2 Origin of Quantum Mechanics II- Inadequacy of Classical Laws of Mechanics

Prof. V.S. Bhasin

epgp books

 

 

 

TABLE OF CONTENTS

 

1.Learning Outcomes

 

2.Introduction

 

3.  Atomic Spectra and Bohr Model of the atom

 

3.1 Emission of Discrete Spectral Lines from Hydrogen Atom

 

3.2 Emission of Discrete Spectral Lines from Hydrogen Atom

 

3.3 Bohr’s Model of Hydrogen Atom

 

3.5 Sommerfeld’s Generalization of Quantum Rules

 

3.6 Space Quantiation and Stern – Gerlach Experiment

 

3.7Limitations of Bohr, Sommerfeld Model

 

4.  Matter Waves : de- Brogle’s Hypothesis

 

4.11 Experimental Verification of Matter Waves

 

5.  Summary

 

1. Learning Outcomes (Times New Roman , size 14)

  • After studying this module, you shall be able to
  • Learn the experimental status on the emission of discreet spectral lines from Hydrogen atom
  • Study the Rutherford Model of the atom
  • Learn Bohr’s model of Hydrogen atom and study the derivation for the expression of the energy levels
  • Know the Stern-Gerlach experiment which gives the experimental support for the predictions of Bohr Model of hydrogen atom Learn Sommerfeld’s generalization of quantum rules given by Bohr and also learn its salient features and finally the limitations of the Bohr-Sommerfeld model Study Space Quantization using Stern-Gerlach Experiment Learn de-Broglie’s Hypothesis and its experimental verification

 

2. Introduction

 

In the preceding module, we studied the quantum nature of electromagnetic radiation at microscopic level. And being aware of the fact that the phenomena of interference, diffraction and polarization of light can be successfully explained by the wave properties, we reached to an important conclusion about the dual nature of electromagnetic radiation, i.e., it has the ability to manifest itself either as waves or as photons. Parallel to these developments, side by side there were equally startling discoveries regarding the atomic structure of matter. In this module, we shall review these attempts and show how these laid the foundations of quantum mechanics. In particular, towards the last section we introduce de-Broglie’s hypothesis of a dual wave-particle character of matter at the atomic level. This hypothesis was experimentally verified later by Davisson and Germer, showing that material particles like electrons possess wave-like properties exhibiting interference and diffraction.

 

3.   Atomic Spectra and Bohr Model of the atom

 

3.1 Emission of Discrete Spectral Lines from Hydrogen Atom

 

By the end of 19th century, a lot of experimental data on the radiation spectrum of atoms was accumulated. In 1885, a major discovery was made by Balmer who studied the spectrum of hydrogen atom and found that in the visible and near ultraviolet regions, this spectrum consists of a series of lines ( denoted by ?? , ??, ??, … ….), now called the Balmer series These lines come closer together as the wavelength decreases till they reach reach a limit at a wavelength λ=3646 ?°. In 1889 J.R. Rydberg found that the apparent regularity in the
lines of Balmer series could be expressed in terms of the wave number, ?⃐ = ?/? . According to Rydberg, Balmer lines are given by

where n=3, 4, 5 . . . . and is a constant for atomic hydrogen. Its value determined from spectroscopic measurements is =RH=1.097*107m-1

Fig.2.1 The Balmer series of atomic hydrogen

 

As shown in Fig.2.1, this spectrum consists of series of lines, denoted by   lying in the visible and near ultraviolet region. Other series of spectra for hydrogen were subsequently discovered. They are known as Lyman, Paschen, Brackett and Pgund series known after their discoverers. It can be noticed that the formula given by Rydberg for the Balmer series could be easily generalized to these series as well by writing the formula in a general form as:


 

 

3.2 The Rutherford Model of the Atom

 

