Quantum mechanics, Physics tutorial

Introduction to Quantum mechanics:

Define: Quantum mechanics is basically a physical science mainly dealing with the behavior of matter and energy on the scale of atoms and subatomic particles or waves.

Quantum mechanics, mainly attempts to explain and account for the properties of atoms and molecules and their components: electrons, protons and neutrons, and other more esoteric particles like quarks and gluons. Such properties comprise the interactions of the particles by one another and by means of electromagnetic radiation (that is, light, X-rays and gamma rays).

The behavior of matter and radiation on the atomic scale frequently looks like strange and the effects of quantum theory are accordingly complex to understand and to accept. Its theories and concepts often conflict by the common-sense notions derived from the observations of the day by day world. There is no reason, though, why the behavior of the atomic world must be conventional to that of the well-known, large-scale world. This is significant to understand that quantum mechanics is a stream or branch of physics and that the business of physics is to explain and account for the way the world - on both the big and the small scale - in reality is and not how one visualizes it or would like it to be.

Difficulties with Classical Physics:

The Classical Physics primarily deals by the macroscopic phenomena. Most of the effects by which the classical theory is mainly concerned are either directly apparent or can be made observable by means of relatively instruments. There is a close link between the world of classical physics and the world of sensory observation.

Throughout the decades of 20th century, Physicists turned their concentration to the study of atomic systems that are inherently unreachable to direct observation. This soon became obvious that the theories, concepts and methods of classical macroscopic physics couldn't be applied directly to the atomic phenomena.

Birth of Quantum Mechanics:

The early growth of atomic theory comprised of efforts to overcome such difficulties via modifying the laws of Classical Physics. Such efforts reached their successful conclusion in the period from the year 1925 to 1930, when a completely new theoretical discipline, Quantum Mechanics was introduced by Schrödinger, Heisenberg, Dirac and others.

The Quantum Mechanics can be considered as the basic theory of atomic phenomena. The experimental data on which it is based are derived from the physical events which lie almost entirely beyond the range of human perception. This is not surprising, that the theory embodies physical theories that are foreign to common daily experience.

The study of quantum mechanics is rewarding for some reasons. At first, it describes the necessary methodology of physics. Second, it has been very successful in giving right results in practically each and every situation to which it has been applied. There is, though, an intriguing paradox. Despite of the overwhelming practical success of the quantum mechanics, the basics of the subject include unresolved problems - in particular, problems regarding the nature of measurement. A necessary characteristic of quantum mechanics is that it is generally not possible, even in principle, to measure a system devoid of disturbing it; the detailed nature of this disturbance and the precise point at which it takes place are vague and controversial. Therefore, quantum mechanics attracted some of the ablest scientists of the 20th century, and they erected what is possibly the finest intellectual structure of the period.

Blackbody radiation and Planck hypothesis:

Basic concepts of blackbody radiation:

The conduction, convection and radiation are the various types of energy transmission.

Conduction is the transfer of heat energy through molecular vibrations not through actual motion of material. For illustration, if you hold one end of an iron rod and the other end of the rod is put on a flame, you will sense hot after some time. We can state that the heat energy reaches your hand by means of heat conduction.

Convection is the transfer of heat by means of actual motion of material. The hot-air furnace, the hot-water heating system and the flow of blood in the body are illustrations.

Radiation is process in which the heat reaching the earth from the sun can't be transferred either by conduction or convection as the space between the earth and the sun comprise of no material medium. The energy is carried through electromagnetic waves which don't need a material medium for propagation. The type of heat transfer is termed as thermal radiation. Blackbody radiation problem was found in the research of thermal radiation.

Blackbody is stated as the body which can absorb all energies which fall on it. This is similar to a black hole. No lights or material can get away from it as long as it is trapped. A big cavity having a small hole on its wall can be taken as a blackbody.

Blackbody radiation: Any radiation which enters the hole is absorbed in the interior of the cavity, and the radiation emitted from the hole is termed as the blackbody radiation.

Planck's Hypotheses:

1) The atoms and molecules compose the blackbody concave can be regarded as the linear harmonic oscillator by electrical charge.

2) The oscillators can just be in a special energy state. All such energies should be the integer multiples of a smallest energy (εo = hν). Thus the energies of the oscillators are E = n hν having n = 1, 2, 3 ...

3) hν was termed photon by Einstein and

E = hv = hc/λ                                                                                                                         

is termed as Planck-Einstein quantization law. Here 'h' is known as Planck constant.

h = 6.626 x 10-34 J.s

h = 4.136 x 10-15 eV.s                                                                                                                       

The relation between joule (J) and Electron Volt (eV) is represented by:

1ev = 1.602 x 10-19 J

1J = 6.242 x 10-18 eV 

Blackbody Radiation and the Quantum Theory:

A blackbody is a surface which:

a) Completely absorbs all the incident radiation.

b) Emits radiation at the utmost possible monochromatic intensity in all directions and at all wavelengths.

