Quantum mechanics
Quantum mechanics [part-1]
The special theory of relativity is about the interdependence of space-time relation with reference to different frames of reference.1905 Einstein. the speed of light is always constant, independent of the frame of reference. It makes no difference whether a source of light is stationary or in relative motion.
Quantum mechanics deals with very fast-moving
and very small objects, the electromagnetic radiation emitted by the
Sun, the photons, as Einstein called it; [like the sub-atomic
particles]; unlike the classical physics of large bodies and of relatively slow motions.
Rutherfords conclusion:
Rutherford's Alpha particle scattering experiment.
-Alpha particles are charged helium atoms.
In doing so he discovered the nucleus of the atom.
The phenomenon of light radiation:
By the end of the 19th century, physicists almost universally accepted the wave theory of light. However, though the ideas of classical physics explain interference and diffraction phenomena relating to the propagation of light, they do not account for the absorption and emission of light.
light also behaves as a particle. dual nature of light. the wave-particle duality. some times light behaves as a wave and sometimes as a particle. but not both at the same time. the particle nature of light is called as, photon.
Photon is a massless particle of energy quantum. Similar to rainwater falling drop by drop. and not continuously.
Quantum mechanics originated from German physicist Planck.
The equation for the quantum of energy is given by the relation,
E = hv where h is a constant called Planck's constant. The value is, h =6.6261 × 10-34 Js.
The photoelectric effect:
Einstein explained photo-electric emission using Planck's concept of energy quanta. light quanta. light is a stream of energy particles called photons.
The phenomenon of emission of electrons from the surface of the metal when the light of suitable frequency falls on it is called the photoelectric effect. The current produced due to emitted electrons is called photocurrent. The photoelectric effect proves the quantum nature of radiation. The relation for a quantum of energy, E = hf is called the Planck-Einstein relation.
Threshold energy. work function. the Planck-Einsein relation, E = hf
Kmax =hf - ɸ Or Kmax + ɸ = hf
Bohr explained the hydrogen atom by placing electrons in discrete energy levels of an atom. the energy of an electron is quantized in an orbital. when an electron jumps from a higher energy state to a lower energy state, it emits energy out as electromagnetic radiation of light. the electron's energy levels are discrete. they are quantized. therefore hydrogen atoms have unique spectral lines.
Max Planck
Planck studied at the Universities of Munich and Berlin, where his teachers included Kirchhoff and Helmholtz, and received his doctorate of philosophy at Munich in 1879.
Planck’s earliest work was on the subject of thermodynamics, an interest he acquired from his studies under Kirchhoff, whom he greatly admired. He published papers on entropy, thermoelectricity, and also on the theory of dilute solutions.
The problems of light radiation processes engaged his attention and he showed that these radiations were to be considered, as electromagnetic in nature. Experimental observations on the wavelength distribution of the energy emitted by a black body as a function of temperature were at variance with the predictions of classical physics.
Planck was
able to deduce the relationship between the energy and the frequency of
radiation. In a paper published in 1900, he announced his derivation of
the relationship: this was based on the revolutionary idea that the
energy emitted by a resonator could only take on, discrete values or
quanta. The energy for a resonator of frequency v is hv where h is a universal constant, now called Planck’s constant. this constant h opened the door for a new theory called quantum mechanics.
This was not only Planck’s most important work but also marked a turning point in the history of physics. The importance of the discovery, with its far-reaching effect on classical physics, was not appreciated at first. However, the evidence for its validity gradually became overwhelming as its application accounted for many discrepancies between observed phenomena and classical theory. Among these applications and developments may be mentioned Einstein’s explanation of the photoelectric effect.
Planck’s work on the quantum theory, as it came to be known, was summarized in two books Thermodynamics (1897) and Theory of heat radiation (1906).
Max Planck and Black-body radiation
The central assumption behind his new derivation, presented to the DPG on 14 December 1900, was the supposition, now known as the Planck postulate, that electromagnetic energy could be emitted only in quantized form, in other words, the energy could only be a multiple of an elementary unit:
E= hv, where h is Planck's constant, ( 6.63×10−34 Js), and ν is the frequency of the radiation. Note that the elementary units of energy discussed here are represented by hν and not simply by ν. Physicists now call these quanta photons, and a photon of frequency ν will have its own specific and unique energy. The total energy at that frequency is then equal to hν multiplied by the number of photons at that frequency.
