CHAPTER 5 Quantum Mechanics – II 229 Zero Point Energy of a Particle Possessing Harmonic Motion and Its Consequence In order to understand the minimum or the zero-point energy of a simple harmonic oscillator, recall the general wavefunction representing all the vibrational states of a simple harmonic oscillator i.e. −푦2/2 휓푛(푦) = 푁푛퐻푛(푦) 푒 (112) Where y is a displacement-based variable with a value equal to √훽x. The constant β depends upon the reduced mass of the oscillator (m) and equilibrium vibrational frequency (ν) as 4휋2푚휈 (113) 훽 = ℎ The symbol 푁푛 and 퐻푛(푦) are the normalization constant and Hermit polynomial for nth state i.e. 1/2 푛 (114) √훽 푛 푦2 푑 −푦2 푁푛 = ( ) 푎푛푑 퐻푛(푦) = (−1) . 푒 . 푒 2푛 푛! √휋 푑푦푛 Also, the general expression for the energies is given below. 1 (115) 퐸 = (푛 + ) ℎ휈 푛 2 Now, for the ground vibrational state (푛 = 0), 푁0 and 퐻0(푦) can be obtained from equation (114) i.e. 훽 1/4 (116) 푁0 = ( ) 푎푛푑 퐻0(푦) = 1 휋 After using the values of y, 푁0 and 퐻0(푦) in equation (112), the ground state function becomes 1/4 훽 2 (117) 휓 (푥) = ( ) . 1. 푒−훽푥 /2 0 휋 Hence, the ground state wave function does not collapse at 푛 = 0, which means that corresponding energy can also be obtained by putting 푛 = 0 in equation (115) i.e. 1 (118) 퐸 = (0 + ) ℎ휈 0 2 or 1 (119) 퐸 = ℎ휈 0 2 The above equation gives the minimum energy which is always possessed by a simple harmonic oscillator. Copyright © Mandeep Dalal 230 A Textbook of Physical Chemistry – Volume I Figure 6. The variation of ground vibrational wavefunction and probability as a function of bond length or displacement in a simple harmonic oscillator. It is well-known that the classical oscillator spends more time in the extreme states i.e. fully compressed and fully expended, and spends the least time with equilibrium bond length. However, as far as the ground vibrational state of the quantum mechanical oscillator is concerned, it spends the most time with equilibrium (because the function is maximum for requ or x = 0), and probability to spend time in compressed and expended mode decreases as the magnitude of compression and expansion increases. Moreover, there is always a limit over the compression as well as over the expansion in the classical oscillator to have a certain amount of energy; however, since the function becomes zero only at infinite displacement, there is no limit over the compression and expansion in the quantum oscillator theoretically. It is also worthy to note that the energy given by the equation (119) is in joules. However, in many textbooks or papers, it is also reported in terms of wavenumbers. To do so, we need first put the value of frequency as 휈 = 푐/휆 and then 1/휆 = 휈̅ in the equation (119) i.e. 1 푐 1 (119) 퐸 = ℎ = ℎ푐휈̅ 0 2 휆 2 Where c is the velocity of light. Now, to convert the zero-point energy in wavenumbers, divide equation (119) by ℎ푐 i.e. 1 ℎ푐휈̅ 휈̅ (120) 휈̅ = = 0 2 ℎ푐 2 Buy the complete book with TOC navigation, Copyright © high resolution images and Mandeep Dalal no watermark. 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Sign Up Copyright© 2019 Dalal Institute • Table of Contents CHAPTER 1 ................................................................................................................................................ 11 Quantum Mechanics – I ........................................................................................................................ 11 Postulates of Quantum Mechanics .................................................................................................. 11 Derivation of Schrodinger Wave Equation...................................................................................... 16 Max-Born Interpretation of Wave Functions .................................................................................. 21 The Heisenberg’s Uncertainty Principle .......................................................................................... 24 Quantum Mechanical Operators and Their Commutation Relations ............................................... 29 Hermitian Operators – Elementary Ideas, Quantum Mechanical Operator for Linear Momentum, Angular Momentum and Energy as Hermitian Operator ................................................................. 52 The Average Value of the Square of Hermitian Operators ............................................................. 62 Commuting Operators and Uncertainty Principle (x & p; E & t) .................................................... 63 Schrodinger Wave Equation for a Particle in One Dimensional Box .............................................. 65 Evaluation of Average Position, Average Momentum and Determination of Uncertainty in Position and Momentum and Hence Heisenberg’s Uncertainty Principle..................................................... 70 Pictorial Representation of the Wave Equation of a Particle in One Dimensional Box and Its Influence on the Kinetic Energy of the Particle in Each Successive Quantum Level ..................... 75 Lowest Energy of the Particle ......................................................................................................... 80 Problems .......................................................................................................................................... 82 Bibliography .................................................................................................................................... 83 CHAPTER 2 ................................................................................................................................................ 84 Thermodynamics – I .............................................................................................................................. 84 Brief Resume of First and Second Law of Thermodynamics .......................................................... 84 Entropy Changes in Reversible and Irreversible Processes ............................................................. 87 Variation of Entropy with Temperature, Pressure and Volume ...................................................... 92 Entropy Concept as a Measure of Unavailable Energy and Criteria for the Spontaneity of Reaction ...........................................................................................................................................................94 Free Energy, Enthalpy Functions and Their Significance, Criteria for Spontaneity of a Process ... 98 Partial Molar Quantities (Free Energy, Volume, Heat Concept) ................................................... 104 Gibb’s-Duhem Equation ................................................................................................................ 108 Problems ........................................................................................................................................ 111 Bibliography .................................................................................................................................