CHAPTER 5 Quantum Mechanics – II 215 The Concept of Degeneracy Among Energy Levels for a Particle in Three Dimensional Box The solution of Schrodinger wave equation for a particle of mass ‘m’ trapped in three dimensional of sides a, b and c with zero potential inside and infinite potential outside provide the total wave function ψ as (27) 8 푛 휋푥 푛푦휋푦 푛 휋푧 휓 (푥, 푦, 푧) = √ 푆푖푛 푥 푆푖푛 푆푖푛 푧 푛푥푛푦푛푧 푎푏푐 푎 푏 푐 Where 푛푥, 푛푦, 푛푦 are the discrete variable whose permitted values from boundary conditions can be 1, 2, 3, 4….∞. The variable x, y and z represent the position of the particle along the corresponding axis. Besides, the expression for total energy is 2 2 2 2 푛 푛푦 푛 ℎ (28) 퐸 = ( 푥 + + 푧) 푛푥푛푦푛푧 푎2 푏2 푐2 8푚 For a cubical box, all the sides become equal (a = b = c). Using this condition in equation (27), the total wave function representing different quantum mechanical states take the following form. (29) 8 푛 휋푥 푛푦휋푦 푛 휋푧 휓 (푥, 푦, 푧) = √ 푆푖푛 푥 푆푖푛 푆푖푛 푧 푛푥푛푦푛푧 푎3 푎 푎 푎 Similarly, the energy expression also changes to ℎ2 (30) 퐸 = (푛2 + 푛2 + 푛2) 푛푥푛푦푛푧 푥 푦 푧 8푚푎2 Now, in order to define various quantum mechanical states, we need to put valid set quantum numbers. The expression for first quantum mechanical and corresponding energy can be obtained by putting 푛푥 = 푛푦 = 푛푧 = 1 in equations (29–30) i.e. (31) 8 휋푥 휋푦 휋푧 3ℎ2 휓 = √ 푆푖푛 푆푖푛 푆푖푛 and 퐸 = 111 푎3 푎 푎 푎 111 8푚푎2 Similarly, the next state with energy can be obtained by putting 푛푥 = 푛푦 = 1 and 푛푧 = 2 in equations (29– 30) i.e. (32) 8 휋푥 휋푦 2휋푧 6ℎ2 휓 = √ 푆푖푛 푆푖푛 푆푖푛 and 퐸 = 112 푎3 푎 푎 푎 112 8푚푎2 If 푛푥 = 푛푧 = 1 and 푛푦 = 2; the wavefunction and energy become Copyright © Mandeep Dalal 216 A Textbook of Physical Chemistry – Volume I (33) 8 휋푥 2휋푦 휋푧 6ℎ2 휓 = √ 푆푖푛 푆푖푛 푆푖푛 and 퐸 = 121 푎3 푎 푎 푎 121 8푚푎2 If 푛푦 = 푛푧 = 1 and 푛푥 = 2, the state with energy becomes (34) 8 2휋푥 휋푦 휋푧 6ℎ2 휓 = √ 푆푖푛 푆푖푛 푆푖푛 and 퐸 = 211 푎3 푎 푎 푎 211 8푚푎2 It can be clearly seen that three quantum mechanical states 휓112, 휓121 and 휓211 possess the same amount of energy (i.e. 6ℎ2/8푚푎2); and therefore, are said to be degenerate. In other words, there are three different ways of existence of the particle inside the box so that the particle possesses 6ℎ2/8푚푎2 energy as total. Figure 2. The energy level diagram representing different quantum mechanical states (in the units of ℎ2/8푚푎2) for a particle trapped in a cubical box. Hence, the degeneracy of the ground state is one i.e. there is only one way for the particle to exist in the box 2 2 to create zero-point energy (3ℎ /8푚푎 ). On the other hand, the degeneracy of first excited stated is 3 as 휓112, 휓121 and 휓211, all have 6 units of energy. Moreover, after careful examination of energy diagram, it can be concluded that degeneracy is 1 if 푛푥 = 푛푦 = 푛푧, 3 if 푛푥 = 푛푦 or 푛푦 = 푛푧 or 푛푥 = 푛푧, and 6 if 푛푥 ≠ 푛푦 ≠ 푛푧. 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 .................................................................................................................................