PPT-On-Q.-M.-Reality-M.-Sc..pptx for MSc

A ppt on Quantum Mechanics

1. QUANTUM MECHANICS- Illusion or Reality ? Prof. D. M. Parshuramkar Dept. of Physics N. H. College, Bramhapuri

2. 1. Classical Mechanics Do the electrons in atoms and molecules obey Newton’s classical laws of motion? We shall see that the answer to this question is “No”. This has led to the development of Quantum Mechanics – we will contrast classical and quantum mechanics.

3. 1.1 Features of Classical Mechanics (CM) 1) CM predicts a precise trajectory for a particle. The exact position ( r )and velocity ( v ) (and hence the momentum p = m v ) of a particle (mass = m ) can be known simultaneously at each point in time. Note: position ( r ),velocity ( v ) and momentum ( p ) are vectors, having magnitude and direction  v = ( v x ,v y ,v z ). position r = ( x , y , z ) velocity v

4. Any type of motion (translation, vibration, rotation) can have any value of energy associated with it – i.e. there is a continuum of energy states. Particles and waves are distinguishable phenomena, with different, characteristic properties and behaviour. Property Behaviour mass momentum Particles position  collisions velocity Waves wavelength  diffraction frequency interference

5. 1.2 Revision of Some Relevant Equations in CM Total energy of particle: E = Kinetic Energy (KE) + Potential Energy (PE) E = ½mv 2 + V  E = p 2 /2m + V ( p = m v ) Note: strictly E, T, V (and r , v , p ) are all defined at a particular time ( t ) – E(t) etc.. T - depends on v V - depends on r V depends on the system e.g. positional, electrostatic PE

6. Consider a 1-dimensional system (straight line translational motion of a particle under the influence of a potential acting parallel to the direction of motion): Define: position r = x velocity v = dx/dt momentum p = mv = m(dx/dt) PE V force F =  (dV/dx) Newton’s 2 nd Law of Motion F = m a = m(dv/dt) = m(d 2 x/dt 2 ) Therefore, if we know the forces acting on a particle we can solve a differential equation to determine it’s trajectory {x(t),p(t)} . acceleration

7. 1.3 Example – The 1-Dimensional Harmonic Oscillator The particle experiences a restoring force ( F ) proportional to its displacement ( x ) from its equilibrium position (x=0). Hooke’s Law F =  k x k is the stiffness of the spring (or stretching force constant of the bond if considering molecular vibrations) Substituting F into Newton’s 2 nd Law we get: m(d 2 x/dt 2 ) =  k x a (second order) differential equation x m F k x = 0 NB – assuming no friction or other forces act on the particle (except F). k

8. Solution: position x(t) = A sin ( t ) of particle frequency  = /2 = (of oscillation) Note: Frequency depends only on characteristics of the system ( m, k ) – not the amplitude ( A )! +A  A x t  time period  = 1/ 

9. Assuming that the potential energy V = 0 at x = 0, it can be shown that the total energy of the harmonic oscillator is given by: E = ½ k A 2 As the amplitude ( A ) can take any value, this means that the energy ( E ) can also take any value – i.e. energy is continuous . At any time ( t ), the position { x(t) } and velocity { v (t) } can be determined exactly – i.e. the particle trajectory can be specified precisely. We shall see that these ideas of classical mechanics fail when we go to the atomic regime (where E and m are very small) – then we need to consider Quantum Mechanics . CM also fails when velocity is very large (as v  c), due to relativistic effects .

10. By the early 20 th century, there were a number of experimental results and phenomena that could not be explained by classical mechanics. a) Black Body Radiation (Planck 1900) 1.4 Experimental Evidence for the Breakdown of Classical Mechanics 2000 4000 6000 2000 K 1750 K 1250 K Energy Radiated Classical Mechanics (Rayleigh-Jeans) “UV Catastrophe” l /nm

11. Planck’s Quantum Theory Planck (1900) proposed that the light energy emitted by the black body is quantized in units of h  ( = frequency of light). E = n h  (n = 1, 2, 3, …) High frequency light only emitted if thermal energy kT  h  . h  – a quantum of energy. Planck’s constant h ~ 6.62610 34 Js If h  0 we regain classical mechanics. Conclusions: Energy is quantized (not continuous). Energy can only change by well defined amounts.

