PVB302-Quantum Mechanical Model Of Hydrogen Atom

Question:

Define the fundamental difference between quantum and classical descriptions for the electron that is bound at the atom’s core.

Describe the meaning of these differences in terms of stability and energy of electrons.

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In general terms, explain the differences between the quantum and classical electron models in terms our ability to determine such things as the electron’s location and its momentum.

As examples, you can use an electron around the hydrogen-atom and an electron with infinite potential to illustrate your point.

Explain why electrons can tunnel through barriers and do things like quantum tunneling.

Answer:

Introduction

After the failed Rutherford model of an Atom proposed in 1911 that was not able explain the stability the atom, Niels Bohr, who is a Danish Physicist suggested in 1913 that electrons could only orbit within discrete orbits or shells having constant radius around nucleus.

Bohr’s Formula gives the radius of these shells. The electron can not exist in these shells.

Friedrich (2006) claims that the model did not fully explain the Hydrogen Spectrum. However, it was able, according to Friedrich (2006) to explain the Zeeman Effect, Heisenberg Uncertainty Principle or atomic spectrum of heavy atoms.

Quantum Mechanics was added to the model.

Quantum Mechanical Model Of Atom

1.Classical & Quantum Mechanical Models

The quantum model of electron describes electron as a wave, with wavefunction square. This gives us the probability to locate the electron at (x) according to Gasiorowicz (2007).

Neils Bohr had 3 postulates concerning the motion of electrons in an atom prior to quantum mechanics.

These postulates were stated in Saraswati (2017).

An electron can rotate around the nucleus within certain fixed orbits that have definite energy. However, there is no radiation.

These orbits are called stationary orbits.

An electron can change the state of an atom to one of low energy and stationary E2 energy. It emits one photon per second.

n = E2-E1/H

where h represents the Planck’s constant.

However, when an electron is at energy energy E1, it can absorb energy (E2-E1) and make the transition from E1 into E2.

Only those orbits can be found in an atom that have the orbital angle momentum (L), of an electron, which is an integral multiple h/ 2p.

L = where the n = 0, 1, 2, ,…..

Griffiths (2004) says that the electron’s wave particle duality tells us that it exhibits both particle and wave natures but not simultaneously.

This dual nature was shown by the double slit experiment.

The screen shows an interference pattern when both slits open. This is even when a single electron passes through one slit at once, confirming the wave nature of the experiment.

The wave nature disappears as soon as we attempt to measure by closing one slit in order to find which electron passes through, and then particle nature emerges.

Bohr’s Postulates were based in the fact that an electron is a particle. It was determined that its energy was comparable to the electron’s energy calculated using Quantum Mechanics methods. The electron was considered a wave, i.e.

E =

Further support for the wave particle duality was provided by deBroglie hypothesis, who proposed that all particles must be associated with a wave named Matter Waves whose wavelength() corresponds to the formula-

Wavelength of matter waves () =h/mv

h – Planck’s Constant

m = the mass of a particle

v is the velocity of a particle.

2. As stated in Griffiths (2013), Maxwell’s Electromagnetic Theory states that an accelerated charged particle radiates energy.

Rutherford’s classical model predicted a circular motion for the electron.

Because it is circular, the electron will accelerate towards its centre. It will then radiate energy. As a result, its radius of circle path will eventually become smaller. This will allow the electron to jump into the nucleus.

After releasing all the energy from the electron, the atom will become unstable and vanish.

Furthermore, this model doesn’t explain the experimentally observed Hydrogen Spectral line.

Figure 1

Bohr proposed that the constraint on the stability of an atom was removed and that Hydrogen Atom could explain the spectra. He postulated that electrons revolve in fixed orbits known as Stationary Orbits, which are fixed orbits with discrete energy levels.

According to Bohr, an atom cannot radiate energy if there is a change from a high to a lower energy level.

He found the formula for radius/velocity of electron in orbit given by Kumar, 2009.

rn =

Figure – 2.

Bohr’s Theory couldn’t explain the atomic spectra non-Hydrogenic nuclei.

The spectral lines that split on the application magnet field (which is nothing but the Zeeman Effect) was also not explained by Bohr’s Theory.