The concept of the energy distribution of blackbody radiation was introduced by Planck and first appeared in the year 1901.

Planck hypothesized that energy can be absorbed or emitted only in discrete units or photons having energy:

E = hν

The constant of proportionality is h = 6.626 × 10-34 J s.

Planck represented that the intensity of radiation emitted through a black body is represented by:

Bλ = c1λ-5/exp (c2/λT) - 1

Here c1 and c2 are constants

c1 = 2πhc2 = 3.74×10-16 Wm-2

c2 = hc/k = 1.44×10-2 mK

The function Bλ is termed as the Planck function.

The Blackbody radiation is isotropic.

If Bλ(T) is plotted as a function of wavelength on the linear scale the resultant spectrum of monochromatic intensity represents the shape described as shown below:

1799_Blackbody radiation.jpg

Blackbody emission (that is, the Planck function) for absolute temperatures as pointed is plotted as a function of the wavelength on a linear scale.

Photoelectric Effect:

Photoelectric effect is the emission of electrons from metal surfaces if electromagnetic radiation of high adequate frequency falls on them. The consequence is given through zinc if exposed to X-rays or ultraviolet. Sodium provides emission with X-rays, ultraviolet and all colors of light apart from orange and red, whereas preparations having caesium respond to infrared and also to high frequency radiation.

The effect was first observed by the German Physicist Heinrich Hertz in the year 1887 throughout the course of his experiments by the first simple radio transmitter. He noticed that a spark jumped more readily between the terminals of his high-voltage source if they were irradiated by means of ultraviolet rays. It was soon exhibited that negatively charged particles were emitted by means of some metals under irradiation. The charge-to-mass ratio of the particles was that of electrons, and physicists soon agreed that the particles were certainly electrons.

Einstein's Interpretation of the Photoelectric Effect:

The wave theory of light can provide no description for the existence of a threshold frequency. Emission of the electron would be expected as soon as adequate wave energy had been absorbed through the surface.

Einstein, in the year 1905, proposed that the description could be found when electromagnetic radiation were to be considered as made up of particles (photons) whose energy was associated to the frequency of the radiation. This was a growth of an earlier proposal through Planck (1902) who was trying to determine a mathematical basis for the curves of the continuous spectrum.

Einstein's photoelectric theory represents:

E = hf

For the energy of a quantum of radiation, that is, a single photon of frequency 'f':

h = 6.63 x 10-34Js, and is termed as Planck's constant.

When is symbolized the work function by Φ, the maximum kinetic energy by which the photoelectron emerges is:

1/2 mV2 = hf - Φ

What is the significance of quantum mechanics?

The given are among the most significant things that quantum mechanics can explain while classical physics can't:

  • Discreteness of energy
  • Wave-particle duality of light and matter
  • Quantum tunneling
  • Heisenberg uncertainty principle
  • Spin of a particle

Discreteness of energy:

Whenever you look at the spectrum of light emitted via energetic atoms (like the orange-yellow light from the sodium vapor street lights, or the blue-white light from the mercury vapor lamps) you will observe that it is comprised of individual lines of dissimilar colors. Such lines represent the discrete energy levels of the electrons in those excited atoms. If an electron in a high energy state jumps down to the lower one, the atom emits a photon of light that corresponds to the correct energy difference of such two levels (that is, conservation of energy). The bigger the energy difference, the more energetic the photon will be and the closer its color will be to the violet end of the spectrum. When electrons were not constrained to discrete energy levels, then the spectrum from an excited atom would be a continuous spread of colors from red to violet devoid of separate lines.

The theory of discrete energy levels can be explained by a 3-way light bulb. A 40/75/115 watt bulb can only shine light at such three wattage's, and if you switch from one setting to the subsequent, the power instantly jumps to the new setting rather than just gradually increasing.

This is the fact that electrons can only exist at discrete energy levels that prevents them from spiraling into the nucleus, as classical physics predicts. And it is this quantization of energy, all along by some other atomic properties which are quantized that provide quantum mechanics its name.

The wave-particle duality of light and matter:

In the year 1690 Christiaan Huygens explained that the light was comprised of waves, whereas in the year 1704 Isaac Newton described that the light was made up of tiny particles. Experiments supported each of their theories. Though, neither a completely-particle theory nor a completely-wave theory could illustrate all of the phenomena related with light! Therefore scientists start to think of light as both a particle and a wave. In the year 1923 Louis de Broglie hypothesized that a material particle could as well demonstrate wavelike properties, and in the year 1927 it was represented (by Davisson and Germer) that electrons can certainly behave similar to waves.