12. Time period of a Simple pendulum Gustav Kirchhoff 1859 : Dark lines of Na seen in solar spectrum are darkened further by interposition of Na – flame in the path of Sun’s ray . Ratio of Emissive power to Absorptive power is independent of the nature of material which is

13. Function of Freq . and Temp . String vibration Phase Space

14. b) Heat Capacities (Einstein, Debye 1905-06) Heat capacity – relates rise in energy of a material with its rise in temperature: C V = (d U /d T ) V Classical physics  C V,m = 3R (for all T ). Experiment  C V,m < 3R ( C V  as T ). At low T , heat capacity of solids determined by vibrations of solid. Einstein and Debye adopted Planck’s hypothesis. Conclusion: vibrational energy in solids is quantized: vibrational frequencies of solids can only have certain values (  ) vibrational energy can only change by integer multiples of h  .

15. c) Photoelectric Effect (Einstein 1905) Ideas of Planck applied to electromagnetic radiation. No electrons are ejected (regardless of light intensity) unless n exceeds a threshold value characteristic of the metal. E k independent of light intensity but linearly dependent on n . Even if light intensity is low, electrons are ejected if n is above the threshold. (Number of electrons ejected increases with light intensity). Conclusion: Light consists of discrete packets ( quanta ) of energy = photons (Lewis, 1922) . h n Metal surface work function = F e  Photoelectrons ejected with kinetic energy: E k = h n - 

16. d) Atomic and Molecular Spectroscopy It was found that atoms and molecules absorb and emit light only at specific discrete frequencies  spectral lines (not continuously!). e.g. Hydrogen atom emission spectrum (Balmer 1885) Empirical fit to spectral lines (Rydberg-Ritz): n 1 , n 2 (> n 1 ) = integers. Rydberg constant R H = 109,737.3 cm -1 (but can also be expressed in energy or frequency units). n 1 = 1  Lyman n 1 = 2  Balmer n 1 = 3  Paschen n 1 = 4  Brackett n 1 = 5  Pfund

17. Revision: Electromagnetic Radiation A – Amplitude l – wavelength n - frequency c = n x l or n = c / l wavenumber n = n / c = 1 / l c (velocity of light in vacuum) = 2.9979 x 10 8 m s -1 .

18. 1.5 The Bohr Model of the Atom The H-atom emission spectrum was rationalized by Bohr (1913): Energies of H atom are restricted to certain discrete values (i.e. electron is restricted to well-defined circular orbits , labelled by quantum number n ). Energy (light) absorbed in discrete amounts ( quanta = photons ), corresponding to differences between these restricted values: E = E 2  E 1 = h h E 1 E 2 h E 1 E 2 Absorption Emission Conclusion: Spectroscopy provides direct evidence for quantization of energies (electronic, vibrational, rotational etc.) of atoms and molecules. n 2 n 1 e  p+

19. Limitations of Bohr Model & Rydberg-Ritz Equation The model only works for hydrogen (and other one electron ions) – ignores e-e repulsion . Does not explain fine structure of spectral lines. Note: The Bohr model (assuming circular electron orbits) is fundamentally incorrect.

20. 2. Wave-Particle Duality Remember: Classically, particles and waves are distinct: Particles – characterised by position, mass, velocity. Waves – characterised by wavelength, frequency. By the 1920s, however, it was becoming apparent that sometimes matter (classically particles) can behave like waves and radiation (classically waves) can behave like particles.

21. 2.1 Waves Behaving as Particles The Photoelectric Effect Electromagnetic radiation of frequency  can be thought of as being made up of particles ( photons ), each with energy E = h  . This is the basis of Photoelectron Spectroscopy ( PES ). Spectroscopy Discrete spectral lines of atoms and molecules correspond to the absorption or emission of a photon of energy h  , causing the atom/molecule to change between energy levels: E = h  . Many different types of spectroscopy are possible.