Soon after the discovery of the electrons by J. J. Thomson in 1897, the first outlines of the atomic structure of matter became known when an atom was supposed to consist of a number negatively charged electrons plus a positively charged residue. However, the picture of the atom became clear, when in 1911, Rutherford’s analysis of experimental data on the scattering of α-particles by thin foils of different elements carried out in 1908 by Geiger and Marsden, arrived at the conclusion that all the positive charge and almost all the mass of the atom is concentrated in a positively charged nucleus of dimensions (≈ −   ) compared with the dimensions of the atom as a whole (≅ −   ). This immediately suggested a planetary model of the atom with its structure very much similar to that of the solar system, with the electrons orbiting around the nucleus (like planets orbiting around the sun). The electrostatic force of attraction between each electron and the nucleus holds the nucleus together. It was soon realized that this picture of the atom presents several difficulties from the point of view of classical theory. In the first place, due to the orbital motion, the electrons would get accelerated and emit radiation continuously. Consequently, the electrons thus loosing energy would spiral inwardly and eventually collapse into the nucleus. This result is contrary to experiment, since the atom is known to be very much stable. Moreover, the frequency of the emitted radiation, which coincides with that of the orbital motion, should continually increase, thereby producing a continuous spectrum , instead of the observed discrete spectral lines. It is thus clear that electromagnetic theory, despite its success in explaining large scale phenomena, could not be applied to the processes at the atomic scale.

 

3.3    Bohr’s Model of Hydrogen Atom

 

In 1913, the Danish physicist Niels Bohr realized that to overcome the situation discussed above, a fairly radical departure from the established principles of classical mechanics and electromagnetism would be required to understand the structure of the atom and the atomic spectra. Accepting the Rutherford model of the atom, he proposed the following basic postulates to explain the dynamical behavior of atoms.

 

Bohr’s Postulates:

 

(1) According to his first postulate the system of electrons and nucleus constituting the atom cannot exist in any arbitrary state of motion allowed by classical theory. It can exist only in certain special (orbital) states, each possible state has discrete value of the total energy. The electron in an atom could revolve in certain stable orbits without the emission of radiant energy. These are called the stationary states of the atom.

 

(2) The electron in the atom revolves around the nucleus only in those orbits for which the angular momentum is an integral multiple of (h/2л). In other words, the angular momentum ( L ) of the orbiting electron is quantized : L = n h /(2 л). This is the hypothesis of quantization of angular momentum.

 

(3) Emission or absorption of radiation takes place when the elecytron makes a quantum jump from one stationary state to another. When the electron jumps from a higher energy state, , to lower energy state, , Bohr (following Planck) suggested that the difference in the energy is radiated as a photon of frequency ν, given by the relation

As a consequence of these postulates and using the classical methods, Bohr obtained the expression for the energies of allowed stationary states in the following way: assuming the nucleus to b very heavy compared to the mass of the electron and is therefore at rest, the electrostatic force of attraction between the proton and the electron is equated with the centripetal force, i.e.,

where 0is the permittivity of free space. Using Bohr’s second postulate, the magnitude of the orbital angular momentum of the electron is quantized giving

 

 

Bohr thus established that the frequencies of spectral lines of any atom can be expressed in terms of the difference between spectral terms (or stationary state energies) characteristic of each atom. In fact , a similar relation, known as the Rydberg-Ritz combination principle, was already observed empirically (1905). The striking agreement of the model with the observed spectral lines of hydrogen-like atoms clearly showed that Bohr’s postulates, though ad-hoc in nature, were basically sound.

 

3.4   Experimental support of Bohr’s Model: Franck-Hertz Experiment

 

The existence of such discrete energy levels was confirmed by the Franck-Hertz experiment on the scattering of a beam of mono energetic electrons by atoms. They noticed that as long as the energy of the electron beam was below a certain minimum value, the scattering was purely elastic. If, however, incident energy exceeds this minimum value, it can still accept only the exact amount ( 1 − 0), or ( 2 − 0) , where , 1 2 where , 1 2 refer respectively to the energies of the ground, first excited and second excited states and the atom can go from the ground state to the first and the second excited states by absorbing this difference of energies. When the incident energy is increased sufficiently, inelastic scattering absorbing the discrete amounts of energy was found to takes place.

 

3.5    Sommerfeld’s Generalization of Quantum Rules

 

Besides the remarkable success achieved in explaining the atomic spectrum, it was felt that the quantization condition of the orbital angular momentum employed in the derivation was valid only in the case of circular orbits and could not be used in more complicated cases. Soon it was noticed that the above condition is in fact a special case of the condition

 

Here q refers to some generalized coordinate and p is the corresponding canonically conjugate momentum. Considering p as a function of the position q of the particle on the trajectory, the integral is to be taken over one period. For example in the case of an electron revolving about the hydrogen nucleus in a plane elliptic orbit, with the nucleus at one focus, the single Bohr condition is replaced by the two conditions:

 

Fig. 2.3 Elliptic path of the atomic electron in the Sommerfeld model.

