Quantum mechanics I

CLASSICAL MECHANICS - FAILURES - PHOTOELECTRIC EFFECT- COMPTON EFFECT - DAVISON AND GERMER EXPERIMENT - HEISENBERG UNCERTAINTY PRINCIPLE - QUANTUM MECHANICS POSTULATES - OPERATORS - COOMMUTATORS

1. Dr.P.GOVINDARAJ   Associate Professor & Head , Department of Chemistry SAIVA BHANU KSHATRIYA COLLEGE ARUPPUKOTTAI - 626101 Virudhunagar District, Tamil Nadu, India QUANTUM MECHANICS I

2. CLASSICAL MECHANICS Velocity of EM radiation ( c )   The distance travelled by EM radiation per unit time interval in the free space is called Velocity of EM radiation The unit of velocity of EM radiation is m/s and the value is equal to 3 x 10 8 m/s Relation between c, λ and  of EM radiation   L et us consider 5 number of waves ( λ ) travelled at particular point per second , the frequency is equal to 5s -1 and the wavelength (distance) is equal to λm . then the distance travelled per second (c) is equal to c = 5 x λ m/s -----(1) When the frequency is 25 s -1 , then the equation becomes c = 25 x λ m/s In general c =  x λ m/s

3. CLASSICAL MECHANICS Electromagnetic spectrum   The electromagnetic spectrum is the range of frequency of electromagnetic radiations and their respective wavelength Definition

4. CLASSICAL MECHANICS Failed to explain the stability of atom 2. Explained successfully the motion of an object which are observable and distinguishable and failed to explain the motion of an object which are not observable and not distinguishable Classical mechanics Failure and success Net charge is equal to 0

5. CLASSICAL MECHANICS According to classical mechanics, Electromagnetic radiation are travel in space in the form of wave and the existence of wave nature in the EM radiation is successfully proved by Interference, Diffraction and polarization experiment of EM radiation Classical Mechanics f ailed to explain photoelectric effect, Compton effect and Black body radiation curves

6. 5. Classical Mechanics failed to explain spectrum of hydrogen atom . Experimentally it is observed that the spectrum of hydrogen atom consists of sets of line which is represented by = 1/ λ = R H where R H is the Rydberg constant   CLASSICAL MECHANICS

7. Black body and black body radiation CLASSICAL MECHANICS Black body is an ideal object which can absorbs and emits all kinds of frequencies (or) wavelengths light radiations and the emission of all kinds of frequencies (or) wavelengths light radiations at particular temperature by the black body is called black body radiation

8. Explanation of black body radiation by Planck’s quantum theory QUANTUM MECHANICS Planck’s theory states that The radiant energy(EM radiation) which is emitted or absorbed by the black body is not continuous ( not in the wave form ) ,but Discontinuous in the form of small discrete pockets of energy each such pockets of energy is called a “Quantum (photon)” The energy of each quantum is directly proportional to the frequency of the radiation E   E = h ------------(1) where E is the energy of each photons of radiation and is expressed in joules ,  is the frequency of the radiation h is the P lanck’s constant and its value is 6.623 x 10 -34 Js

9. QUANTUM MECHANICS Since  = , the equation (1) becomes E = --------(2) A body can emit or absorb energy only in whole number multiplies of quantum, i.e., 1h , 2 h , 3 h  ………..n h . Energy in fractions of a quantum cannot be lost or absorbed. This is known as quantization of energy Based on his theory, Planck obtained the following expression for energy density of black body radiation E()d = x --------(3) where E()d  is the energy density of black body radiation k is the Boltzmann constant  

10. QUANTUM MECHANICS The above equation (3) adequately explain black body radiation curves at all wavelength obtained at different temperature as shown in figure

11. PHOTOELECTRIC EFFECT The photoelectric effect refers to the emission or ejection of electrons from the surface of generally a metal in response to incident light

12. PHOTOELECTRIC EFFECT On plotting kinetic energy of photoelectrons against frequency of incident light radiation gives straight line shown in the figure This plot indicate that the kinetic energy of photoelectron increases as the frequency of incident radiation increases and on extra plotting the straight line towards x-axis meet at point ‘a’ at which there is no photoelectric effect The frequency of incident radiation at point ‘a’ is called threshold frequency  a