22. The Compton Effect (1923) Experiment: A monochromatic beam of X-rays (  i ) = incident on a graphite block. Observation: Some of the X-rays passing through the block are found to have longer wavelengths (  s ).   i  s Intensity   i  s

23. Explanation: The scattered X-rays undergo elastic collisions with electrons in the graphite. Momentum (and energy) transferred from X-rays to electrons. Conclusion: Light (electromagnetic radiation) possesses momentum. Momentum of photon p = h/  Energy of photon E = h = hc/  Applying the laws of conservation of energy and momentum we get:  i  s  e  p=h/  s p=m e v

24. 2.2 Particles Behaving as Waves Electron Diffraction (Davisson and Germer, 1925) Davisson and Germer showed that a beam of electrons could be diffracted from the surface of a nickel crystal. Diffraction is a wave property – arises due to interference between scattered waves. This forms the basis of electron diffraction – an analytical technique for determining the structures of molecules, solids and surfaces (e.g. LEED ). NB: Other “particles” (e.g. neutrons, protons, He atoms) can also be diffracted by crystals.

25. 2.3 The De Broglie Relationship (1924) In 1924 (i.e. one year before Davisson and Germer’s experiment), De Broglie predicted that all matter has wave-like properties. A particle, of mass m , travelling at velocity v , has linear momentum p = mv . By analogy with photons, the associated wavelength of the particle (  ) is given by:

26. 3. Wavefunctions A particle trajectory is a classical concept. In Quantum Mechanics, a “particle” (e.g. an electron) does not follow a definite trajectory { r (t), p (t)} , but rather it is best described as being distributed through space like a wave . 3.1 Definitions Wavefunction (  ) – a wave representing the spatial distribution of a “particle”. e.g. electrons in an atom are described by a wavefunction centred on the nucleus.  is a function of the coordinates defining the position of the classical particle: 1-D  (x) 3-D  (x,y,z) =  ( r ) =  (r, ,) (e.g. atoms)  may be time dependent – e.g.  (x,y,z,t)

27. The Importance of   completely defines the system (e.g. electron in an atom or molecule). If  is known, we can determine any observable property (e.g. energy, vibrational frequencies, …) of the system. QM provides the tools to determine  computationally, to interpret  and to use  to determine properties of the system.

28. 3.2 Interpretation of the Wavefunction In QM, a “particle” is distributed in space like a wave. We cannot define a position for the particle. Instead we define a probability of finding the particle at any point in space. The Born Interpretation (1926) “The square of the wavefunction at any point in space is proportional to the probability of finding the particle at that point.” Note: the wavefunction (  ) itself has no physical meaning.

29. 1-D System If the wavefunction at point x is  (x) , the probability of finding the particle in the infinitesimally small region ( dx ) between x and x+dx is: P(x)    (x) 2 dx   (x) – the magnitude of  at point x . Why write    2 instead of  2 ? Because  may be imaginary or complex   2 would be negative or complex. BUT: probability must be real and positive ( 0  P  1 ). For the general case, where  is complex (  = a + i b) then:    2 = * where * is the complex conjugate of  . ( * = a – i b) (NB ) probability density

30. 3-D System If the wavefunction at r = (x,y,z) is  ( r ) , the probability of finding the particle in the infinitesimal volume element d  (= dxdydz) is: P( r )    ( r ) 2 d  If  ( r ) is the wavefunction describing the spatial distribution of an electron in an atom or molecule, then:   ( r ) 2 =  ( r ) – the electron density at point r

31. 3.3 Normalization of the Wavefunction As mentioned above, probability: P( r )    ( r ) 2 d What is the proportionality constant? If  is such that the sum of   ( r ) 2 at all points in space = 1, then: P(x) =   (x) 2 dx 1-D P( r ) =   ( r ) 2 d  3-D As summing over an infinite number of infinitesimal steps = integration , this means: i.e. the probability that the particle is somewhere in space = 1. In this case,  is said to be a normalized wavefunction .

32. How to Normalize the Wavefunction If  is not normalized, then: A corresponding normalized wavefunction (  Norm ) can be defined: such that: The factor ( 1/A ) is known as the normalization constant (sometimes represented by N ).

