Quantum Mechanics (Unit IV) PDF

Quantum Mechanics (Unit IV) Please read from any standard text book Engineering Physics, Second Edition (H.K.Malik & A.K.Singh). : Chapter 15 (Development of Quantum Mechanics) and Chapter 16 (Quantum Mechanics) Both Chapters Included Why Quantum Mechanics?????? Failure of Classical M...

Quantum Mechanics (Unit IV) Please read from any standard text book Engineering Physics, Second Edition (H.K.Malik & A.K.Singh). : Chapter 15 (Development of Quantum Mechanics) and Chapter 16 (Quantum Mechanics) Both Chapters Included Why Quantum Mechanics?????? Failure of Classical Mechanics Need for Another Mechanics (Theory) ?????????????? Development of Quantum Mechanics  Classical Mechanics-A Brief History Behind QM Newton, Sir Isaac, PRS, (1643 – 1727), English physicist and mathematician Euler, Leonhard Lagrange, Joseph Louis Hamilton, William Rowan (1805 -- (1707 -- 1783), (1736 -- 1813), 1865), Swiss Italian-French mathematician, Irish mathematician and 3 mathematician. astronomer and physicist. astronomer.  Classical Electrodynamics Coulomb, Charles Biot, Jean Baptiste Ampere, Andre Faraday, Michael Lorentz, Hendrik Augustin (1736 – (1774 --1862), French Marie (1775 -- 1836), (1791 -- 1867), Antoon (1853 -- 1806), French Physicist; French Physicist English Physicist 1928), Dutch physicist Savart, Félix (1791 -- Physicist 1841), French Physicist Maxwell, James Clerk (1831 – 1879), Scottish physicist 4  Experiments and Ideas Prior to Quantum Theory (Before 1913) Blackbody Radiation and Quanta of Energy Ultra-violate Catastrophe Failure of Classical Physics Blue, Green, Red: Experimentally Observed Spectrum. Black: Predicted By Rayleigh-Jeans Rayleigh-Jeans Law There will be infinite amount of energy at higher frequency (UV-Region) or at higher temperature: Wrong, it never happens. 𝑬 𝝂 𝒅𝝂 ∝ 𝝂𝟐 𝑲𝑻𝒅𝝂 Planck (1858 -- 1947), German physicist. Planck's law of black body radiation (1900) Planck’s assumption (1900): Radiation of a given frequency ν could only be emitted and absorbed in “quanta” of energy E=hν  Failure of Classical Theory 1 Classical theory suggested a “UV catastrophe,” leading to obviously nonsensical infinite energy radiating from hot body. Max Planck solved this problem by postulating light quanta (now often called the father of quantum mechanics).  Radiation interaction with matter: Photoelectric Effect and Quanta of Light  In 1839, Alexandre Edmond Becquerel observed the photoelectric effect via an electrode in a conductive solution exposed to light. In 1905, Albert Einstein proposed the well-known Einstein's equation for 1 photoelectric effect. Wins Nobel Prize me vk2  Ek  h  W (1921) 2 In 1916, Robert Andrews Millikan finished a decade-long experiment to confirm Einstein’s explanation of photoelectric effect. Wins Electric Charge of an electron Nobel Prize (1923)  Photoelectric Effect A Photocell is Used to Study the Photoelectric Effect  Photoelectric Effect Larger light intensity means larger number of photons at a given frequency (Energy).  Photoelectric Effect The photoelectric effect provides evidence for the particle nature of light. It also provides evidence for quantization. The electrons will only be ejected once the threshold frequency is reached. Below the threshold frequency, no electrons are ejected. Above the threshold frequency, the number of electrons ejected depend on the intensity of the light.  Failure of Classical Theory 2 Why was red light incapable of knocking electrons out of certain materials, no matter how bright (more intensity) – yet blue light could readily do so even at modest intensities – called the photoelectric effect – Einstein explained in terms of photons, and won Nobel Prize Atomic Structure  Nuclear atom model (1911): Ernest Rutherford Rutherford, Ernest, FRS (1871 -- 1937), New Zealand-English nuclear physicist.  Failure of Classical Theory 3 This means an electron should fall into the nucleus. Classical Electrodynamics: Charged particles radiate EM New mechanics is energy (photons) when their velocity vector changes (e.g. they needed! accelerate). Atomic Spectra  Spectroscopy Balmer, Johann Jakob (1825 -- 1898), from n ≥ 3 to n = 2 Swiss mathematician and an honorary physicist. Balmer series visible spectrum (1885) Rydberg formula for hydrogen (1888) Rydberg, Johannes Rydberg formula for all Robert (1854 -- 1919), Swedish physicist. hydrogen-like atom (1888) Failure of Classical Theory 4 What caused spectra of atoms to contain discrete “lines” – it was apparent that only a small set of optical frequencies (wavelengths) could be emitted or absorbed by atoms Each atom has a distinct “fingerprint” Light only comes off at very specific wavelengths – or frequencies – or energies Note that hydrogen (bottom), with only