1. Explain what is fundamentally different between the classical and quantum descriptions of the electron that is bound to the atom’s nucleus and describe the role that wave-particle duality plays.
2. Describe what these differences mean in terms of the atom’s stability and the energy of the electron.
3. In a general sense, describe and explain what the differences between the classical and quantum models of the electron lead to in terms of our ability to know such things as the electron’s location and its momentum. Use the electron around the hydrogen atom and an electron in an infinite potential well as examples to demonstrate your point.
4. Explain why electrons can do things like quantum tunnel through potential barriers yet we cannot.
Quantum Mechanical Model of Atom
After the failure of the Rutherford model of an atom proposed in 1911 which was not able to explain the stability of the atom; Niels Bohr, a Danish Physicist, suggested in 1913 that the electrons could only orbit in discrete orbits or shells with a constant radius around the nucleus. The electron could not exist in between these shells and occupy only those shells whose radius is given by Bohr’s Formula. According to Friedrich (2006), though the model successfully explained the Hydrogen Spectra precisely, it was not able to explain Zeeman Effect, Heisenberg Uncertainty Principle and atomic spectra of heavy atoms. Further, the model was modified by the Introduction of new field of Physics namely Quantum Mechanics.1.Classical and Quantum Mechanical Models: According to Verma (1993) , the Classical Model for an atom suggests that the electron revolving around the nucleus is a particle and the force governing its motion is Electrostatic force of attraction. On the other hand, the quantum model for electron defines electron as a wave with wavefunction square of which gives us the probability of locating the electron at the position (x) as given in Gasiorowicz (2007). Before the evolution of quantum mechanics, Neils Bohr gave 3 postulates regarding motion of electron in an atom. As stated in Saraswati (2017), these were-
An electron can revolve around the nucleus in certain fixed orbits of definite energy without emission of any radiant energy. Such orbits are called stationary orbits.
An electron can make a transition in the atom from a stationary state of high energy E2 to a state of low energy E1 and in doing so, it emits a single photon with frequency,
ν = E2-E1/H
where h is the Planck's constant.
Conversely, on absorbing an energy (E2−E1) when the electron is at energy E1, the electron can make a transition from E1 to E2.
Only those orbits are allowed in the atom corresponding to which the orbital angular momentum (L) of the electron is an integral multiple of h/ 2π,
L = where n = 0, 1, 2,.....
According to Griffiths (2004), the wave particle duality of electron tells us that the electron exhibit both particle and wave nature but not simultaneously. This dual nature was demonstrated by the famous double slit experiment. The screen shows the interference pattern when both the slits are open (even when single electron passes through the slit at a time) confirming its wave nature. As soon as we try to make a measurement by closing one slit so that we could find through which slit the electron passes, the wave nature vanishes and particle nature comes into picture.
Bohr’s Postulates were based on the fact that the electron is a particle and its energy was calculated which was found to be similar to the energy of the electron calculated using Quantum Mechanics methods considering electron as a wave i.e.,
The wave particle duality was further backed up by deBroglie hypothesis who claimed that every particle must be associated with a wave called Matter Waves whose wavelength() is given by the formula-
Wavelength of matter wave () =h/mv
h - Planck’s Constant
m - the mass of the particle
v - velocity of the particle.
2.As stated in Griffiths (2013), according to Maxwell’s Electromagnetic Theory, an accelerated charge particle radiate energy. Rutherford’s classical model suggested the circular motion of the electron. Due to its circular motion, the electron will accelerate toward the centre and thus it will radiate energy which implies it will loose energy and its radius of circular path will eventually get smaller and smaller resulting in the jump of electron into the nucleus. Therefore, the atom is not stable and will vanish after radiating all its energy of the electron. Moreover, this model didn’t explain the various Hydrogen Spectral lines which were observed experimentally.
Classical Model of Atom and its Problems
Figure - 1
Bohr removed constraint of unstability of the atom and also explained the spectra produced by Hydrogen Atom by postulating that electron revolve in fixed orbits called Stationary Orbit of fixed orbits with fixed discrete energy levels. According to him, an atom can only radiate energy whenever there is a transition from a high energy level to a low energy level. He calculated the formula for radius and velocity of the electron in the orbit given by (Kumar, 2009),
rn = and
Figure – 2
But Bohr’s Theory was not able to explain the atomic spectra of non-Hydrogenic atoms. Also, it didn’t explained the spectral lines splitting on the application of magnetic field which is nothing but Zeeman Effect. It was later explained by the quantum mechanical model taking into consideration the electron’s spin in the atom. (Kumar, 2018)
3.The particle nature of the electron gave simultaneous measurement of the momentum and position of the particle. For an electron moving in a circular orbit of Hydrogen atom, we know its radius exactly i.e., rn = ( Z = 1) and thus ?r = 0. However, since it is moving in a circular orbit, it cannot have any radial velocity, and thus pr = 0 and ?pr = 0. Thus, we have simultaneous exact knowledge of both r and pr which violates the uncertainty principle.(Ghoshal, 2010)
Now, considering the quantum mechanical model of Hydrogen atom i.e., electron in an infinite potential well, we consider electron as a wave whose wavefuntion and energy is given by-
= sin ( ) n = 1, 2, 3,...