33. 3.4 Quantization of the Wavefunction The Born interpretation of  places restrictions on the form of the wavefunction: (a)  must be continuous (no breaks) ; (b) The gradient of  ( d  /dx ) must be continuous (no kinks) ; (c)  must have a single value at any point in space; (d)  must be finite everywhere ; (e)  cannot be zero everywhere. Other restrictions ( boundary conditions ) depend on the exact system. These restrictions on  mean that only certain wavefunctions and  only certain energies of the system are allowed.  Quantization of   Quantization of E

34. 3.5 Heisenberg’s Uncertainty Principle “It is impossible to specify simultaneously, with precision, both the momentum and the position of a particle*” ( * if it is described by Quantum Mechanics) Heisenberg (1927) D p x .D x  h / 4 p (or /2 , where  = h /2 p ). D x – uncertainty in position D p x – uncertainty in momentum (in the x-direction) If we know the position ( x ) exactly, we know nothing about momentum ( p x ). If we know the momentum ( p x ) exactly, we know nothing about position ( x ). i.e. there is no concept of a particle trajectory {x(t),p x (t)} in QM (which applies to small particles). NB – for macroscopic objects, D x and D p x can be very small when compared with x and p x  so one can define a trajectory. Much of classical mechanics can be understood in the limit h  0 .

35. How to Understand the Uncertainty Principle To localize a wavefunction (  ) in space (i.e. to specify the particle’s position accurately, small D x ) many waves of different wavelengths (  ) must be superimposed  large D p x ( p = h /  ). The Uncertainty Principle imposes a fundamental (not experimental) limitation on how precisely we can know (or determine) various observables.  2 ~ 1

36. Note – to determine a particle’s position accurately requires use of short radiation (high momentum) radiation. Photons colliding with the particle causes a change of momentum ( Compton effect )  uncertainty in p .  The observer perturbs the system. Zero-Point Energy (vibrational energy E vib  0 , even at T = 0 K) is also a consequence of the Uncertainty Principle: If vibration ceases at T = 0, then position and momentum both = 0 (violating the UP). Note: There is no restriction on the precision in simultaneously knowing/measuring the position along a given direction ( x ) and the momentum along another, perpendicular direction ( z ): D p z .D x = 0

37. Similar uncertainty relationships apply to other pairs of observables. e.g. the energy ( E ) and lifetime (  ) of a state: D E . D    (a) This leads to “lifetime broadening” of spectral lines: Short-lived excited states (  well defined, small D  ) possess large uncertainty in the energy ( large D E ) of the state. Broad peaks in the spectrum. Shorter laser pulses (e.g. femtosecond, attosecond lasers) have broader energy (therefore wavelength) band widths. (1 fs = 10 15 s, 1 as = 10 18 s)

38. 4. Wave Mechanics Recall – the wavefunction (  ) contains all the information we need to know about any particular system. How do we determine  and use it to deduce properties of the system? 4.1 Operators and Observables If  is the wavefunction representing a system, we can write: where Q – “observable” property of system (e.g. energy, momentum, dipole moment …) – operator corresponding to observable Q.

39. This is an eigenvalue equation and can be rewritten as: ( Note:  can’t be cancelled). Examples: d/dx ( e ax ) = a e ax d 2 /dx 2 ( sin ax ) = a 2 sin ax operator Q acting on function  (eigenfunction) function  multiplied by a number Q (eigenvalue)

40. To find  and calculate the properties (observables) of a system: 1. Construct relevant operator 2. Set up equation 3. Solve equation  allowed values of  and Q . Quantum Mechanical Position and Momentum Operators 1. Operator for position in the x - direction is just multiplication by x 2. Operator for linear momentum in the x-direction:  (solve first order differential equation   , p x ).

41. Constructing Kinetic and Potential Energy QM Operators 1. Write down classical expression in terms of position and momentum. 2. Introduce QM operators for position and momentum. Examples 1. Kinetic Energy Operator in 1-D CM  QM 2. KE Operator in 3-D CM QM 3. Potential Energy Operator (a function of position)  PE operator corresponds to multiplication by V(x) , V(x,y,z) etc. “del-squared” partial derivatives operate on (x,y,z)

42. 4.2 The Schr ö dinger Equation (1926) The central equation in Quantum Mechanics. Observable = total energy of system. Schr ö dinger Equation Hamiltonian Operator E Total Energy where and E = T + V . SE can be set up for any physical system. The form of depends on the system. Solve SE   and corresponding E .

