Chapter 1

Introduction

Man’s mind once stretched by a new idea, never regains its original dimension.

—Oliver Wendell Holmes

1.1  MOTIVATION

Physics deals with the study of behaviour of physical systems under various conditions. It boils down to the study of nature of interactions between particles—the constituents of physical systems—and the consequences thereof. At macroscopic level, almost all observed phenomena have been explained by invoking the laws of Newtonian mechanics, which is generally known as ‘classical mechanics’.

The very natural question which comes to our mind is ‘What is the need of any other mechanics (e.g. quantum mechanics) when very profound and well-tested mechanics (i.e. classical mechanics) is already there?’ For centuries, classical mechanics has been applied successfully to explain each and every observable phenomenon at macroscopic level. Well, towards the end of nineteenth century and the beginning of twentieth century, some phenomena, such as spectral distribution of blackbody radiations, atomic spectra, photoelectric effect, and so on, could not be explained on the basis of classical mechanics and classical electrodynamics (classical mechanics applied to the dynamics of charged particles). Slowly, but surely, it became clear from various experimental results that classical mechanics can be applied fairly well to explain the phenomenon involving macroscopic bodies, but when it comes to the phenomenon involving microscopic particles, classical mechanics can not explain these. We shall encounter, as we proceed in this chapter, various examples to this effect. We shall then conclude that a modified mechanics has to be used to explain these phenomena involving microscopic particles. This modified mechanics is nothing but ‘quantum mechanics’. In fact, we shall see in this book that quantum mechanics is the sole basis of our present understanding of all physical phenomena on atomic scale. Not only quantum mechanics helps in understanding all phenomena at microscopic level, it shows the way to develop new technologies. Quantum mechanics has had applications in developing numerous technologies in the past such as (i) semiconductor devices, optoelectronic, and laser devices, (ii) biotechnology, including understanding and modifying molecules, such as DNA, and cells, we wish to understand and modify on an atomic scale. Recently, quantum mechanics is being used in developing and understanding nanotechnology (the technology at nano scale).

1.2  THE USUAL (CLASSICAL) MECHANICS

When we deal with motion of large (macroscopic) objects, such as balls, and are interested in dynamics of the objects (without any internal motion or deformation), we simply apply Newton’s laws of motion, or more rigorous Hamilton’s equations of motion. For a known force working on the body, we can find out its positions (i.e. the position of its centre of mass) at a later time t if its position is known initially at time t = 0.

In classical mechanics, the motion of a body in the inertial frame of reference is described by the differential equation,

where F is the force, p = [mv = m (dx/dt)] is the linear momentum and v (= dx/dt) is the velocity of the moving body in the direction of unit vector e_x.

Let us consider the simple example of the motion of a particle of mass m under the constant gravitational field exerting a force mg on the particle. Let us take x-axis in the downward vertical direction. Let x₁ be the initial position where the particle is left with initial velocity zero, then Eq. (1.1) may be easily solved for F = mg to give final position of the particle x(t) as a function of time t as

The results of this equation can easily be verified experimentally. For example, an experiment of free fall of a ball may be performed in a region of negligible air resistance. We can measure the distance travelled by the ball in a given time. To measure the distance travelled, we will have to observe the initial position of the ball (at t = 0) and the final position (at time t) of the ball.

1.3  THE PROCESS OF OBSERVATION

To observe directly the position of the ball at any instant, one shall have to use light. The light falls on the ball, gets reflected, and enters the retina of the observer. Thus the position of the ball is observed. There could be other sophisticated methods of taking observation, but basically all processes shall use radiation probe at one stage or the other. We can not have an observation without the interaction of a radiation probe with the ball. The fact remains that the ball is experiencing only one force; the gravitational force in the above example of free fall of the ball, and the force of radiation used for observing it, not to speak, is totally negligible. The experimental observations completely agree with the theoretical result of the Newtonian mechanics.