 

making it possible to apply Bohr’s postulates to other more general atomic systems.

 

The quantization of the elliptical orbits of the hydrogen atom by Sommerfeld is worth mentioning because it gives, for the first time, the first example of two general properties which persist in quantum mechanics. First, the property of degeneracy of energy levels of systems. Sommerfeld noticed that the quantized elliptical orbits in a given plane are characterized by two quantum numbers, nr and k, representing the radial and angular parts of the orbital motion. He observed however that the energies of the quantum states depend only on the sum n= nr +k of these quantum numbers. Thus for a given value of the principal quantum number n, there are different states, i.e., elliptical orbits of different eccentricities corresponding to k=1,2,……n, all of these have the same energy En . For example, for n=2, we can have k=1,2, giving 2-fold degeneracy and for n=3, there is 3-fold degeneracy. Such a system with energy level En , when the quantum orbits in one plane are considered, is said to be n-fold degenerate. It has now been understood that this degeneracy in the case of hydrogen atom essentially arises due to the dynamical nature, i.e., 1/r dependence of the electrostatic potential.

 

The second general property, which is built in the treatment given by Sommerfeld is the removal of degeneracy when the symmetry is broken. This is due to the fact that the variation of the speed of the electron in hydrogen atom as it moves along an elliptical orbit induces corresponding changes in its mass given by the theory of relativity. This mass variation, Sommerfeld showed, gives rise to a slow precession of the orbit in its own plane, i.e., it causes the change of orientation of the ellipse at a steady state. As a result the energy associated with each orbit is changed by a small amount depending on the quantum number k. As a consequence

Fig, 2.4     Splitting of the energy levels of hydrogen atom due to relativistic effect.

 

of this, the nth Bohr level splits into n-closely-spaced levels, thereby removing the degeneracy. The spectral lines, split in this way, give rise to what is called a fine structure.

 

3.6   Space Quantiation and   Stern – Gerlach Experiment

 

In the above, while discussing the degeneracy of the nth Bohr level, we considered the orbits in one plane only. In other words, we considered only the orbits with respect to a particular direction for the angular momentum vector, which is normal to the plane of the orbit. In fact, there is further degeneracy arising from the possibility of various orientations for the angular momentum vector with respect to a fixed axis. This axis may refer to the direction of some externally applied field (say, for instance, the magnetic field).

Fig. 2.5 Space quantization of the angle θ. For l=2, five values of θ are possible.

 

Specifically, the component of the angular momentum parallel to the axis has to be m , quantum number m taking integer values only. For orbital angular momentum of magnitude l, the values which m can take are limited to m= -l, -l+1, -l+2, ………..l-2, l-1and l, i.e., (2l+1) values. This is called space quantization.

 

Fig.2.6 Schematic diagram of Stern- Gerlach Experiment showing splitting of the beam of atoms between the pole pieces of electromagnet producing inhomogeneous magnetic field.

 

This was the underlying principle of Stern-Gerlach experiment. A collimated beam of silver atoms was passed through an inhomogeneous field produced by an electromagnet. When the atomic beam was passed between the pole pieces traversing the whole length of varying strength of the field, the beam split into two parts, one part deflected upwards and the other downwards. While this splitting of the beam into two directions demonstrated confirmation of space quantization, it is worth mentioning that this appearance of two directions resulting in the corresponding two values for z could be explained only after the discovery of the spin of the electron. The spin quantum number characterizing the spinning motion of the electron has the value ½, unlike the integral values of the orbital angular momentum. This has two values of projections:

3.7 Limitations of Bohr, Sommerfeld Model

 

It was felt from the beginning that old quantum theory of Bohr and Sommerfeld is, in fact, a strange mixture of incorporating two diagonally opposite views: while at one hand, it invokes quantum mechanical concepts, at the same time, it uses the methods of classical mechanics. It is, therefore, not really a fundamental theory. The domain of the Bohr-Sommerfeld quantum rules is limited to periodic or multiply-periodic motions and therefore has limited applicability. While the postulates, on which this theory is based do provide a basis for understanding the stability of the atoms and the observed features of atomic spectra, the question, which remains unanswered, is : why certain orbits should be completely stable and others not allowed to exist?