13. BOHR’S THEORY OF HYDROGEN ATOM

14. HYDROGEN SPECTRA EXPLAINED BOHR CONCEPT The lines which arise due to the transitions from higher energy levels to the first energy level are grouped as Lyman series Similarly , the lines obtained as a result of transitions of electrons from higher energy levels to second, third , fourth and fifth energy levels gives rise to Balmer , Paschen , Brackett and Pfund series respectively

15. HYDROGEN SPECTRA EXPLAINED BOHR CONCEPT The Wave number of the each spectral lines can be calculated by Rydberg equation

16. COMPTON EFFECT Definition: The increase in wavelength of x-rays after scattering from the surface of an object is known as the Compton effect

17. COMPTON EFFECT Explanation: When X-rays of high energy are allowed to fall on an object (carbon block or light element), an electron is ejected from the carbon block and X-rays are scattered from their original path. In this process, a photon from the incident X-rays collides with the loosely bound electron in the carbon block and, after receiving an impact energy from the photon, the electron which was initially at rest gains some velocity and hence moves below the direction of the incident X-rays while the photon is deflected above. Thus, X-rays are scattered from their original path as a result of the collision between the photon and the electron.

18. COMPTON EFFECT So that, the scattered X-rays have lower energy than the incident x-ray i.e., the scattered X-rays have longer wavelength than incident x-ray If the wavelength of the incident X-rays and the scattered X-rays are  and ’ respectively, then h  and h’ are the energies associated with photon of the incident and scattered X-rays respectively Thus the decrease in energy is equal to ( h  - h  ’), since h  > h  ’. This energy ( h  - h ’) is transferred to the electron at rest and therefore, the electron gain some velocity and moves from its path in this process The difference in ’ and  (i.e., ’ - ) is called Compton shift and is denoted by ∆ . According to Compton, the Compton shift is expressed as ∆  = (1 - cosθ )  

19. COMPTON EFFECT According to Compton, the Compton shift is expressed as ∆  = (1 - cosθ ) Where θ is the angle between the incident and the scattered X-rays m is the rest mass of the electron c is the velocity of light According to this equation, Compton shift is independent of the nature of the object and the wavelength of incident X-rays but depends only on the scattering angle θ  

20. COMPTON EFFECT Case 1 : θ = 0 , i.e., the scattered radiation is parallel to the incident radiation. In this case, cos 0 = 1 so that ∆  = 0, i.e., there is no Compton shift ∆  = (1 - 1) = 0 Case 2 : θ = 90 , i.e., the scattered radiation is perpendicular to the incident radiation. In this case, cos 9 = so that ∆  = (1 - 0) = Case 3 : θ = 180 , i.e., the scattered radiation is in the direction opposite to the incident radiation. In this case, cos 180 = -1 so that ∆  = (1 – (-1)) =  

21. de-Broglie suggested that All matter particles in motion ( eg : electrons, protons, neutrons, atoms or molecules etc.) have a dual character, i.e., all matter particle possess characteristics of both a material particle and a wave The wave associated with matter particles is called de-Broglie’s matter wave de-BROGLIE’S CONCEPT OF MATTER WAVES Diagrammatic representation of de-Broglie matter wave

22. According to Einstein, the energy of an electron (X- ray) is E = mc 2 ---------(1) Where m ------ mass of the electron c ------ velocity of electron According to Plank's , the energy of light radiation (X-ray) is E = h  ----------(2) Where h ------- Plank’s constant  ------- Frequency of light radiation de-BROGLIE’S CONCEPT OF MATTER WAVES Derivation of de-Broglie wave equation:

23. Equating equation (1) and (2), we get mc 2 = h  ---------(3) Substitute  = in equation (3), we get mc 2 = h mc =  = This equation is known as de – Broglie equation   de-BROGLIE’S CONCEPT OF MATTER WAVES