But now, suppose we go on reducing the size of the ball which ultimately reduces to a tiny particle (molecule or atom). We will definitely like to apply Newton’s laws of motion to this atom. And a force of magnitude F in the x-direction on this particle (atom) of mass m, shall again give the position as a function of time t as

We would like to verify this result experimentally. Naturally, as in the case of a ball, we shall use the radiation probe to find the position of the tiny particle at a particular time. But now, the effect of the radiation probe on the tiny particle is not negligible. This effect shall really disturb the motion of the particle from what it would have been under the force F only. It is a complex situation. The theoretical result [Eq. (1.3)] is obtained under the condition that only a single force F is applied on the particle. Whereas, when experiment is done to observe the position of the particle at some time t, additional—uncontrolled and undetermined—force of radiation field also works on the particle. We can try to minimize this additional force but it can not be made zero, as we have to observe its position, any way.

One thing becomes clear from the above observation: to verify experimentally the result of Newton’s equation of motion, either we will have to include (in the equation of motion) the effect of radiation field on the particle along with the force F, or design an experiment where there is no effect on the motion of particle of any agency used in the measurement process. The second possibility seems to be impossible. To realize the first possibility, the effect of the radiation field used in the process of measurement may not be known exactly, and so may not be included in the equation of motion in an exact way. It simply means, in case of microscope particles the results of classical mechanics can not be verified experimentally, or whatever is observed experimentally is not the result of classical equation of motion of the particle under the influence of a known force F. It becomes necessary to have a different mechanics for microscope particles where the equation of motion should include, possibly in an approximate way, the effect of observation on the particle. This mechanics may not give exact results for the motion of microscopic particles. But even approximate or probable results should be welcomed under the circumstances mentioned above. This new mechanics, which will be developed in the coming Chapters, is called as quantum mechanics and the new equation of motion is called as Schrodinger equation. In the next section we put forward arguments, in brief, to suggest the form of the new mechanics.

1.4  THE NEW MECHANICS

It may seem from the discussions of previous Section that we can simply start with Newton’s equation of motion (of the microscopic particle under the given force) and by adding the additional force present during the measurement process (which may not be known precisely), we may get the new (imprecise) equation of motion. But, let us pause here for a moment. Are we not implicitly assuming that a microscopic particle is in the form of a very tiny ball with well-defined position and momentum (though, may be unknown)? This is what we always imagine of a microscopic particle (based on our experience with macroscopic objects like balls or dust particles). In fact, we should not be sure of it, unless we experimentally observe it so. Therefore, let us go to the shelter of experimental results on microscopic particles.

As we shall discuss in details in Chapters 2 and 3, the results of all experiments performed (to find shape and size of the microscopic particles) confirm that microscopic objects (like electrons, neutrons, protons, atoms etc. and e.m. radiations) manifest themselves in the form of waves: sometimes in the form of extended waves and sometimes in the form of compact waves.

Now, this is something totally new. On the basis of the arguments put forward in Section 1.3, we may expect that identical particles under identical conditions when observed through different types of probes, may yield different results (as the particle shall be perturbed differently in different experiments). But, we never expected that the experimental results on microscopic particles may be so drastic as to suggest that these (particles) are propagating in the form of waves. It may seem still more surprising to find that different experiments suggest these particles propagating in different form of waves; some experiments confirming the particles in the beam as travelling in the form of plane propagating waves, while others confirming the particles travelling in the form of very compact (localized) waves. It simply means at microscopic level, THE NATURE manifests itself in a totally different way than what it does at macroscopic level. And we have to accept this fact. We shall, therefore, have to analyze the physical world at microscopic level in accordance with the stated fact. Thus, we shall have to find the equation of motion of these microscopic objects which are not in the form of tiny balls but are in the form of waves. These waves, called as matter waves, which represent microscopic particles, are different from the ordinary (mechanical) waves like sound waves or waves on a (vibrating) string. It means that the microscopic objects should be described through some sort of wave mechanics. The wave mechanics of these (matter) waves has to be different from the wave mechanics of ordinary (mechanical) waves. Before we discuss what form this ‘wave mechanics’ is going to take, we shall describe, in brief, some characteristics of these waves (details are given in Chapter 2).