Planck and Nernst, seeking to clarify the increasing number of contradictions, organized the First Solvay Conference (Brussels 1911).
In recognition of Planck's fundamental contribution to a new branch of physics, he was awarded the Nobel Prize in Physics for 1918 (he actually received the award in 1919).
Albert Einstein
In 1896 Albert Einstein entered the Swiss Federal Polytechnic School in Zurich to be trained as a teacher in physics and mathematics. In 1901, the year he gained his diploma, he acquired Swiss citizenship and, as he was unable to find a teaching post, he accepted a position as technical assistant in the Swiss Patent Office. In 1905 he obtained his doctor’s degree.
The photoelectric effect is a phenomenon where electrons are emitted from the metal surface when the light of sufficient frequency is incident upon. The concept of the photoelectric effect was first documented in 1887 by Heinrich Hertz and later by Lenard in 1902. But both the observations of the photoelectric effect could not be explained by Maxwell’s electromagnetic wave theory of light. Hertz (who had proved the wave theory) himself did not pursue the matter as he felt sure that it could be explained by the wave theory.
Einstein resolved this problem using Planck’s revolutionary idea that light behaves as a particle. The energy carried by each particle of light (called quanta or photon) is dependent on the light’s frequency (ν) as E = hν
Where h = Planck’s constant = 6.6261 × 10-34 Js.
Light, Einstein said, is a beam of particles whose energies are related to their frequencies according to Planck's formula. When the light beam is directed at a metal, the photons collide with the atoms. If a photon's frequency is sufficient to knock off an electron, the collision produces the photoelectric effect.
Since
light is bundled up into photons, Einstein theorized that when a photon
falls on the surface of a metal, the entire photon’s energy is
transferred to the electron. light is a stream of particles, the
photons.
A part of this energy is used to remove the electron from the metal atom’s grasp and the rest is given to the ejected electron as kinetic energy. Electrons emitted from underneath the metal surface lose some kinetic energy during the collision. But the surface electrons carry all the kinetic energy imparted by the photon and have the maximum kinetic energy.
We can write this mathematically as:
The energy of photon = energy required to eject an electron (work function) + Maximum kinetic energy of the electron
E = W + KE
hv = W + KE
KE = hv – w
The thresh hold energy needed to liberate an electron is called work function. this defines the minimum critical frequency needed to liberate photo-electrons.
An increase in the intensity of the same monochromatic light (so long as the intensity is not too high, which is proportional to the number of photons impinging on the surface in a given time), increases the rate at which electrons are ejected—the photoelectric current I—but the kinetic energy of the photoelectrons and the stopping voltage remain the same. For a given metal and frequency of incident radiation, the rate at which photoelectrons are ejected is directly proportional to the intensity of the incident light.
Photoemission from atoms, molecules, and solids
Electrons that are bound in atoms, molecules, and solids each occupy distinct states of well-defined binding energies. When light quanta deliver more than this amount of energy to an individual electron, the electron may be emitted into free space with excess (kinetic) energy that is hv higher than the electron's binding energy. The distribution of kinetic energies thus reflects the distribution of the binding energies of the electrons in the atomic, molecular or crystalline system: an electron emitted from the state at binding energy Eb is found at kinetic energy Ek = hv - Eb. This distribution is one of the main characteristics of the quantum system and can be used for further studies in quantum chemistry and quantum physics.
The Compton Effect
The Compton effect is the term used for an unusual result observed when X-rays are scattered on some materials. By classical theory, when an electromagnetic wave is scattered off atoms, the wavelength of the scattered radiation is expected to be the same as the wavelength of the incident radiation. Contrary to this prediction of classical physics, observations show that when X-rays are scattered off some materials, such as graphite, the scattered X-rays have different wavelengths from the wavelength of the incident X-rays. This classically unexplainable phenomenon was studied experimentally by Arthur H. Compton and his collaborators, and Compton gave its explanation in 1923.
To explain the shift in wavelengths measured in the experiment, Compton used Einstein’s idea of light as a particle. The Compton effect has a very important place in the history of physics because it shows that electromagnetic radiation cannot be explained as a purely wave phenomenon. The explanation of the Compton effect gave a convincing argument to the physics community that electromagnetic waves can indeed behave like a stream of photons, which placed the concept of a photon on firm ground.
