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\(\S1.1\) An Overview of Classical Concepts

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 <<QM X-Roads-01>>

1. An Overview of Classical Concepts
    Introduction;Thought experiments with waves;Thought experiments with bullets;
    Waves versus particles. 

2. Structure and Formulations of Physical Theories
   Structure of physical theories; Formulations of classical mechanics;Thermodynamics and
   statistical mechanics.

3. Inadequacy of Classical Theories


From times of Newton, it has been debated whether light is  like a beam of corpuscles or a is it a wave? It was believed that only one of these properties could be associated with a beam of light as the characteristic properties of classical particles and waves appeared to be incompatible. We briefly recall the differences between waves and particles.

A  particle cannot be subdivided, whereas one can {\it subdivide} a wave indefinitely. For a wave, the intensity can be reduced continuously without any limit. The waves transfer energy and momentum in a continuous fashion. For a beam of particles, the intensity cannot be be reduced indefinitely, beyond having one particle at ``one time``. An exchange of energy momentum from particles takes place in  a discrete fashion.

A particles is localised in space where as the waves are, generally, not localised. A particle follows a well defined trajectory, but no trajectory can be associated with waves. A particle cannot bend round a corner, whereas waves exhibit the phenomena of diffraction.

The waves show phenomena of interference. Thus two waves of intensity $I$, meeting at a point, can give rise to  resultant wave of intensity between $0$ and $4I$. Clearly a similar situation for particles is not possible; two bullets meeting at point cannot destroy each other, nor do they create bullets giving rise to four bullets. A beam of bullets will not show phenomena of polarisation exhibited by (transverse) waves.

For a complete specification of the state of a particle we require a finite number of coordinates and momenta. The waves are specified by infinite number of ''coordinates'', an amplitude at each point of the medium. All waves, for example sound waves, can be thought of as a collection of an infinite  number of oscillators. For sound waves these oscillators are just that particles of the medium. Light can also be thought of as a collection of infinite number oscillators, but without any medium being associated with it.

Some of the differences between wave and particle natures, outlined above, will be highlighted with the help of some {\it Gedanken experiments} ( thought experiments) in the next few sections.

Thought experiments with  waves 

Single Slit Experiment with Waves

The interference and diffraction are characteristic properties of waves. When a wave falls on a slit, a diffraction pattern is observed as the waves can bend round the corners of the slit. Consider the experimental set up of Fig.1 where a wave travelling along the $z$- axis passes through a slit and hits a screen where the intensity of waves is measured as a function of $x$. If $a$ is the slit width and $\lambda$ is the wave length, the intensity as a function of $x$ is given by
\begin{equation} I = I_0\left(\frac{\sin \alpha}{\alpha}\right)^2, \label{EQ01} \end{equation}

\begin{equation} \alpha =\frac{\pi a}{\lambda} \sin \theta, \qquad \sin\theta \approx \tan\theta =\frac{x}{L}, \end{equation}

and $L$ is the distance of the screen from the slit. The waves not only reach the point $x = 0$, but a significantly large part of the wave is received for large values of $x$. If the intensity of the incident wave is reduced, one would continue to get the same pattern, even for arbitrarily low values of the intensity of the incident waves.

Double Slit Experiment With Waves

In a double slit arrangement, a part of the incident wave travels through each slit and on reaching a point, at position \(\vec{r}\),  on the screen, the two parts interfere and produce an interference pattern. This is because the waves travelling through different paths have a phase difference $\delta$  and the resultant amplitude $\phi$ at any point is given by the sum of the amplitudes, $\phi_1, \phi_2$ of the waves coming through the two slits :
\begin{eqnarray} \phi_1(\vec{r}, t) &=& A_1 \exp(i\vec{k}\cdot \vec{r})\\ \phi_2(\vec{r}, t) &=& A_2 \exp(i\vec{k} \cdot\vec{r} + i\delta), \\ \phi  &=& \phi_1+ \phi_2. \end{eqnarray}

The resultant intensity is then given by
\begin{eqnarray} I_{12} = |\phi_1 + \phi_2|^2 = |\phi_1|^2 + |\phi_2|^2 + (\phi_1^*\phi_2 + \phi_1\phi_2^*). \end{eqnarray}

Here the first, the second and the third terms give the intensity due to the first slit, the second slit and the interference term, respectively. When the intensity of the incident waves is reduced, we continue to get the same intensity distribution pattern on the screen. Assuming that the detectors placed on the screen can measure arbitrarily low intensity by collecting waves received over a large period of time, the intensity pattern would look the same irrespective of the intensity of the incident beam.

Double slit experiment with only one slit open at a time

Let us now consider a variation of the double slit interference experiment, keeping only one slit open at a time. Let us assume that only one slit is kept open for time $0 < t < T$ and the second slit is closed. During the time $T < t < 2T$, the second  slit is kept open and the first slit is kept closed. In this case what intensity pattern do you expect to see on the screen? Would one get the interference pattern
\begin{equation} I_{12} = |\phi_1 + \phi_2|^2 \end{equation}

or simply the sum of intensities due to the two slits:
\begin{equation} I = I_1 + I_2 = |\phi_1|^2 + |\phi_2|^2? \end{equation}

Interference can take place only between parts of a wave arriving simultaneously at a point with a phase difference. This happens when different parts of a wave arrive taking different paths. When only each slit is kept open for half the time, we expect to observe the intensity pattern $I_1 + I_2$ which is just the sum of intensities due  to the two individual slits. While there is no doubt about the conclusion, the issue has to be settled only by doing an experiment. Primary reference for a discussion of waves and particle concepts is \cite{Feyn3}.