one electron and one proton, emits several wavelengths Old Quantum Theory (1913 -- 1924)  Bohr's model of atomic structure, 1913 The electron's orbital angular momentum is quantized Bohr, Niels Henrik David (1885 -- 1962), Danish physicist. The theory that electrons travel in discrete orbits around the atom's nucleus. The idea that an electron could drop from a higher-energy orbit to a lower one, emitting a photon (light quantum) of discrete energy (this became the basis for quantum theory). Bohr’s theory in 1 page 1 e2 mv 2 circular motion:  4 0 r 2 r quantization of angular momentum: mvn rn  nKh 2 1 2 1 e  1  me 4 1 2 total energy: En  mvn     2 4 0 rn  4 0  2 K 2 2 h n 2 Quantum predictions must match classical results for large n 2  1  me 4 freq. of radiation at n 1: h  En 1  En    3 2 2  4 0  n K h 2 vn  1  me 4 freq. of classical circular motion:    2 rn  4 0  n3 2 K 3 h3 K  1 / (2 ) mvn rn  nh / (2 )  n  Summary Electron Transitions Failures of the Bohr Model It fails to provide any understanding of why certain spectral lines are brighter than others. There is no mechanism for the calculation of transition probabilities. The Bohr model treats the electron as if it were a miniature planet, with definite radius and momentum. This is in direct violation of the uncertainty principle which dictates that position and momentum cannot be simultaneously determined. The Bohr model gives us a basic conceptual model of electrons orbits and energies. The precise details of spectra and charge distribution must be left to quantum mechanical calculations, as with the Schrödinger equation.  de Broglie Wave Prince de Broglie gets his Ph.D. de Broglie matter wave hypothesis (1923): All matter has a wave-like nature (wave-particle duality) and that the wavelength and momentum of a particle are related by a simple equation.  de Broglie Wave Every particle or system of particles can be defined in quantum mechanical terms – and therefore have wave-like properties The quantum wavelength of an object is:  = h/p (p is momentum) – called the de Broglie wavelength typical macroscopic objects – masses ~ kg; velocities ~ m/s p  1 kg·m/s –   10-34 meters (very small to even detect!!!!!) typical “quantum” objects: – electron (10-30 kg) at thermal velocity (105 m/s)   10-8 m – so  is 100 times larger than an atom: very relevant to an electron! de Broglie waves of electron  Wave-Particle Duality Particle Nature of Light Wave Nature of Light Every Quantum Mechanical Particle (not just light or photon) behaves like a wave and a particle simultaneously Interference, Diffraction, Polarization: Wave nature Compton Effect, Photoelectric effect, Blackbody Radiation: Particle Nature Check this link for Particle-Wave Duality explanation 1 https://www.youtube.com/watch?v=NvzSLByrw4Q 2. https://www.youtube.com/watch?v=Q1YqgPAtzho  Wave-Particle Duality Particle Nature Wave Nature Heisenberg Uncertainty Principle The process of measurement involves interaction – this interaction necessarily “touches” the subject – by “touch,” we could mean by a photon of light The more precisely we want to know where something is, the “harder” we have to measure it – so we end up giving it a kick So we must unavoidably alter the velocity of the particle under study – thus changing its momentum If x is the position uncertainty, and p is the momentum uncertainty, then inevitably, xp  h/2  Heisenberg Uncertainty Principle Et  Measurement disturbes the system Phase Velocity and Group Velocity  Phase Velocity and Group Velocity E  E 0 cosk(x  ct) E0 = wave amplitude (related to the energy carried by the wave). 2 k  2 ˜ = angular wavenumber  (λ = wavelength;  ˜ = wavenumber = 1/λ) Alternatively: E  E 0 cos(kx  t)  Where ω = kc = 2πc/λ = 2πf = angular frequency (f = frequency)  Phase Velocity and Group Velocity E  E 0 cosk(x  ct);   k(x  ct) The argument of the cosine function represents the phase of the wave, ϕ, or the fraction of a complete cycle of the wave. In-phase waves Line of equal phase = wavefront = contours of maximum field Out-of-phase waves  The Phase Velocity How fast is the wave traveling? Velocity is a reference distance divided by a reference time. The phase velocity is the wavelength / period: v =  / t Since f = 1/t: v = f In terms of k, k = 2/ , and the angular frequency,  = 2/ t, this is: v =/k  The Group Velocity This is the velocity at which the overall shape of the wave’s amplitudes, or the wave ‘envelope’, propagates. (= signal velocity) Here, phase velocity = group velocity (the medium is non-dispersive) Dispersion: phase/group velocity depends on frequency Black dot moves at phase velocity. Red dot moves at group velocity. This is normal dispersion (refractive index decreases with increasing λ)  Dispersion: phase/group velocity depends on frequency Black dot moves at group velocity. Red dot moves at phase velocity. This is anomalous dispersion (refractive index increases with increasing λ)  Phase Velocity and Group Velocity The group velocity is the velocity with which the envelope of the wave packet, propagates through space. The phase velocity is the velocity at which the phase of any one frequency component of the wave will propagate. You could pick one particular phase of the wave (for example the crest) and it would appear to travel at the phase velocity.  