24. Limitation of de-Broglie wave equation : This equation is applicable only for small particles (microscopic particles) like electrons, neutrons, atoms etc. and has no significance for large particles (macroscopic particles) Examples: a ) let us consider an electron in a hydrogen atom with mass (m = 9.108 x 10 -31 kg) and moving with a velocity 2.188 x 10 6 ms -1 . Then  = 0.3323 x 10 -9 m This value of  is large and comparable to the wavelength of X-rays and hence can be measured   de-BROGLIE’S CONCEPT OF MATTER WAVES

25. b) Let us consider a large particle like a stone in motion of mass of 1 g (= 1 x 10 -3 kg ) and moving with a velocity 1ms -1 . Then  = m This value of  is shorter than the wavelength of any electromagnetic radiations and hence cannot be measured by any method. Thus, de-Broglie wave equation has no significance for large particles.   de-BROGLIE’S CONCEPT OF MATTER WAVES

26. MATTER WAVE ELECTROMAGETIC WAVE Not radiated into empty space Radiated into empty space Velocity is not same as that of light i.e., its value is lesser than the velocity of light Velocity is same as that of light. The value is 3 x 10 8 ms -1 Not emitted by oscillating charged particle but associated with particles which are travel in space Emitted by oscillating charged particle There is no electric and magnetic field Possess both electric and magnetic field Eg : Electron revolving around nucleus of an atom Eg : Radio waves DIFFERENCE BETWEEN MATTER WAVES AND EM WAVES

27. EXPERIMENTAL VERIFICATION OF MATTER WAVE CONCEPT DAVISSON – GERMER EXPERIMENT In this experiment, electrons were emitted from a hot filament and were accelerated by a potential (∆ϕ) ranging between 40 and 68 volts before striking a nickel plate, as shown in the diagram

28. The impact of electron resulted in the production of diffraction pattern which were similar to those given by X-rays under similar conditions. Since X-rays possess wave character the experiment give direct evidence for wave character of electrons. The intensity of electron waves scattered by the nickel plate at different angles were measured. It was found that the reflection was most intense and took place at an angle of 50 when electrons were accelerated through 54 volts. Substituting the value of ∆ ϕ as 54 volts in the following equation, the wavelength of the electron wave calculated to be 1.668 A. This wavelength also lies in the range of X-rays DAVISSON – GERMER EXPERIMENT  =  

29. DAVISSON – GERMER EXPERIMENT The wavelength of the scattered electron also calculated by the Bragg equation (n  = 2dsin θ ) by substituting n (a whole number for a reflection of maximum intensities), d (the distance between successive lattice plane of the nickel crystal ) and θ (angle of diffraction) and the value also comes out to be 1.668 A. These experimental results confirmed that the electron has wave nature as that of X-rays

30. HEISENBERG UNCERTAINTY PRINCIPLE Statement : It is impossible to predict both position and momentum of an electron simultaneously Explanation : Consider photons of light radiation incident on a moving reasonable size particle, its position and velocity will not be changed by the impact of light photons Hence the position and velocity of the particle will be exactly predicted

31. HEISENBERG UNCERTAINTY PRINCIPLE But in the case of very minute particle like electron the path and velocity of an electron will be changed due to the impact of light photons used to observe it i.e., For an electron moving in x- direction, the Heisenberg uncertainty principle is expressed Mathematically as , (∆x)(∆p) ≥ h/4 п

32. HEISENBERG UNCERTAINTY PRINCIPLE i.e., For an electron moving in x- direction, the Heisenberg uncertainty principle is expressed Mathematically as (∆x)(∆p) ≥ h/4 п where ∆ x is the uncertainty in position ∆p is the uncertainty in momentum If ∆x is very small, i.e., the position of a particle is known more or less exactly ∆p would be large, i.e., uncertainty with regard to momentum will be large Similarly, if an attempt is made to measure exactly the momentum of the particle the uncertainty with regards to position will become large

33. POSTULATES OF QUANTUM MECHANICS The physical state of a system at time t is described by the wave function ψ ( x,t ) The wave function ψ ( x,t ) and its first and second derivatives, ∂ ψ ( x,t ) /∂x and ∂ 2 ψ ( x,t ) / ∂ x 2 are continuous, finite and single valued for all values of x. Also, the wave function ψ ( x,t ) is normalized ----------(1) where ψ* is the conjugate complex of ψ formed by replacing i with – i whenever it occurs i n the function ψ ( i = ) A physically observable quantity can be represented by a Hermitian operator. An operator is said to be Hermitian if it satisfies the following condition  