Compton effect is a process in which x-rays collide with electrons and are scattered.
By the early 20th century, research into the interaction of X-rays with the matter was well underway. It was observed that when X-rays of a known wavelength interact with atoms, the X-rays are scattered through an angle θ and emerge at a different wavelength related to θ. The experiments had found that the wavelength of the scattered rays was longer (corresponding to lower energy) than the initial wavelength.
In 1923, Compton published a paper in the Physical Review that explained the X-ray shift by attributing particle-like momentum to light quanta (Einstein had proposed light quanta in 1905 in explaining the photo-electric effect, but Compton did not build on Einstein's work). The energy of light quanta depends only on the frequency of the light. In his paper, Compton derived the mathematical relationship between the shift in wavelength and the scattering angle of the X-rays by assuming that each scattered X-ray photon interacted with only one electron. His paper concludes by reporting on experiments that verified his derived relation:
λ' - λ = h (1-cosθ) /me c
- In the Compton effect, X-rays scattered off some materials have different wavelengths than the wavelength of the incident X-rays. This phenomenon does not have a classical explanation.
- The Compton effect is explained by assuming that radiation consists of photons that collide with weakly bound electrons in the target material. Both electron and photon are treated as relativistic particles. Conservation laws of the total energy and of momentum are obeyed in collisions.
- Treating the photon as a particle with a momentum that can be transferred to an electron leads to a theoretical Compton shift that agrees with the wavelength shift measured in the experiment. This provides evidence that radiation consists of photons.
- Compton scattering is an inelastic scattering, in which scattered radiation has a longer wavelength than the wavelength of incident radiation.
The de Broglie hypothesis
De Broglie had intended a career in humanities,
and received his first degree in history. Afterwards he turned his
attention toward mathematics and physics and received a degree in
physics. With the outbreak of the First World War in 1914, he offered his services to the army in the development of radio communications.
His 1924 thesis Recherches sur la théorie des quanta (Research on the Theory of the Quanta) introduced his theory of electron waves. This included the wave-particle duality theory of matter, based on the work of Max Planck and Albert Einstein on light. This research culminated in the de Broglie hypothesis stating that any moving particle or object had an associated wave. De Broglie thus created a new field in physics.
Louis Victor de Broglie was a French physicist and aristocrat who made groundbreaking contributions to quantum theory. In his 1924 Ph.D. thesis, he postulated the wave nature of electrons and suggested that all matter has wave properties. This concept is known as the de Broglie hypothesis, an example of wave-particle duality, and forms a central part of the theory of quantum mechanics.
In 1923, Louis de Broglie proposed a hypothesis to explain the theory of the atomic structure. By using a series of substitutions de Broglie hypothesizes particles to hold properties of waves. Within a few years, de Broglie's hypothesis was tested by scientists shooting electrons and rays of lights through slits. What scientists discovered was the electron stream acted the same way as light, proving de Broglie correct.
He gave the relation λ = h/mv, where λ is the wavelength, h is Planck's constant, m is the mass of a particle, moving at a velocity v. de Broglie suggested that particles can exhibit properties of waves.
where mv = p is the momentum of the particle. The above equation is called de Broglie equation and 'λ' is called the de Broglie wavelength. Thus the significance of the de Broglie equation lies in the fact that it relates the particle character with the wave character of matter.
Louis
de Broglie, in 1924, stated that a wave is associated with a moving
particle (i.e. matter) and so named these waves as matter waves. He
proposed that just like the light has dual nature, electrons also have
wave-like properties.
The wavelength of a moving particle is given by, λ = h/p
where h is Planck's constant and p is the momentum of the moving particle.
de Broglie equation states that a matter can act as waves much like light and radiation, which also behave as waves and particles. The equation further explains that a beam of electrons can also be diffracted just like a beam of light. In essence, the de Broglie equation helps us understand the idea of moving particles of matter having a wavelength.
