by Moniharu

Quantum Mechanics Physics

Waves and Particles : Planck’s Quantum Theory

Light travels in the form of small discrete packets of energy called quanta .

It is given as ,

E v dv = ‘(8pihv^3)/(c^3)’

This is known as Planck’s radiation law . This law can be expressed in terms of wavelength as ,

v = ‘C/lambda’ or | dv | = ‘|-(C/lambda^3)dlambda|’

= ‘(C/lambda^2dlambda)’

‘:.’ ‘E_lambda’ ‘dlambda’ = ‘(8pihc)/(lambda^5)((c^3)/(H^3))’ ‘1/{(exp((hc)/(lambdakT)-1))}’ ‘(C/lambda^2dlambda)’

( OR )

‘E_lambda’ ‘dlambda’ = ‘(8pihc)/(lambda^5)’ ‘1/(exp((hc)/(lambdakT)-1))’ ‘dlambda’

Assumptions

A black body radiator contains simple harmonic oscillators of possible frequencies .

The oscillators cannot emit their or absorb energy continuously . This is contrary to electromagnetic theory which allows a continuous emission or absorbs of energy .

Emission or absorption of energy takes place in discrete amounts i.e., energy of oscillator is quantized . The energy of an atomic oscillator of frequency can have certain values like 0 , hv , 2 hv . This is integral multiple of a small unit energy eV called he quantum or photon . In general eV is called the quantum or photon . It is given by

E = n h v . n = positive integer and h = Planck’s constant

De Broglie Hypothesis

According to the Planck and Einstein theories , the energy of a photon whose frequency v is expressed as

E = h v

where , h = Planck’s constant

According to Einsteins mass energy relation

E = m c2 , where m = mass of the photon , c = velocity of a photon

From equations (1) and (2) , we get ,

hv = mc2

‘(hc)/lambda’ = mc2

‘lambda’ = ‘h/(mc)’

Momentum of a photon , p = mc

‘:.’ ‘lambda’ = ‘h/p’

In the same way according to De Broglie hypothesis , if an electron of charge e , mass m , is moving with a velocity v in the presence of potential V , then the wavelength associated with that electron can be expressed as ,

‘lambda’ = ‘h/(mV)’ = ‘h/p’

The above equation is called De Broglie’s equation .

And the energy of the electron in terms of potential can be expressed as E = eV and the kinetic energy of electron is given as ,

E = ‘1/2’ mv2

eV = ‘1/2’ mv2

m e V = ‘1/2’ m2 v2

2 m e V = m2 v2 [ since p = mv ]

2 m e V = p2

By substituting above equation in ‘lambda’ = ‘h/p’ , we get

‘lambda’ = ‘h/sqrt(2meV)’

Matter Waves

Louis Victor De broglie , in 1923 , proposed that material particles , such as electrons , atoms etc., exhibit wave nature . These waves are called matter waves or De broglie waves . De broglie hypothesis was conformed experimentally a few years later . Thus , as per this hypothesis matter has wave particle dual nature .

Properties

A matter wave is not a physical phenomenon . They are pilot waves , in the sense that their only function is to pilot or guide the material particles . Hence they are also treated as probability waves .

These waves are not electromagnetic waves either .

The concept of matter waves is given for particles that are in mation . If a particle is at rest then there is no meaning of matter waves associated with it .

Smaller is the velocity of the particle , greater is the wavelength associated with it .

Lighter is the particle , greater is the wavelength of the matter wave .

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Quantum mechanics (QM — also known as quantum physics, or quantum theory) is a branch of physics which deals with physical phenomena at nanoscopic scales where the action is on the order of the Planck constant. It departs from classical mechanics primarily at the quantum realm of atomic and subatomic length scales. Quantum mechanics provides a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. Quantum mechanics provides a substantially useful framework for many features of the modern periodic table of elements including the behavior of atoms during chemical bonding and has played a significant role in the development of many modern technologies.

In advanced topics of quantum mechanics, some of these behaviors are macroscopic (see macroscopic quantum phenomena) and emerge at only extreme (i.e., very low or very high) energies or temperatures (such as in the use of superconducting magnets). For example, the angular momentum of an electron bound to an atom or molecule is quantized. In contrast, the angular momentum of an unbound electron is not quantized. In the context of quantum mechanics, the wave–particle duality of energy and matter and the uncertainty principle provide a unified view of the behavior of photons, electrons, and other atomic-scale objects.

The mathematical formulations of quantum mechanics are abstract. A mathematical function, the wavefunction, provides information about the probability amplitude of position, momentum, and other physical properties of a particle. Mathematical manipulations of the wavefunction usually involve bra–ket notation which requires an understanding of complex numbers and linear functionals. The wavefunction formulation treats the particle as a quantum harmonic oscillator, and the mathematics is akin to that describing acoustic resonance. Many of the results of quantum mechanics are not easily visualized in terms of classical mechanics. For instance, in a quantum mechanical model the lowest energy state of a system, the ground state, is non-zero as opposed to a more “traditional” ground state with zero kinetic energy (all particles at rest). Instead of a traditional static, unchanging zero energy state, quantum mechanics allows for far more dynamic, chaotic possibilities, according to John Wheeler.

The earliest versions of quantum mechanics were formulated in the first decade of the 20th century. About this time, the atomic theory and the corpuscular theory of light (as updated by Einstein)[1] first came to be widely accepted as scientific fact; these latter theories can be viewed as quantum theories of matter and electromagnetic radiation, respectively. Early quantum theory was significantly reformulated in the mid-1920s by Werner Heisenberg, Max Born and Pascual Jordan, (matrix mechanics); Louis de Broglie and Erwin Schrödinger (wave mechanics); and Wolfgang Pauli and Satyendra Nath Bose (statistics of subatomic particles). Moreover, the Copenhagen interpretation of Niels Bohr became widely accepted. By 1930, quantum mechanics had been further unified and formalized by the work of David Hilbert, Paul Dirac and John von Neumann[2] with a greater emphasis placed on measurement in quantum mechanics, the statistical nature of our knowledge of reality, and philosophical speculation about the role of the observer. Quantum mechanics has since permeated throughout many aspects of 20th-century physics and other disciplines including quantum chemistry, quantum electronics, quantum optics, and quantum information science. Much 19th-century physics has been re-evaluated as the “classical limit” of quantum mechanics and its more advanced developments in terms of quantum field theory, string theory, and speculative quantum gravity theories. https://www.youtube.com/watch?v=ZsVGut7G-dU

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