43. Examples 1. Particle Moving in 1-D  (x) The form of V(x) depends on the physical situation: Free particle V(x) = 0 for all x. Harmonic oscillator V(x) = ½ k x 2 2. Particle Moving in 3-D  (x,y,z) SE  or Note: The SE is a second order differential equation

44. 4.3 Particle in a I-D Box System Particle of mass m in 1-D box of length L . Position x = 0 L . Particle cannot escape from box as PE V(x)=  for x = 0, L (walls). PE inside box: V(x)= 0 for 0< x < L . 1-D Schr ö dinger Eqn. (V = 0 inside box).   x L PE ( V )

45. This is a second order differential equation – with general solutions of the form:  = A sin kx + B cos kx SE   (i.e. E depends on k ).

46. Restrictions on  In principle Schr ö dinger Eqn. has an infinite number of solutions. So far we have general solutions: any value of {A, B, k}  any value of {  ,E} . BUT – due to the Born interpretation of  , only certain values of  are physically acceptable : outside box ( x<0, x>L ) V =   impossible for particle to be outside the box     2 = 0   = 0 outside box.  must be a continuous function  Boundary Conditions  = 0 at x = 0  = 0 at x = L .

47. Effect of Boundary Conditions x = 0  = A sin kx + B cos kx = B  =  B =   = A sin kx for all x x = L  = A sin kL = sin kL = 0  kL = n  n = 1, 2, 3, … (n  0, or  = 0 for all x) 1 A=0 ?  (or  = 0 for all x) sin kL = 0 ? 

48. Allowed Wavefunctions and Energies k is restricted to a discrete set of values: k = n  /L Allowed wavefunctions:  n = A sin(n  x/L) Normalization: A =  (2/L)  Allowed energies: 

49. Quantum Numbers There is a discrete energy state ( E n ), corresponding to a discrete wavefunction (  n ), for each integer value of n . Quantization – occurs due to boundary conditions and requirement for  to be physically reasonable (Born interpretation). n is a Quantum Number – labels each allowed state (  n ) of the system and determines its energy ( E n ). Knowing n , we can calculate  n and E n .

50. Properties of the Wavefunction Wavefunctions are standing waves: The first 5 normalized wavefunctions for the particle in the 1-D box: Successive functions possess one more half-wave ( they have a shorter wavelength). Nodes in the wavefunction – points at which  n = 0 (excluding the ends which are constrained to be zero). Number of nodes = (n-1)  1  0;  2  1;  3  2 …

51. Curvature of the Wavefunction If y = f(x) dy/dx = gradient of y (with respect to x). d 2 y/dx 2 = curvature of y. In QM Kinetic Energy  curvature of  Higher curvature  (shorter  )  higher KE For the particle in the 1-D box (V=0): KE

52. Energies E n  n 2 /L 2  E n  as n (more nodes in  n ) E n  as L (shorter box) n (or L )  curvature of  n   KE  E n  L   1  2 node E L 1 E L 2

53. E n  n 2  energy levels get further apart as n Zero-Point Energy (ZPE) – lowest energy of particle in box: CM E min = 0 QM E = 0 corresponds to  = 0 everywhere (forbidden). E n 1 2 3

54. If V(x) = V  0 , everywhere in box, all energies are shifted by V . E 1 E 2 V = 0 V E 1  =E 1 +V E 2  =E 2 +V V 

55. Density Distribution of the Particle in the 1-D Box The probability of finding the particle between x and x+dx (in the state represented by  n ) is: P n (x) =  n (x) 2 dx = ( n (x)) 2 dx ( n is real)  Note: probability is not uniform  n 2 = 0 at walls (x = 0, L) for all  n .  n 2 = 0 at nodes (where  n = 0).   2  2

56. 4.4 Further Examples (a) Particle in a 2-D Square or 3-D Cubic Box Similar to 1-D case, but    (x,y) or  (x,y,z) . Solutions are now defined by 2 or 3 quantum numbers e.g. [  n,m , E n,m ]; [  n,m,l , E n,m,l ]. Wavefunctions can be represented as contour plots in 2-D (b) Harmonic Oscillator Similar to particle in 1-D box, but PE V(x) = ½kx 2 (c) Electron in an Atom or Molecule 3-D KE operator PE due to electrostatic interactions between electron and all other electrons and nuclei.

