Basics of Quantum Mechanics- Why Quantum Physics? - • Classical mechanics (Newton's mechanics) and Maxwell's equations (electromagnetics theory) can explain MACROSCOPIC phenomena such as motion of billiard balls or rockets. • Quantum mechanics is used to explain microscopic phenomena such as photon-atom scattering and flow of the electrons in a semiconductor. • QUANTUM MECHANICS is a collection of postulates based on a huge number of experimental observations. • The differences between the classical and quantum mechanics can be understood by examining both • The classical point of view • The quantum point of view
Basics of Quantum Mechanics- Quantum Point of View - • Quantum particles can act as both particles and waves WAVE-PARTICLE DUALITY • Quantum state is a conglomeration of several possible outcomes of measurement of physical properties Quantum mechanics uses the language of PROBABILITY theory (random chance) • An observer cannot observe a microscopic system without altering some of its properties. Neither one can predict how the state of the system will change. • QUANTIZATION of energy is yet another property of "microscopic" particles.
The idea of duality is rooted in a debate over the nature of light and matter dating back to the 1600s, when competing theories of light were proposed by Huygens and Newton. Sir Isaac Newton 1643 1727 light consists of particles Christiaan Huygens Dutch 1629-1695 light consists of waves
If h is the Planck constant Then Louis de BROGLIE French (1892-1987) Max Planck (1901) Göttingen
Soon after theelectron discovery in 1887 - J. J. Thomson (1887) Some negative part could be extracted from the atoms • Robert Millikan (1910) showed that it was quantified. • Rutherford (1911) showed that the negative part was diffuse while the positive part was concentrated.
Quantum numbers In mathematics, a natural number (also called counting number) has two main purposes: they can be used for counting ("there are 6 apples on the table"), and they can be used for ordering ("this is the 3rd largest city in the country").
Atomic Spectroscopy Absorption or Emission Johannes Rydberg 1888 Swedish
Bohr and Quantum Mechanical Model Wavelengths and energy • Understand that different wavelengths of electromagnetic radiation have different energies. • c=vλ • c=velocity of wave • v=(nu) frequency of wave • λ=(lambda) wavelength
Bohr also postulated that an atom would not emit radiation while it was in one of its stable states but rather only when it made a transition between states. • The frequency of the radiation emitted would be equal to the difference in energy between those states divided by Planck's constant.
E2-E1= hv h=6.626 x 10-34 Js = Plank’s constant E= energy of the emitted light (photon) v = frequency of the photon of light • This results in a unique emission spectra for each element, like a fingerprint. • electron could "jump" from one allowed energy state to another by absorbing/emitting photons of radiant energy of certain specific frequencies. • Energy must then be absorbed in order to "jump" to another energy state, and similarly, energy must be emitted to "jump" to a lower state. • The frequency, v, of this radiant energy corresponds exactly to the energy difference between the two states.
Orbitals and quantum numbers • In the Bohr model, the electron is in a defined orbit • Schrödinger model uses probability distributions for a given energy level of the electron. • Solving Schrödinger's equation leads to wave functions called orbitals • They have a characteristic energy and shape (distribution).
The Bohr model used a single quantum number (n) to describe an orbit, the Schrödinger model uses three quantum numbers: n, l and ml to describe an orbital • The lowest energy orbital of the hydrogen atom has an energy of -2.18 x 1018 J and the shape in the above figure. Note that in the Bohr model we had the same energy for the electron in the ground state, but that it was described as being in a defined orbit.
The principle quantum number 'n' • Has integral values of 1, 2, 3, etc. • As n increases the electron density is further away from the nucleus • As n increases the electron has a higher energy and is less tightly bound to the nucleus
The azimuthal or orbital (second) quantum number 'l' • Has integral values from 0 to (n-1) for each value of n • Instead of being listed as a numerical value, typically 'l' is referred to by a letter ('s'=0, 'p'=1, 'd'=2, 'f'=3) • Defines the shape of the orbital
The magnetic (third) quantum number 'ml' Has integral values between 'l' and -'l', including 0 Describes the orientation of the orbital in space For example, the electron orbitals with a principle quantum number of 3
the third electron shell (i.e. 'n'=3) consists of the 3s, 3p and 3dsubshells (each with a different shape) • The 3s subshell contains 1 orbital, the 3p subshellcontains 3 orbitals and the 3d subshell contains 5 orbitals. (within each subshell, the different orbitals have different orientations in space) • Thus, the third electron shell is comprised of nine distinctly different orbitals, although each orbital has the same energy (that associated with the third electron shell) Note: remember, this is for hydrogen only.
Practice: • What are the possible values of l and ml for an electron with the principle quantum number n=4? • If l=0, ml=0 • If l=1, ml= -1, 0, +1 • If l=2, ml= -2,-1,0,+1, +2 • If l=3, ml= -3, -2, -1, 0, +1, +2, +3
Problem #2 • Can an electron have the quantum numbers n=2, l=2 and ml=2? • No, because l cannot be greater than n-1, so l may only be 0 or 1. • ml cannot be 2 either because it can never be greater than l
In order to explain the line spectrum of hydrogen, Bohr made one more addition to his model. He assumed that the electron could "jump" from one allowed energy state to another by absorbing/emitting photons of radiant energy of certain specific frequencies. Energy must then be absorbed in order to "jump" to another energy state, and similarly, energy must be emitted to "jump" to a lower state. The frequency, v, of this radiant energy corresponds exactly to the energy difference between the two states. Therefore, if an electron "jumps" from an initial state with energy Ei to a final state of energy Ef, then the following equality will hold: (delta) E = Ef - E i = hv To sum it up, what Bohr's model of the hydrogen atom states is that only the specific frequencies of light that satisfy the above equation can be absorbed or emitted by the atom.
