phi(x)\right|^2\, \mbox{d}x\leq |\alpha|^2 \int_{\mathbb{R}} \left|\psi(x)\right|^2 \, \mbox{d}x + |\beta|^2 \int_{\mathbb{R}} \left|\phi(x)\right|^2 \, \mbox{d}x < \infty \end{equation}for every pair of functions $\psi(x)$ and $\phi(x)$ in $\mathcal{S}^2(\mathbb{R})$, and pair of complex numbers $\alpha$ and $\beta$. That is if $\psi(x)$ and $\phi(x)$ are in $\mathcal{S}^2(\mathbb{R})$, then the function $\alpha\psi(x)+\beta\phi(x)$ is again in $\mathcal{S}^2(\mathbb{R})$. By the quantum superposition principle, we identify the set $\mathcal{S}^2(\mathbb{R})$ to comprise a set of legitimate pure states of the quantum particle. Notice that we did not say "to comprise the whole class of pure states of the particle." The reason is that it is not yet the Hilbert space which is supposed to already contain all available pure states of the particle. To obtain the Hilbert space, we first need to turn $\mathcal{S}^2(\mathbb{R})$ into a pre-Hilbert space by assigning an inner product to it. Let $\left<\cdot\left|\cdot\right>\right.$ be the still undetermined inner product. The inner product must incorporate the square integrability condition. For every $\psi(x)$ in $\mathcal{S}^2(\mathbb{R})$ we define its norm or length by
\begin{equation}\label{normcond}
\left|\left|\psi\right|\right|^2 = \int_{\mathbb{R}}|\psi(x)|^2\, \mbox{d}x .
\end{equation}The inner product must necessarily be consistent with this definition of the norm. For a Hilbert space, the relationship between the norm and the inner product is given by $\left|\left|\psi\right|\right|^2 = \left<\psi\left|\psi\right>\right.$. This is a necessary condition but not sufficient to determine the appropriate inner product.

The inner product is ultimately determined by the quantum principle of equivalence of mutual exclusivity and orthogonality of states. This requires us to define when two or more wave-functions are mutually exclusive. Let us consider two wave-functions $\psi_1(x)$ and $\psi_2(x)$ that are non-vanishing only in the intervals $\Delta_1$ and $\Delta_2$, respectively. Let us assume that the intervals do not overlap. If the particle is prepared in the wave-function $\psi_1(x)$, then the probability of finding the particle in the interval $\Delta_2$ is given by $\int_{\Delta_2}|\psi_1(x)|^2 \mbox{d}x=0$; the probability vanishes because $\psi_1(x)$ vanishes everywhere in $\Delta_2$. That is if the particle is prepared in $\psi_1(x)$, then any measurement of the position of the particle excludes any result that is consistent with any result when the particle is prepared in the state $\psi_2(x)$. The converse of the statement is true. This leads us to the identification of mutually exclusive states: non-overlapping wave-functions are mutually exclusive.

The definition of the inner product must then be consistent with this definition of mutually exclusive states. For every pair of wave-functions $\psi(x)$ and $\phi(x)$, we define the functional $\int_{\mathbb{R}}F(x) \psi^*(x) \phi(x)\, \mbox{d}x$ where the function $F(x)$ is positive everywhere in the real line. This functional satisfies all the axioms of an inner product. Moreover, we find that when $\psi(x)$ and $\phi(x)$ are non-overlapping the value of the functional is zero. Hence it is consistent with the definition of mutual exclusivity of two non-overlapping wave-functions. However, it is arbitrary because of the arbitrariness of $F(x)$. The inner product is fixed by imposing the norm condition \eqref{normcond}, which requires that $F(x)=1$. Finally we identify the inner product to be given by
\begin{equation}
\left<\psi\left|\phi\right>\right. = \int_{\mathbb{R}}\psi^*(x) \phi(x)\, \mbox{d}x .
\end{equation}Form this we have the metric
\begin{equation}
||\psi-\phi||^2 = \int_{\mathbb{R}}|\psi(x)-\phi(x)|^2\, \mbox{d}x .
\end{equation} And we have completed the construction of the pre-Hilbert space of the pure states of the particle. The Hilbert space is finally obtained by completing the pre-Hilbert space $\mathcal{S}^2(\mathbb{R})$, by adjoining to it all limit points of the pre-Hilbert space. The Hilbert space is denoted by $\mathcal{L}^2(\mathbb{R})$, the Lebesgue square integrable complex valued functions. This Hilbert space is infinite dimensional and separable.

Quantum Particle in Three Dimensional Space

In the foregoing we considered a quantum particle restricted along a line. When the particle can move in the entire spatial space, the Hilbert space must be appropriately modified. The Hilbert space, denoted by $\mathcal{L}^2(\mathbb{R}^3)$, consists of all complex valued functions $\psi(\vec{r})$ satisfying the condition
\begin{equation}
\int_{\mathbb{R}^3} |\psi(\vec{r})|^2\, \mbox{d}^3 r < \infty . \end{equation}The inner product is given by \begin{equation} \left<\psi\left|\phi\right>\right. = \int_{\mathbb{R}^3} \psi^*(\vec{r})\phi(\vec{r})\, \mbox{d}^3 r ,
\end{equation}for all pairs of wave-functions $\psi(x)$ and $\phi(x)$ in $\mathcal{L}^2(\mathbb{R}^3)$. The inner product again is consistent with the condition of mutual exclusivity of two wave-functions that are non-overlapping, with the interpretation that the probability of finding the particle in the region $V$ is given by
\begin{equation}
P_{\psi}(V) = \int_{V} |\psi(\vec{r})|^2\, \mbox{d}^3 r,
\end{equation}provided that $\psi(\vec{r})$ is normalized.