The quantum mechanical model, which takes into consideration electron spin in the atom, later explained it.

3.The particle nature of an electron allowed simultaneous measurement the momentum and the position of the particle.

An electron moving in a circular orbit of Hydrogenatom atom has its radius precisely.

Because it is in a circular orbit, however, its radial velocity cannot be greater than 0.

This violates the uncertainty principle because we simultaneously have exact knowledge of pr and r.

We now consider the quantum mechanical model Hydrogen atom, which is electron in an infinite potential well. Electron is a wave whose energy and wavefuntion are given by-

= n = 1, 2 ,….

Bransden and Joachan (2003,) state that one sinusoidal wave has an exact measurable wavelength (). This means that the electron presented to deBroglie by a sine wave is nothing but the matterwave. It also has a precise momentum.

A single sine wave, however, continues in both directions. The wave is not located anywhere.

Therefore, it is impossible to know the exact position of an electron.

A resultant wave is a combination of several sine waves each with a unique wavelength.

Addition of more sine waves together creates a localized resultant wave that is more precise and gives us less uncertainty about where the electron is located.

The wavelength that meets deBroglie’s formula to calculate electron’s momentum is not known. However, the resultant waves are made up of multiple wavelengths (wavelengths of the sinusoidal wave) which exist simultaneously.

The electron’s momentum is therefore uncertain to some degree.

If more sine waves are added, the wave will give the electron more localization. However, there will be more uncertainty in its momentum and wavelength represented by the wave.

Therefore, quantum mechanical models satisfy the Heisenberg uncertainty principle

4.According Griffiths (2004), figure -3 shows the diagram for a rectangular potential barrier. It extends from x=0 to x=a.

The potential of the barrier remains constant and equal to U0.

The U potential is 0.

Figure – 3.

Let a stream E of particles of energy be incident from left onto the barrier surface at 0.

The two following cases are possible:

E > U0 will mean that classical mechanics says that particles will be transmitted completely and no reflection can occur.

Quantum mechanically, however, there is always a chance that x=0 or x=a.

If E U0, classically, the particles will be completely reflected and therefore cannot penetrate the barrier.

However, quantum mechanically there is still a possibility of penetration and the appearance of particles within region III.

Quantum Mechanical Tunnelling Effect refers to the finite probability of transmission though the barrier for E U0.

After solving the Schrodinger Equation E U0, we’ll find that the transmission coefficient to allow the particle to pass through the barrier is by-

T =

Here, =; m – mass of the particle

It is assumed that the particle can only be transmitted through the potential barrier U0 in height and width, regardless of whether E U0.

This cannot be explained using classical theory.

Transmission coefficient depends upon four factors, namely, the width of the barrier (a), the mass and energy of the particles (m) and the barrier height(U0). We know the mass of human beings is larger than that of the electron (me), which means that mhb> me.

For humans, this means that m large value is.

T = The value of the number that is close to zero

The wavefunction for electron is not affected by a barrier.

So, while electrons can quantum tunnel through potential obstacles, we cannot.

Conclusion

Quantum Mechanics, a new field in Physics, was created by the astonishing discovery of the dual nature electrons.

With quantum techniques, many difficult problems could be solved.

It is important to remember that quantum mechanics only applies to very small particles, such as the proton and electron.

Quantum Mechanics allows us to predict probabilistic results, unlike Newtonian mechanics.

Many technologies have been made possible by this new field of science, including Scanning tunnelling microscope that utilizes the tunnelling effect and Quantum Dots and Quantum Computations.

Quantum Mechanics can be used to solve the mystery of the structure of atoms.

Refer to

Joachan C.J.

Physics of Atoms, Molecules.

London: Pearson

Theoretical Atomic Physics.

New Jersey Wiley

Ghoshal S.N.

Atomic Physics.

Griffiths D.J.

Introduction to Electrodynamics.

London: Pearson Education

Griffiths D.J.

Introduction to Quantum Mechanics.

London: Pearson Education

Fundamentals of Quantum Mechanics.

Cambridge University Press

Atomic and Molecular Physics.

Saraswati V. (2017) Quantum Mechanics Atomic and Molecular Physics.

Delhi: Himanshu Publications

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