Atomic Spectroscopy Absorption or Emission -R/72 -R/62 -R/52 -R/42 Johannes Rydberg 1888 Swedish -R/32 IR -R/22 VISIBLE -R/12 UV Emission Quantum numbers n, levels are not equally spaced R = 13.6 eV
Vacuum Photoelectric Effect (1887-1905) discovered by Hertz in 1887 and explained in 1905 by Einstein. Albert EINSTEIN (1879-1955) Heinrich HERTZ (1857-1894)
Compton effect 1923playing billiards assuming l=h/p Arthur Holly Compton American 1892-1962
Davisson and Germer 1925 Clinton Davisson Lester Germer In 1927 Diffraction is similarly observed using a mono-energetic electron beam Bragg law is verified assuming l=h/p
Wave-particle Equivalence. • Compton Effect (1923) • Electron Diffraction Davisson and Germer (1925) • Young's Double Slit Experiment Wave–particle duality In physics and chemistry, wave–particle duality is the concept that all matter and energy exhibits both wave-like and particle-like properties. A central concept of quantum mechanics, duality, addresses the inadequacy of classical concepts like "particle" and "wave" in fully describing the behavior of small-scale objects. Various interpretations of quantum mechanics attempt to explain this apparent paradox.
Young's Double Slit Experiment Mask with 2 slits Screen
Young's Double Slit Experiment This is a typical experiment showing the wave nature of light and interferences. What happens when we decrease the light intensity ? If radiation = particles, individual photons reach one spot and there will be no interferences If radiation particles there will be no spots on the screen The result is ambiguous There are spots The superposition of all the impacts make interferences
Young's Double Slit Experiment Assuming a single electron each time What means interference with itself ? What is its trajectory? If it goes through F1, it should ignore the presence of F2 Mask with 2 slits Screen
Young's Double Slit Experiment There is no possibility of knowing through which split the photon went! If we measure the crossing through F1, we have to place a screen behind. Then it does not go to the final screen. We know that it goes through F1 but we do not know where it would go after. These two questions are not compatible • Two important differences with classical physics: • measurement is not independent from observer • trajectories are not defined; hn goes through F1 and F2 both! or through them with equal probabilities! Mask with 2 slits Screen
de Broglie relation from relativity Popular expressions of relativity: m0 is the mass at rest, m in motion E like to express E(m) as E(p) with p=mv Ei + T + Erelativistic + ….
de Broglie relation from relativity Application to a photon (m0=0) To remember To remember
Useful to remember to relate energy and wavelength Max Planck
A New mathematical tool: Wave functions and Operators Each particle may be described by a wave function Y(x,y,z,t), real or complex, having a single value when position (x,y,z) and time (t) are defined. If it is not time-dependent, it is called stationary. The expression Y=Aei(pr-Et) does not represent one molecule but a flow of particles: a plane wave
Wave functions describing one particle To represent a single particle Y(x,y,z) that does not evolve in time, Y(x,y,z) must be finite (0 at ∞). In QM, a particle is not localized but has a probability to be in a given volume: dP= Y* Y dV is the probability of finding the particle in the volume dV. Around one point in space, the density of probability is dP/dV= Y* Y Y has the dimension of L-1/3 Integration in the whole space should give one Yis said to be normalized.
Normalization An eigenfunction remains an eigenfunction when multiplied by a constant O(lY)= o(lY) thus it is always possible to normalize a finite function Dirac notations <YIY>
Mean value • If Y1 and Y2 are associated with the same eigenvalue o: O(aY1 +bY2)=o(aY1 +bY2) • If not O(aY1 +bY2)=o1(aY1 )+o2(bY2) we define ō = (a2o1+b2o2)/(a2+b2) Dirac notations
Introducing new variables Now it is time to give a physical meaning. p is the momentum, E is the Energy H=6.62 10-34 J.s
Plane waves This represents a (monochromatic) beam, a continuous flow of particles with the same velocity (monokinetic). k, l, w, n, p and E are perfectly defined R (position) and t (time) are not defined. YY*=A2=constant everywhere; there is no localization. If E=constant, this is a stationary state, independent of t which is not defined.
Correspondence principle 1913/1920 For every physical quantity one can define an operator. The definition uses formulae from classical physics replacing quantities involved by the corresponding operators Niels Henrik David Bohr Danish 1885-1962 QM is then built from classical physics in spite of demonstrating its limits
Operators p and H We use the expression of the plane wave which allows defining exactly p and E.
Momentum and Energy Operators Remember during this chapter
Stationary state E=constant Remember for 3 slides after
Kinetic energy Classical quantum operator In 3D : Calling the laplacian Pierre Simon, Marquis de Laplace (1749 -1827)
Correspondence principleangular momentum Classical expression Quantum expression lZ= xpy-ypx