Well, if we look at the logic put forward in Section 1.3 (that the microscopic particle is under different types of forces when observed with different types of probes), it seems befitting that the waves (representing the microscopic particles) show characteristics which depend upon what type of apparatus (i.e., what type of probe) is used to observe these waves/particles. Let us consider, for example, a mono-energetic beam of identical particles (say electrons) and observe these through the use of two-slit arrangement. In this case, we find fringe pattern on the screen (see Section 2.7), which forces us to conclude that each electron in the beam is moving in the form of a plane propagating wave with (well defined) wave length, λ. From the experimental data of fringe width β, distance d between two slits and distance D of the screen from the slit system, the value of λ can be easily determined. As we shall see in detail in Chapter 3, the de Broglie relation connects wave length λ (of the wave) to the linear momentum p (of the particle/wave) through the relation p = (h/λ) = ħk[ħ =(h/2π), k = (2π/λ)]. So, the two-slit experiment not only confirms that the (mono-energetic) electron beam is travelling in the form of plane propagating wave, it also measures the linear momentum of the particle moving in the form of plane wave.

Thus the two-slit apparatus when interacting with the mono-energetic beam of free particles (propagating in the form of plane waves) gives information about the (wavelength and hence of) linear momentum of the particles in the beam. This is the experimental information. Let us take this case as a proto-example and try to find out the equation of motion of the (microscopic) particles moving freely. The equation of motion of the particles moving freely (in the form of plane waves) has to be such that the solution of this equation of motion gives information (about the linear momentum of the particle) that can be verified by the experimental result (of two-slit set up). There should be a one-to-one correspondence: (i) on one hand, we perform two–slit experiment on the beam of particles to get the value of linear momentum of the particles; (ii) on the other hand, we have an equation of motion which gives value of the linear momentum of the particle (propagating in the form of plane wave).

You remember, we started with the fact that Newton’s equation of motion is not applicable to (as it is not experimentally verifiable for) microscopic particles due to (comparable and not exactly known) additional forces during the process of observation.

Now, we see an opportunity in the above mentioned example of a freely moving beam of particles and two-slit apparatus giving the value of the linear momentum of the particles. The opportunity is in the form of taking the equation of motion as an operator equation, where an operator operating on a function representing the propagating wave, gives the value of the linear momentum of the particle/wave. If this proposition is possible, then the so-called operator equation or more generally operator mechanics shall replace classical mechanics, at least for the present case of freely moving particles.

On one hand, the beam of mono-energetic (freely moving) particles (propagating in the form of plane waves in x-direction) meet the two-slit apparatus which gives information about the linear momentum of the particle. On the other hand, the propagating wave represented by the (wave) function (say) ψ(x, t) and the corresponding operator (i.e., the operator corresponding to the physical quantity of interest, which is linear momentum in the present case) operating on the wave function ψ(x, t) gives the value of the linear momentum of the particle. So, the operator equation for the particular case under consideration is like this:

‘Operator (of linear momentum) operating on the wave function gives the value of the linear momentum and leaves the wave function unchanged’.

or

or

Here (denoting the operator for the x-component of linear momentum) operates on the wave function ψ(x, t) and gives the eigenvalue p (of operator in state ψ(x, t). The equation of motion in the form of operator equation (1.4), the backbone of operator mechanics, is called as the Schrodinger equation. The operator mechanics is called as quantum mechanics for reasons which will become clear in Chapters 2 to 5.

We have written operator equation (1.4) corresponding to the linear momentum operator for a particular case of freely moving particles. In this case, particles have well-defined value of linear momentum [which may be determined by the two-slit experiment or by the operator equation (1.4)]. We may generalize the operator equation (1.4) for any operator. Let us consider a general case of a particle moving in the potential field V(r). In this case, the particle is not having a constant value of its linear momentum but has constant value of its total energy E. Its total energy E may be written as

Therefore, the operator Ĥ (called as Harniltonian operator) corresponding to total energy may be written as

and the corresponding operator equation may be written as

or

which is Schrodinger equation.

The sequence of arguments put forward in this section to arrive at the Schrodinger equation (1.8) is, in fact, a brief summary of what is described in details in Chapters 2 to 4.