A “Wave Packet” ____________________________________ How do you construct a wave packet?  What happens when you add up waves? ________________________________ The Superposition principle Adding up waves of different frequencies..... ____________________________________  Group velocity Group velocity of a wave is the velocity with which the variations in the shape of the wave's amplitude (known as the modulation or envelope of the wave) propagate through space. The group velocity is defined by the equation where: vg is the group velocity; ω is the wave's angular frequency; k is the wave number. The function ω(k), which gives ω as a function of k, is known as the dispersion relation.  Birth of QM The necessity for quantum mechanics was thrust upon us by a series of observations. The theory of QM developed over a period of 30 years, culminating in 1925-27 with a set of postulates. QM cannot be deduced from pure mathematical or logical reasoning. QM is not intuitive, because we don’t live in the world of electrons and atoms. QM is based on observation. Like all science, it is subject to change if inconsistencies with further observation are revealed.  The wave function,  De Broglie waves can be represented by a simple quantity , called a wave function, which is a complex function of time and position A particle is completely described in quantum mechanics by the wave function A specific wave function for an electron is called an orbital The wave function can be used to determine the energy levels of an atomic system https://www.youtube.com/watch?v=vShpwplJyXk First Postulate of Quantum Mechanics Every physically-realizable state of the system is described in quantum mechanics by a state function  that contains all accessible physical information about the system in that state. https://www.youtube.com/watch?v=EmNQuK- E0kI https://www.youtube.com/watch?v=KKr91v7y LcM First Postulate of Quantum Mechanics Physically realizable states  states that can be studied in laboratory Accessible information  the information we can extract from the wave function State function  function of position, momentum, energy that is spatially localized. First Postulate of Quantum Mechanics If 1 and 2 represent two physically-realizable states of the system, then the linear combination   c11  c 2  2 where c1 and c2 are arbitrary complex constants, represents a third physically realizable state of the system. Note: Wave function (x,t)  position and time probability amplitude Quantum mechanics describes the outcome of an ensemble of measurements, where an ensemble of measurements consists of a very large number of identical experiments performed on identical non-interacting systems, all of which have been identically prepared so as to be in the same state. Second Postulate of Quantum Mechanics If a system is in a quantum state represented by a wave function , then 2 PdV   dV is the probability that in a position measurement at time t the particle will be detected in the infinitesimal volume dV. 2  ( x, t )  position and time probability density The importance of normalization follows from the Born interpretation of the state function as a position probability amplitude. According to the second postulate of quantum mechanics, the integrated probability density can be interpreted as a probability that in a position measurement at time t, we will find the particle anywhere in space. Second Postulate of Quantum Mechanics Therefore, the normalization condition for the wave function is: 2 *  PdV    ( x , y , z ) dV   ( x, y, z ) ( x, y, z )dV  1  Probability of finding the quantum mechanical particle at a particular region of space at a particular time. Limitations on the wave function: Only normalized functions can represent a quantum state and these are called physically admissible functions. State function must be continuous and single valued function. State function must be a smoothly-varying function (continuous derivative).  