A Particle in a Box

For a particle confined in the interval $[a,b]$, where $a < b$, the Hilbert space consists of all square integrable complex valued functions in the interval $[a,b]$, i.e. those $\psi(x)$'s satisfying the condition \begin{equation} \int_a^b |\psi(x)|^2 \mbox{d} x < \infty . \end{equation} The Hilbert space is denoted by $\mathcal{L}^2[a,b]$. In this Hilbert space, the inner product is given by \begin{equation} \left<\psi\left|\phi\right>\right. = \int_{a}^b \psi^*(x)\phi(x)\, \mbox{d}x,
\end{equation}for all pairs of wave-functions $\psi(x)$ and $\phi(x)$ in $\mathcal{L}^2[a,b]$. The same interpretation holds for the wave-functions $\psi(x)$ of the Hilbert space $\mathcal{L}^2[a,b]$ as those in the Hilbert spaces above.

Separable, Infinite Dimensional Hilbert Spaces and $l^2$

The Hilbert spaces $\mathcal{L}^2(\mathbb{R})$, $\mathcal{L}^2(\mathbb{R}^3)$ and $\mathcal{L}^2[a,b]$ are examples of separable and infinite dimensional Hilbert spaces. They represent distinct physical systems in different configuration spaces. However, each of them and any separable infinite dimensional Hilbert space is isomorphic to the Hilbert space $l^2$. Two Hilbert spaces are isomorphic if there is a one-to-one correspondence between vectors of the Hilbert spaces; and two such Hilbert spaces are mathematically equivalent. In an earlier post we have addressed the question whether two isomorphic Hilbert spaces represent the same quantum system or not.

The isomorphism of a separable and infinite dimensional Hilbert space $\mathcal{H}$ and $l^2$ can be established as follows. $\mathcal{H}$, being separable, possesses a countable set of orthonormal basis vectors, say $\{\phi_1,\phi_2,\phi_3,\dots\}$. Then every vector $\psi$ of $\mathcal{H}$ has the Fourier expansion
\begin{equation}
\psi=\sum_{k=1}^{\infty} a_k \phi_k,\;\;\; a_k = \left<\phi_k\left|\psi\right>_{\mathcal{H}}\right.,
\end{equation}in which $\left<\cdot|\cdot\right>_{\mathcal{H}}$ is the inner product in $\mathcal{H}$. Note that the complex Fourier coefficients $a_k$ are uniquely determined by $\psi$, so that we have the one-to-one correspondence
\begin{equation}
\psi \longleftrightarrow (a_1,a_2,a_3,\dots)
\end{equation} Also that $\psi$ belongs to $\mathcal{H}$ implies that it has a finite norm, that is
\begin{eqnarray}\label{xxx}
||\psi||^2 &=& \left<\psi\left|\psi\right>\right.
&=& \sum_{k=1}^{\infty} |a_k|^2 < \infty, \end{eqnarray} where we have used the orthonormality of the basis vectors to arrive at the right hand side of the equation. Now for an arbitary $\varphi$ in $\mathcal{H}$ with the Fourier expansion \begin{equation} \varphi=\sum_{k=1}^{\infty} b_k \phi_k,\;\;\; b_k = \left<\phi_k\left|\varphi\right>_{\mathcal{H}}\right.,
\end{equation}we have the inner product
\begin{equation}\label{inner1}
\left<\psi\left|\varphi\right>_{\mathcal{H}}\right. = \sum_{k=1}^{\infty} a_k^* b_k ,
\end{equation}where we have invoked again the orthonormality of the basis vectors.

We now recall that a vector $\zeta$ of $l^2$ is a sequence of complex numbers, $\zeta=(\alpha_1,\alpha_2,\alpha_3,\dots)$, with the sequence satisfying
\begin{equation}\label{xxxx}
\sum_{k=1}^{\infty} |\alpha_k|^2 < \infty. \end{equation} For arbitrary pair of vectors $\zeta=(\alpha_1,\alpha_2,\alpha_3,\dots)$ and $\eta=(\beta_1,\beta_2,\beta_3)$ in $l^2$, the inner product is given by \begin{equation}\label{inner2} \left<\zeta\left|\eta\right>_{l^2}\right. = \sum_{k=1}^{\infty} \alpha_k^* \beta_k .
\end{equation}

The fact that an arbitrary vector $\psi$ of $\mathcal{H}$ is uniquely identified with its complex Fourier coefficients $(a_1,a_2,a_3,\dots)$ and that these coefficients satisfy the same condition \eqref{xxx} as those elements of $l^2$ \eqref{xxxx} allow us to set up a one-to-one correspondence between vectors of $\mathcal{H}$ and vectors of $l^2$. Moreover, the inner product in $l^2$ \eqref{inner2} reproduces the inner product \eqref{inner1} in $\mathcal{H}$, so that both Hilbert space have the same mathematical structures.