Thought experiments with  bullets

Single slit experiment
Waves can turn around the corners but not particles, certainly not the bullets. Consider an experiment with bullets fired along the $z$- axis from a gun and hitting a wall with a slit. Next to the wall there is a screen where all bullets are stopped and are collected in containers placed at different positions. The number of bullets, $N(x)$, received  in a container placed at $x$, in a given time $t$, is counted. We plot the number of bullets as function of $x$. Assuming an ideal parallel beam of bullets, travelling in the $z$- direction, only the containers with positions in the range $−d\le x \le d$ will receive the bullets, where $2d$ is the slit width. The container placed at far away positions will not receive any bullet. We should expect to see a distribution as in Fig.1.2. The bullets cannot bend round the corners of the slit and cannot reach containers at positions $|x| > d$.


Double slit experiment

If a double slit experiment is performed with two slits, one would get a count distribution pattern, $N(x)$, which is just the sum of the patterns corresponding to the two individual slits.
\begin{equation} N(x) = N_1(x) + N_2(x). \end{equation}

Note that each bullet must pass through one of the slits, unlike waves where parts of the incident wave could pass through each slit and produce an interference pattern. The result will remain unchanged if the experiment with the bullets is repeated with each slit open only half the time.

A probabilistic description

The classical theory is deterministic in the sense that, knowing the initial position and momentum of a bullet, one can predict where it will hit the screen. Now imagine a gun firing the bullets randomly at different angles, with a maximum number of bullets along the $z$- axis. In this case, the bullets would reach positions outside the range $(-d, d)$ and the number of bullets reaching a point $x$ will be as shown above. % TODO explain as above?? %FIXME Assuming that the angle between the $z$- axis and the initial direction of a bullet is random, and not known a priori, it will not be possible to predict where the bullet would hit the screen. For a small number of bullets the distribution $N(x)$ of the bullets on the screen cannot be predicted. But, if the experiment is repeated sufficiently many times we expect to see the same distribution $N(x)$, irrespective of the rate of firing the bullets.In such a case, how do we describe the outcome of an experiment with a single bullet? We associate a probability distribution $P(x)$ that a bullet will hit the screen at a position $x$. If $N_0$ is the total number of bullets in the experiment, then

\begin{equation} P(x) =\frac{ N(x)}{N_0} \end{equation}

In a double slit experiment we would expect to see a distribution

$P_\text{12}(x)$: \begin{equation} P_\text{12}(x) = P_1(x) + P_2(x), \end{equation}


$P_\text{12}(x) =$ Probability that a bullet reaches position $x$ when both the slits are open,\\

$P_1(x) =$ Probability that a bullet reaches position $x$ when only the first slit is open,\\

$P_2(x) =$ Probability that a bullet reaches position $x$ when only the second slit is open.\\

The outcome of an experiment with waves can also be described in probabilistic terms. We normalise the intensity of waves at a position $x$ by the intensity of the incident waves, $I_0$, and associate a probability $P(x) = I(x)/I_0$, which would give the intensity pattern as a function of $x$. Here it must be emphasised that a probabilistic description for waves is purely a matter of taste or convenience, but for bullets we are forced to introduce the probabilistic description because of incomplete information about the initial momentum of the bullets. Even in this case, the classical theory is deterministic in the sense that each bullet takes a well defined trajectory which can be predicted if the initial position and momentum are known accurately. A knowledge of the probability distribution of the angle at which bullet is fired, will in principle permit us to compute $P(x)$ and the number of bullets reaching a container at position $x$.


Waves versus particles

An  analysis of single and double slit experiments with waves and beams of particles  brings out the fact that a wave bends round the corners, whereas particles do not. Interference phenomenon seen in the double slit experiment emphasises that a part of a wave passes thorough one slit and another part passes through another slit and an interference pattern is produced when parts of waves from different paths meet on the screen. The bullets (particles) have a well defined trajectory and for a beam of particles, each particle must pass through one of the slits and no interference pattern is produced. While in interaction with other physical systems, the waves transfer energy and momentum in a continuous fashion, in contrast, the particles transfer energy and momentum in a discrete manner. Another important difference between the waves and particles is that a wave can be subdivided indefinitely by cutting down the intensity. For a beam of particles, the intensity cannot be reduced beyond having a single particle at a time.

The particle and the wave aspects appear to be incompatible and it would seem impossible that they could coexist for a physical system. However, the nature has its own way and it is amazing that all particles show a wave nature and all waves have a particle nature. First indications came from the photoelectric effect from its explanation in 1905 by Einstein. Next came the matter wave hypothesis by de Broglie and its confirmation by Davisson Germer experiments and it was established that the electrons exhibit wave nature. It is now accepted that all matter has a wave nature associated with it. This lead to acceptance of 'every body' having a dual nature, behaving like both a wave and a particle.

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