= n = 1, 2, 3,....
According to Bransden and Joachan (2003), a single sinusoidal wave has a precise measurable wavelength (), so the electron presented by a sine wave which is nothing but the matter wave suggested by deBroglie has a precise or definite momentum. But a single sine wave keeps going in both the directions i.e., the wave is not localized anywhere. So, the position of the electron is totally uncertain. When several sine waves having unique wavelength each are added together, we get a resultant wave that is localized to some extent. Adding more sine waves together gives us more localized resultant wave , and it also gives less uncertainty about the electron's location. It is not clear which wavelength satisfies deBroglie's formula for the calculation of electron’s momentum, since, the resultant wave consists of a range of wavelengths ( wavelengths of the sinusoidal waves) existing simtaneously. Thus, this gives us uncertainty about the electron's momentum to some extent. When more sine waves are added to the wave, the wave after addition will give more localization of the electron but there will also be more uncertainty in the momentum and wavelength of the electron represented by the resultant wave . Therefore, the quantum mechanical model satisfies the Heisenberg uncertainty principle
4.According to Griffiths (2004), the diagram of a rectangular potential barrier is represented in figure - 3 which extends from x = 0 to x = a. The potential of the barrier is constant and equal to U0. On the left and right side of the barrier the potential U = 0.
Figure – 3
Considering that a stream of particles of energy E be incident from left on the barrier surface at x = 0. The following two cases arise-
- If E > U0, then according to classical mechanics the particles will be wholly transmitted and no reflection is possible. But quantum mechanically there is always some probability at x = 0 and x = a
- If E < U0, then classically the particles will be wholly reflected and hence penetration through the barrier is impossible. But quantum mechanically there is always some probability of penetration into the barrier and appearance of the particles in region III. This finite probability of transmission through the barrier even for E < U0is called the Quantum Mechanical Tunnelling Effect.
After solving the Schrodinger Equation for the case E < U0 , we will find that the transmission coefficient for the given particle to pass through the potential barrier is given by-
Here, = ; m - mass of the particle
From T 0 implies that there is a finite probability of transmission of the particle through the potential barrier of height U0 and width a even if E < U0 . This cannot be accounted for in view of classical theory. Transmission coefficient depends on 4 factors i.e., width of barrier (a), mass of particle (m), Energy of the particle (E), and the barrier height (U0) We know that mass of human beings (mhb) is much larger than the mass of the electron (me), i.e., mhb >> me. . So, in case of humans, m large value , so,
sinh2() = = >> 1
T = which will be nearly zero for
While for electron, it the wavefunction will retain its amplitude even after penetrating a small barrier.
Hence, electrons can do quantum tunnel through potential barriers yet we cannot.
The spectacular discovery of dual nature of electron gave rise to a new field of Physics called Quantum Mechanics. Many complex problems of the nature could be solved now using quantum techniques. One thing that should be kept in mind that the quantum mechanics is applicable only for very small particles like proton, electron, etc. Quantum Mechanics gives us probabilistic results unlike the Newtonian mechanics. Numerous technology have been given using this new field of science like Scanning Tunnelling Microscope which uses tunnelling effect of the electron, Quantum Dots and Quantum Computations and many more. In the end , we can conclude that the riddle of the structure of atoms is successfully explained with the help of Quantum Mechanics.
Bransden B.H. and Joachan C.J. (2003). Physics of Atoms and Molecules. 2nd Edition. London: Pearson
Friedrich H. (2006). Theoretical Atomic Physics. 3rd Edition. Berlin: Springer
Gasiorowicz S. (2007). Quantum Physics. 3rd Edition. New Jersey: Wiley
Ghoshal S.N. (2010). Atomic Physics. Delhi: S.Chand
Griffiths D.J. (2013). Introduction to Electrodynamics. 4th Edition. London: Pearson Education
Griffiths D.J. (2004). Introduction to Quantum Mechanics. 2nd Edition. London: Pearson Education
Kumar A. (2018). Fundamentals of Quantum Mechanics. Cambridge: Cambridge University Press
Kumar R. (2009). Atomic and Molecular Physics. Meerut: Campus Books
Saraswati V. (2017) Quantum Mechanics Atomic and Molecular Physics. Delhi: Himanshu Publications
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