34. POSTULATES OF QUANTUM MECHANICS -----------(2 ) where i and are the wave functions representing the physical state of the quantum system, such as particle, an atom or a molecule The allowed values of an observable A are the eigenvalues, a i , in the operator equation i = a i i the above equation is known as eigenvalue equation. Here is the operator for the observable and i is an Eigen function of with eigenvalue a i , In other words, measurement of the observable A yields the Eigen value a i The average value <A> of an observable A, corresponding to the operator is obtained from the relation  

35. POSTULATES OF QUANTUM MECHANICS <A> = where the function is assumed to be normalized in accordance with equation (1). Thus the average value of the x-coordinate is given by <x> = The quantum mechanical operators corresponding to the observables are constructed by writing the classical expressions in terms of the variables and converting the expressions to the operators  

36. POSTULATES OF QUANTUM MECHANICS The wave function ψ ( x,t ) is a solution of the time-dependent Schrodinger equation ∂ψ ( x,t )/ ∂t where is the Hamiltonian operator of the system  

37. OPERATORS Definition : An operator is a mathematical tool which operate on a function results another function i.e., (operator).(function) = Another function The function on which the operation is carried out is called an operand. The operator written alone has no significance Example : d(x 3 )/dx = 3x 2 Here d/dx stands for differentiation with respect to x is the operator, x3 is the operand and 3x2 is the result of the operation Similarly, taking the square ()2, taking square root , multiplication by a constant KX etc., are different operations which can be carried on any function (i.e., the operand )

38. ALGEBRA OF OPERATORS The operators follow certain rules similar to those of the algebra. A few of these are given below Addition and Subtraction of operators: If and are two different operators and f is the operand, then ( + ) f = + f and ( ) f = f 2. Multiplication of operators Multiplication of operators is represented f i.e., First the function f is operated by the operator result another function f’ , and then f’ is operated by the operator gives final result, say f’’ i.e., f = f’ = f’’ otherwise f = f ’’ If f = f, then the operators are said be commutator If f ≠ f, then the operators are said to be non commutator  

39. ALGEBRA OF OPERATORS 3. Linear operators An operator is said to linear operator if it. Satisfy the f ollowing condition (f + g) = f + g where f and g are different functions i.e., linear operators are operators which operate on the sum of two functions gives the same result as the sum of two results obtained by carrying out the same operations on two function separately Example: d/dx and d 2 /dx 2 are linear operators whereas taking the square and taking the square root are non linear operators  

40. ALGEBRA OF OPERATORS 3. Laplacian operators ∇ 2 It is mathematically defined as ∇ 2 = + + we know that , Schrodinger wave equation is + + + ψ = 0 in terms of laplacian operator , SWE is written as ∇ 2 + ψ = 0  

41. QUANTUM MECHANICAL OPERATORS Derivation of linear momentum operator : A beam of electrons travelling along the x-direction can be treated as a wave propagating along the x-axis Taking these waves to be sinusoidal, the wave function may be written as = A sin ---------(1) where  is the wavelength An equivalent form for the time-independent wave is = c exp [ ] ---------(2) = c exp [ ] = ----------(3)  

42. QUANTUM MECHANICAL OPERATORS Using the de-Broglie’s equation (  = h/ P x ) equation (3) becomes = = = ( = -----------(4) Equation (4) is the form of operator equation i.e., Eigen equation where P x is the linear momentum ( is the operator for linear momentum  

43. QUANTUM MECHANICAL OPERATORS i.e., and Similarly and and  

44. QUANTUM MECHANICAL OPERATORS Derivation of Kinetic energy operator : We know that the SWE is + + + ψ = 0 -----------(1) where E – V = Kinetic energy (K.E) On rearranging the equation (1), we get ψ = - ( + + ) ψ = - ( + + ) - ( + + ) = ψ  

45. QUANTUM MECHANICAL OPERATORS - ( + + ) = ψ - ( + + ) ψ = ψ - ψ = ψ where is the Laplacian operator ( - ) ψ = ψ -----------(2) Equation (2) is in the form of quantum mechanical operator (Eigen equation) in which (E-V) is the Kinetic energy and - is the kinetic energy operator i.e., = -  