Third Postulate of Quantum Mechanics Every observable in quantum mechanics is represented by an operator which is used to obtain physical information about the observable from the state function. For an observable that is represented in classical physics by a function Q(x,p), the corresponding   operator is Q( x , p). Observable Operator  Position x Momentum    p i x Energy 2 2 2 p    E   V ( x)    V ( x) 2m 2m x 2 Fourth Postulate of Quantum Mechanics 1926 Erwin Schrödinger proposed an equation that describes the evolution of a quantum- mechanical system  SWE which represents quantum equations of motion, and is of the form:  2  2  2 2     V ( x)( x, t )    V ( x)( x, t )  i 2m x 2  2m x 2  t This work of Schrödinger was stimulated by a 1925 paper by Einstein on the quantum theory of ideal gas, and the de Broglie theory of matter waves. Note: Examining the time-dependent SWE, one can also define the following operator for the total energy:   E  i t Fourth (Fundamental) postulate of Quantum mechanics: Examining the time-dependent SWE, one can also define the following operator for the Fourth Postulate of Quantum Mechanics total energy:   E  i t Fourth (Fundamental) postulate of Quantum mechanics: The time development of the state functions of an isolated quantum system is governed     by the time-dependent SWE H  i / t , where H  T  V is the Hamiltonian of the system. Note on isolated system: The TDSWE describes the evolution of a state provided that no observations are made. An observation alters the state of the observed system, and as it is, the TDSWE can not describe such changes. Non-Relativistic Schrödinger Equation Time independent, E is a constant, Hamiltonian (Ĥ) is independent of time. Same as eigenvalue equation of Ĥ, where E is the Eigenvalue of Ĥ Time dependent, Hamiltonian (Ĥ) is a function of time. It gives up time evolution of wave function.  Particle in a box Region I Region II Region III V(x)=∞ V(x)=0 V(x)=∞ 0 L x The Particle does not exist outside the box or the wave function (ψ=0) is zero. V(x)=0 for L>x>0 V(x)=∞ for x≥L, x≤0 Time independent Schrödinger equation,   2 d 2 ( x) 2  V ( x)  E 2m dx  Particle in a box Schrödinger equation of the particle inside the box, Where, A & B are constants Boundary conditions of the potential dictate that the wave function must be zero at x = 0 and x = L. B=0 and kL = nπ Therefore, the wave function becomes,  Particle in a box After normalizing the wave function inside the box, Therefore, L  x sin 2kx  A   1 2 2 4k  0 2 L  2 A   1 A  2 L  Quantized Energy The quantized wave number now becomes Solving for the energy yields Note that the energy depends on the integer values of n. Hence the energy is quantized and nonzero. The special case of n = 0 is called the ground state energy.  Quantum Tunnelling Quantum tunneling, also known as tunneling (in USA) is a quantum mechanical phenomenon whereby a wave function can propagate through a potential barrier. The transmission through the barrier can be finite and depends exponentially on the barrier height and barrier width. The wave function may disappear on one side and reappear on the other side. The wave function and its first derivative are continuous. Tunneling occurs with barriers of thickness around 1–3 nm and smaller. https://www.youtube.com/watch?v=RF7 dDt3tVmI A simulation of a wave packet incident on a potential barrier. In relative units, the barrier energy is 20, greater than the mean wave packet energy of 14. A portion of the wave packet passes through the barrier.  Applications of Quantum Tunnelling  Scanning Tunnelling Microscope.  Nuclear Fusion: Quantum tunnelling is a crucial part of nuclear fusion. The average temperature of a star’s core is usually not sufficient for atomic nuclei to overcome the Coulomb barrier and kick start thermonuclear fusion. The tunnelling increases the chances of infiltrating this barrier. Though the probability is still low, the huge number of nuclei in the stellar core is enough to drive a steady fusion reaction.  Electronics: Tunneling is a frequent source of current leakage in very-large- scale integration (VLSI) electronics. The VLSI electronics experience substantial power loss and heating effects that cripple such devices. It is usually considered the lower threshold on how microelectronic device elements can be created. Tunnelling is also a basic technique employed to set the floating gates in flash memory. Cold emission, tunnel junction, quantum-dot cellular automata, tunnel diode, and tunnel field-effect transistors are some of the main electronic processes or devices that use quantum tunneling.  Applications of Quantum Tunnelling Quantum Biology: Quantum tunnelling is one of the core quantum phenomena in quantum biology. It is essential for both proton tunnelling and electron tunnelling. Electron tunnelling is a critical factor in numerous biochemical redox reactions (cellular respiration, photosynthesis) and enzymatic catalysis. Proton tunnelling also has a key role in spontaneous DNA mutation.