How can somewhat be both a particle and a wave at similar time? For one thing, it is wrong to think of light as a stream of particles moving up and down in a wavelike mode. In reality, light and matter exist as particles; what behaves similar to a wave is the probability of where that particle will be. The reason light at times appears to act as a wave is because we are observing the accumulation of many of the light particles distributed over the probabilities of where each and every particle could be.

For illustration, assume that we had a dart-throwing machine which had a 5% chance of striking the bulls-eye and a 95% chance of striking the outer ring and no chance of striking any other place on the dart board. Now, assume that we let the machine throw 100 darts, keeping all of them stuck in the board. We can observe each and every individual dart (so we recognize they behave similar to a particle) however we can as well see a pattern on the board of a large ring of darts surrounding a small cluster in the middle. This prototype is the accumulation of the individual darts over the probabilities of where each and every dart could have landed, and represents the 'wavelike' behavior of the darts.

Quantum tunneling:

This is one of the most fascinating phenomena to come up from the quantum mechanics; devoid of it computer chips would not exist, and a 'personal' computer would possibly take up a whole room. As described above, a wave finds out the probability of where a particle will be. When that probability wave encounters an energy barrier most of the wave will be reflected back, however a small part of it will 'leak' to the barrier. When the barrier is small adequate, the wave that leaked via will continue on the other side of it. Even although the particle does not encompass adequate energy to get over the barrier, there is still a small probability which it can 'tunnel' via it!

Let's state you are throwing a rubber ball against a wall. You recognize that you do not have adequate energy to throw it through the wall, so you for all time expect it to bounce back. The Quantum mechanics, though, states that there is a small probability that the ball could go right through the wall (devoid of damaging the wall) and carry on its flight on the other side! With something as big as a rubber ball, although, that probability is very small that you could throw the ball for billions of years and never see it goes via the wall. However with something as tiny as an electron, tunneling is a daily occurrence.

On the flip side of the tunneling, if a particle encounters a drop in energy there is a small probability that it will be reflected. In another words, if you were rolling a marble off a flat level table, there is a small chance that whenever the marble reached the edge it would bounce back rather than dropping to the floor! Again, for something as big as a marble you will probably never see somewhat similar to that happen, however for photons (the mass less particles of light) it is an extremely real occurrence.

Heisenberg uncertainty principle:

People are well-known by measuring things in the macroscopic world around them. Somebody pulls out a tape measure and finds out the length of a table. A state trooper aims his radar gun at a car and recognizes what direction the car is traveling, and also how fast. They get the information they want and do not worry whether the measurement itself has modified what they were measuring. After all, what would be the sense in finding out that a table is 80 cm long if the act of measuring it modified its length!

Werner Heisenberg was the very first to recognize that some pairs of measurements have an intrinsic uncertainty related with them. For example, if you have a very good idea of where something is positioned, then, to a certain degree, you should encompass a poor idea of how fast it is moving or in what direction. We do not observe this in daily life as any inherent uncertainty from Heisenberg's principle is well in the acceptable accuracy we wish.

Heisenberg's uncertainty principle wholly flies in the face of the classical physics. After all, the very basis of science is the capability to measure things accurately, and now quantum mechanics is stating that it is not possible to get such measurements precise! Though the Heisenberg uncertainty principle is a fact of nature, and it would be not possible to form a measuring device that could get around it.

Spin of a particle:

In the year 1922 Otto Stern and Walther Gerlach performed an experiment whose results could not be described by classical physics. Their experiment pointed out that atomic particles have an intrinsic angular momentum, or spin, and that this spin is quantized (that is, it can only encompass certain discrete values). Spin is a totally quantum mechanical property of a particle and can't be described in any way through classical physics.

This is significant to realize that the spin of an atomic particle is not a measure of how it is spinning! However, it is not possible to tell whether something as small as an electron is spinning at all! The term 'spin' is just a convenient manner of talking about the intrinsic angular momentum of the particle.

Magnetic resonance imaging (or MRI) employs the fact that beneath certain conditions the spin of hydrogen nuclei can be 'flipped' from one state to the other. By measuring the position of such flips, a picture can be formed of where the hydrogen atoms (mostly as a part of water) are in a body. As tumors tend to encompass a different water concentration from the surrounding tissue, they would stand out in such a picture.

What is the Schrödinger equation?

Each and every quantum particle is characterized through a wave function. In the year 1925 Erwin Schrödinger developed the differential equation that explains the evolution of those wave functions. By employing Schrödinger's equation scientists can determine the wave function that resolves a specific problem in quantum mechanics. Unfortunately, it is generally not possible to determine an exact solution to the equation, so certain suppositions are used in order to get an approximate answer for the specific problem.

1669_schrodinger-equation.gif

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