46. QUANTUM MECHANICAL OPERATORS Operator for total energy : We know that an operator for kinetic energy is = = - ------------(1) Where E is the total energy V is the Potential energy is the Laplacian operator On rearranging equation (1), we get = - + V Where E is the operator for total energy and is called Hamiltonian operator = - + V  

47. QUANTUM MECHANICAL OPERATORS Operator for position and potential energy : The quantum mechanical operators for the position and potential energy are the same as their corresponding classical mechanical quantities a) Position operator: i.e., The position operator ( ) is the multiplication of x for locating the position of a microscopic particle along x-axis i.e., = multiplication by   PROPERTY CLASSICAL QUANTITY QUANTUM MECHANICAL OPERATOR OPERATION Position Multiplication by x PROPERTY CLASSICAL QUANTITY QUANTUM MECHANICAL OPERATOR OPERATION Position Multiplication by x

48. QUANTUM MECHANICAL OPERATORS b) Potential energy operator: i.e., The potential energy of the electron in an atom depends upon the distance between the electron and nucleus and the potential energy operator ( ) is the multiplication by V (x) along the x-axis (one dimensional) and for three dimension, multiplication by V ( x,y,z ) i.e., = multiplication by V (x) = multiplication by V ( x,y,z )   PROPERTY CLASSICAL QUANTITY QUANTUM MECHANICAL OPERATOR OPERATION Potential energy (energy of attraction of an electron by an atomic nucleus) V (x) Multiplication by V x PROPERTY CLASSICAL QUANTITY QUANTUM MECHANICAL OPERATOR OPERATION Potential energy (energy of attraction of an electron by an atomic nucleus) V (x) Multiplication by V x

49. QUANTUM MECHANICAL OPERATORS b) Angular momentum operator: According to classical mechanics, angular momentum (L) is the vector property and it is the cross product of the vector properties of and i.e., ------------(1) Where is the distance between the revolving electrons and nucleus and it is expressed in terms of Cartesian co-ordinates as = ix + jy + kz --------------(2) i , j, k are the unit vectors along x, y and z axis is the linear momentum of the electron and it is expressed as = iP x + jP y + kP z ---------------(3)   P

50. QUANTUM MECHANICAL OPERATORS Substituting (2) and (3) in (1) we get ) ----------(4) The cross product in equation (4) is performed using the matrix concept i.e., = = i (y P z - zP y ) – j ( P z - zP x ) + k ( P y - yP x ) = i (y P z - zP y ) + j ( zP x - P z ) + k ( P y - yP x ) -----------(5) The angular momentum operator is obtained on substituting position operators for , y and z and linear momentum operators for P x , P y and P z in equation (5)  

51. QUANTUM MECHANICAL OPERATORS = i ( - ) + j ( - ) + k ( - ) -------------( 6) = i x + j y + k z -------------(7) Where x = - = - --------------(8) = - On substituting the position operators and linear momentum operators in equation (8), we get x = y - z x = [ y - z , similarly = [ z - x , = [ x - y -------------( 9)  

52. QUANTUM MECHANICAL OPERATORS In equation (7), x , and are the angular momentum operators in terms of Cartesian co-ordinates On taking square of equation (7), we get 2 = 2 x + 2 y + 2 z  

53. COMMUTATOR The commutator of the two operators and is defined as [ ] = - = Irrespective of the order of operations of operator and on an operand  result same value  =   -  = On removing  , we get - = 0 i.e., [ ] = 0   Definition:

54. Evaluation of commutators : 1. Show that the position and linear momentum operators are not commutator The position operator ( ) is the multiplication of x for locating the position of a microscopic particle along -axis i.e., = multiplication of Linear momentum operator is , As per commutator definition with respect to operand  [ , ]  =  -  =  - ( )   COMMUTATOR

55. [ , ]  = -  , so [ , ] = - Hence and are not commutator 2. Show that [ , ] = -1 with respect to operand  By definition , [ , ]  = - ( ) = - -  [ , ]  = -  Removing  from both sides of this equation , we get [ , ] = - 1   COMMUTATOR

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