 

4    Matter Waves : de- Brogle’s Hypothesis

 

As a consequence of this hypothesis, de-Broglie explained the Bohr model of the atom by visualizing that an electron orbiting around a nucleus could be thought of as having wave-like properties and showed that it will be observed only in situations that permit a standing wave around the nucleus.

 

Thus using Bohr’s first postulate of hydrogen atom, angular momentum L of the electron in a circular orbit of radius r is

He, in fact, derived the Bohr-Sommerfeld quantization rules showing that quantized energy levels appear analogous to the normal modes of a vibrating string.

 

4.1 Experimental Verification of Matter Waves

 

The wave-like nature associated with electrons was demonstrated by showing that a beam of electrons could exhibit diffraction just like a beam of light. Electron diffraction experiments carried out by Davisson and Germer, in 1927 and by G.P Thompson, in 1928, who observed diffraction effects with beams of electrons scattered by crystals. A schematic arrangement of the experimental arrangement is as shown in the figure.

Fig. 2.8 Schematic diagram of the Davisson Germer experiment. Monoenergetic beam of electrons is scattered by a crystal and detected by the detector D .

 

 

Electrons emitted by a tungsen filament are accelerated to a desired velocity by applying suitable voltage. This collimated beam is made to fall on the surface of nickel crystal. The electrons are scattered in all directions by the atoms of the crystal. The intensity of the electron beam, scattered in a given direction is measured by the electron detector. The detector can be moved on a circular scale and is connected to a sensitive galvanometer to record the current. The whole apparatus is enclosed in an evacuated chamber. The intensity of the scattered beam is measured for different values of angle of scattering θ. The variation of the intensity I of the scattered electrons with the angle of scattering θ is obtained for different accelerating voltages.

 

It was noticed that a strong peak appeared in the intensity I of the scattered electrons for an accelerating voltage was of 54 V at a scattering angle θ=50°. The appearance of the peak in a particular direction is due to the constructive interference from different layers of the regularly spaced atoms of the crystal.

 

The de-Broglie wavelength λ associated with electrons is given by

 

 

This is found to be in rather good agreement between the theoretical value and the experimentally observed value of the de-Broglie wavelength.

 

The de-Broglie hypothesis has been basic to the development of modern quantum mechanics. Actually, immediately after de-Broglie put forth this hypothesis, Erwin Schrodinger proposed that the behavior of matter waves associated with material particles is described by a wave function , which represents the matter wave. The motion of atomic particle, free or subject to a force, is governed by a certain differential equation. Schrodinger was thus able to explain the basic fact of quantization as a consequence of the wave nature of matter and to replace the ad-hoc quantization rules of the old quantum theory.

 

5. Summary:

  • After studying this module, you shall be able to
  • Learn the experimental status on the emission of discreet spectral lines from Hydrogen atom
  • Study the Rutherford Model of the atom
  • Learn Bohr’s model of Hydrogen atom and study the derivation for the expression of the energy levels
  • Know the Stern-Gerlach experiment which gives the experimental support for the predictions of Bohr Model of hydrogen atom
  • Learn Sommerfeld’s generalization of quantum rules given by Bohr and also learn its salient features and finall the limitations of the Bohr-Sommerfeld model
  • Study Space Quantization using Stern-Gerlach Experiment
  • Learn de-Broglie’s Hypothesis and its experimental verification

table>

you can view video on Origin of Quantum Mechanics II-Inadequacy of ClassicalLaws of Mechanics

References

 

For More Details ( on this topic and other topics discussed in Text Module) See

  1.  L.D. Landau & E.M.Lifshitz (1977). Quantum Mechanics: Non-Relativistic Theory. Vol. 3 (3rd ed.). Pergman Press
  2. Messiah, Quantum Mechanics, Vol. 1, Wiley, NY, 1961
  3. Quantum Mechanics by Bransden and Joachain, Pearson Education Ltd(2000).
  4. Quantum Mechanics ,Vol. I, by C. Cohen- Tanoudji , John Wiley & Sons
  5. A Text Book of Quantum Mechanics by P M Mathews and K Venkatesan; Tata McG raw-Hill Publishing Co. Ltd

 

For General Study on Origins of Quantum Theory:

 

1 Introducing Quantum Theory by J P Mc Evoy, University of Pittsburgh

2. The Origins of the Quantum Theory by Cathryn Carson