Experimental Confirmation
In 1927, physicists Clinton Davisson and Lester Germer, of Bell Labs, performed an experiment where they fired electrons at a crystalline nickel target. The resulting diffraction pattern matched the predictions of the de Broglie wavelength. De Broglie received the 1929 Nobel Prize for his theory (the first time it was ever awarded for a Ph.D. thesis) and Davisson/Germer jointly won it in 1937 for the experimental discovery of electron diffraction (and thus the proving of de Broglie's hypothesis).
The DPG held a celebration, during which the Max-Planck medal (founded as the highest medal by the DPG in 1928) was awarded to French physicist Louis de Broglie.
Erwin Schrodinger
In the first years of his career Schrödinger became acquainted with the ideas of the old quantum theory, developed in the works of Max Planck, Albert Einstein, Niels Bohr, Arnold Sommerfeld, and others. This knowledge helped him work on some problems in theoretical physics, but the Austrian scientist at the time was not yet ready to part with the traditional methods of classical physics.
In autumn 1922 he analyzed the electron orbits in an atom from a geometric point of view, using methods developed by the mathematician Hermann Weyl (1885–1955). This work, in which it was shown that quantum orbits are associated with certain geometric properties, was an important step in predicting some of the features of wave mechanics.
Earlier in the same year he created the Schrödinger equation of the relativistic Doppler effect for spectral lines, based on the hypothesis of light quanta and considerations of energy and momentum. He liked the idea of his teacher Exner on the statistical nature of the conservation laws, so he enthusiastically embraced the articles of Bohr, Kramers, and Slater, which suggested the possibility of violation of these laws in individual atomic processes (for example, in the process of emission of radiation).
In January 1926, Schrödinger published in Annalen der Physik the paper "Quantisierung als Eigenwertproblem" (Quantization as an Eigenvalue Problem) on wave mechanics and presented what is now known as the Schrödinger equation. In this paper, he gave a "derivation" of the wave equation for time-independent systems and showed that it gave the correct energy eigenvalues for a hydrogen-like atom. This paper has been universally celebrated as one of the most important achievements of the twentieth century and created a revolution in most areas of quantum mechanics and indeed of all physics and chemistry.
A second paper was submitted just four weeks later that solved the quantum harmonic oscillator, rigid rotor, and diatomic molecule problems and gave a new derivation of the Schrödinger equation. A third paper, published in May, showed the equivalence of his approach to that of Heisenberg and gave the treatment of the Stark effect. A fourth paper in this series showed how to treat problems in which the system changes with time, as in scattering problems. In this paper, he introduced a complex solution to the wave equation in order to prevent the occurrence of fourth and sixth-order differential equations. (This was arguably the moment when quantum mechanics switched from real to complex numbers.) When he introduced complex numbers in order to lower the order of the differential equations, something magical happened, and all of wave mechanics was at his feet. (He eventually reduced the order to one.) These papers were his central achievement and were at once recognized as having great significance by the physics community.
Schrödinger was not entirely comfortable with the implications of quantum theory referring to his theory as “wave mechanics.” He wrote about the probability interpretation of quantum mechanics, saying: "I don't like it, and I'm sorry I ever had anything to do with it." (Just in order to ridicule the Copenhagen interpretation of quantum mechanics, he contrived the famous thought experiment called Schrödinger's cat paradox.)
Erwin Schrödinger proposed the quantum mechanical model of the atom, which treats electrons as matter waves. ... The square of the wave function, ψ2, represents the probability of finding an electron in a given region within the atom.
Key points
Louis de Broglie proposed that all particles could be treated as matter waves with a wavelength λ, given by the following equation: λ= h / mv
- Erwin Schrödinger proposed the quantum mechanical model of the atom, which treats electrons as matter waves.
- Schrödinger's equation, H^ψ =Eψ, can be solved to yield a series of wave function ψ, each of which is associated with an electron binding energy, E.
- The square of the wave function, ψ, squared, represents the probability of finding an electron in a given region within the atom.
- An atomic orbital is defined as the region within an atom that encloses where the electron is likely to be 90% of the time.
- The Heisenberg uncertainty principle states that we can't know both the energy and position of an electron. Therefore, as we learn more about the electron's position, we know less about its energy, and vice versa.
- Electrons have an intrinsic property called spin, and an electron can have one of two possible spin values: spin-up or spin-down.
- Any two electrons occupying the same orbital must have opposite spins.