57. yudh 57 A SUMMARY OF DUAL ITY OF NATURE Wave particle duality of physical objects LIGHT Wave nature -EM wave Particle nature -photons Optical microscope Interference Convert light to electric current Photo-electric effect PARTICLES Wave nature Matter waves -electron microscope Particle nature Electric current photon-electron collisions Discrete (Quantum) states of confined systems, such as atoms.

58. Yodh 58 QUNATUM MECHANICS: ALL PHYSICAL OBJECTS exhibit both PARTICLE AND WAVE LIKE PROPERTIES. THIS WAS THE STARTING POINT OF QUANTUM MECHANICS DEVELOPED INDEPENDENTLY BY WERNER HEISENBERG AND ERWIN SCHRODINGER. Particle properties of waves: Einstein relation: Energy of photon = h (frequency of wave). Wave properties of particles: de Broglie relation: wave length = h/(mass times velocity) Physical object described by a mathematical function called the wave function . Experiments measure the Probability of observing the object.

59. Yodh 59 A localized wave or wave packet: Spread in position Spread in momentum Superposition of waves of different wavelengths to make a packet Narrower the packet , more the spread in momentum Basis of Uncertainty Principle A moving particle in quantum theory

60. Yodh 60 ILLUSTRATION OF MEASUREMENT OF ELECTRON POSITION Act of measurement influences the electron -gives it a kick and it is no longer where it was ! Essence of uncertainty principle.

61. Yodh 61 Classical world is Deterministic: Knowing the position and velocity of all objects at a particular time Future can be predicted using known laws of force and Newton's laws of motion. Quantum World is Probabilistic: Impossible to know position and velocity with certainty at a given time. Only probability of future state can be predicted using known laws of force and equations of quantum mechanics. Observer Observed Tied together

62. Yodh 62 BEFORE OBSERVATION IT IS IMPOSSIBLE TO SAY WHETHER AN OBJECT IS A WAVE OR A PARTICLE OR WHETHER IT EXISTS AT ALL !! QUANTUM MECHANICS IS A PROBABILISTIC THEORY OF NATURE UNCERTAINTY RELATIONS OF HEISENBERG ALLOW YOU TO GET AWAY WITH ANYTHING PROVIDED YOU DO IT FAST ENOUGH !! example: Bank employee withdrawing cash, using it ,but replacing it before he can be caught ... CONFINED PHYSICAL SYSTEMS – AN ATOM – CAN ONLY EXIST IN CERTAIN ALLOWED STATES ... . THEY ARE QUANTIZED

63. Yodh 63 COMMON SENSE VIEW OF THE WORLD IS AN APPROXIMATION OF THE UNDERLYING BASIC QUANTUM DESCRIPTION OF OUR PHYSICAL WORLD ! IN THE COPENHAGEN INTERPRETATION OF BOHR AND HEISENBERG IT IS IMPOSSIBLE IN PRINCIPLE FOR OUR WORLD TO BE DETERMINISTIC ! EINSTEIN, A FOUNDER OF QM WAS UNCOMFORTABLE WITH THIS INTERPRETATIO N Bohr and Einstein in discussion 1933 God does not play dice !

64. Einstein-Podosky-Rosen (EPR)Paradox .Quantum entanglement .Double slit Exp. :Quantum description of Nature is incomplete.

65. What is QM trying to tell us? Bohr: In our description of Nature , the purpose is not to disclose the real essence of the phenomena but only to track down , so far as it is possible, relations between the manifold aspects of our experience.

66. 19 th century problem : What is Electrodynamics trying to tell us? Fields in empty space have physical reality ; the medium that supports them does not

67. 20 th century problem : What QM is trying to tell us ? Correlations have physical reality ; that which that correlate does not. Correlation between energy states is the reality ; not the energy states.

68. Planck’s guiding spirit: There are absolute laws in Nature that must be simple and logical.

69. Concluding Remark: QM has been an unqualified success in quantitatively describing the atomic and sub-atomic world, its interpretative aspects have not been satisfactory.

70. THANKS..