Chem 50101: Quantum & Statistical Mechanics

 

Year: Fifth

Course Code: Chem.-5101

Type: Theory

Marks: (75+25) = 100

Credit: 4

Course Title: Quantum Mechanics and Statistical Mechanics

Exam-2022, 2023

Objective of the Course: To understand, determine and solve the mechanism and functions of quantum mechanical operators.

Course Teacher: 

 

Course Content/Description:        

ILO: Upon completion of this course students will be able to-

1. Quantum Mechanical Operators:

Concept of operators, Quantum mechanical operators and classical mechanical variables. Linear, Laplacian, Hamiltonian, Momentum, Position operators. Orthogonality, orthogonality and normalization of wave functions, Kronecker’s delta. Commuting operators, Hermitian operators and their significances. Postulates of quantum mechanics, Eigen functions and Eigen values. Dirac bracket notations. Approximate solution of Schrodinger equation: Perturbation method; Variation method and their applications. [20 lectures]

determine & compare operators & theories of quantum mechanical system.

2. Statistical Mechanics:

a) Probability and frequency; Macrostates & Microstates; Cell, position space; Momentum space, Phase space & Phase cells; Fundamental postulates of statistical mechanics; Ensembles: Microcanonical, Canonical & Grand Canonical. 

b) Statistical mechanics: Systems of independent particles, Fermi-Dirac, Bose-Einstein and Boltzmann distribution laws and their comparison. [18 lectures]

determine & compare theories of statistical mechanics.

3. Partition Functions:

Partition functions: relation to thermodynamic functions. Evaluation of partition functions: translational, rotational, vibrational, electronic and nuclear partition functions. [12lectures]

evaluate & compare different partition functions.

4. Characteristicsof Crystalline Solids: 

Thermal characteristics of crystalline solids: Einstein and Debye theories and their comparison. [10 lectures]

characterize & compare theories of crystalline solids.

Required texts/Resources:

  1. Quantum Chemistry, D.A. McQuarrie, University Science Books, USA, 2003.
  2. Quantum Chemistry, B.K. Sen, Tate McGraw-Hill, New Delhi, 1996. 
  3. Molecular Quantum Mechanics (3rd Edition), P.W. Atkins and R.S. Friedman, Oxford University Press, 1997. 
  4. Quantum Chemistry Through Problems and Solutions, R.K. Prasad, New Age International Publishers, New Delhi, 1997. 
  5. Quantum Chemistry, I.A. Levin, Prentice Hall, 1995, 
  6. Statistical Mechanics, D.A. McQuarrie, University Science Books, 2003.
  7. Statistical Thermodynamics, M.C. Gupta, New Age International Publishers, New Delhi, 1995. 
  8. Physical Chemistry (7th Edition), J. de Paula, P. Atkins, Oxford University Press, 2003. 
  9. Introduction to Quantum Mechanics, A. C. Phillips, John Wiley & Sons, Inc., (https://www.fisica.net/mecanica-quantica/Phillips%20-%20Introduction%20to%20Quantum%20Mechanics.pdf)
  10. Quantum Mechanics A Modern and Concise Introductory Course, Daniel R. Bes, Third Edition. https://www.pdfdrive.com/quantum-mechanics-a-modern-and-concise-introductory-course-e184539188.html

Information about assignments and marks: Continuous assessment: Total marks: 25 (class attendance = 10 + class tests & assignments = 15); Final exam marks: 75; Total 60 lectures.


Concept of operators, 


Quantum mechanical operators and classical mechanical variables. 


Linear operators, Laplacian operators, Hamiltonian operators, Momentum operators, Position operators. 

As the first ex­am­ple, while a math­e­mat­i­cally pre­cise value of the po­si­tion $x$ of a par­ti­cle never ex­ists, in­stead there is an $x$-​po­si­tion op­er­a­tor ${\widehat x}$. It turns the wave func­tion $\Psi$ into $x\Psi$:  

\begin{displaymath} \Psi(x,y,z,t) \quad \begin{picture}(100,10) \put(50,12){... ...t(0,2){\vector(1,0){100}} \end{picture} \quad x \Psi(x,y,z,t) \end{displaymath}(3.3)

The op­er­a­tors ${\widehat y}$ and ${\widehat z}$ are de­fined sim­i­larly as ${\widehat x}$.

In­stead of a lin­ear mo­men­tum $p_x$ $\vphantom0\raisebox{1.5pt}{$=$}$ $mu$, there is an $x$-​mo­men­tum op­er­a­tor 

\begin{displaymath} \fbox{$\displaystyle {\widehat p}_x = \frac{\hbar}{{\rm i}} \frac{\partial}{\partial x} $} \end{displaymath}(3.4)

that turns $\Psi$ into its $x$-​de­riv­a­tive: 
\begin{displaymath} \Psi(x,y,z,t) \quad \begin{picture}(100,25) \put(50,21){... ...00}} \end{picture} \quad \frac{\hbar}{{\rm i}}\Psi_x(x,y,z,t) \end{displaymath}(3.5)

The con­stant $\hbar$ is called “Planck's con­stant.” (Or rather, it is Planck's orig­i­nal con­stant $h$ di­vided by $2\pi$.) If it would have been zero, all these trou­bles with quan­tum me­chan­ics would not oc­cur. The blobs would be­come points. Un­for­tu­nately, $\hbar$ is very small, but nonzero. It is about 10$\POW9,{-34}$ kg m$\POW9,{2}$/s.

The fac­tor ${\rm i}$ in ${\widehat p}_x$ makes it a Her­mit­ian op­er­a­tor (a proof of that is in de­riva­tion {D.9}). All op­er­a­tors re­flect­ing macro­scopic phys­i­cal quan­ti­ties are Her­mit­ian.

The op­er­a­tors ${\widehat p}_y$ and ${\widehat p}_z$ are de­fined sim­i­larly as ${\widehat p}_x$: 

\begin{displaymath} \fbox{$\displaystyle {\widehat p}_y = \frac{\hbar}{{\rm i}... ...t p}_z = \frac{\hbar}{{\rm i}} \frac{\partial}{\partial z} $} \end{displaymath}(3.6)

The ki­netic en­ergy op­er­a­tor ${\widehat T}$ is: 

\begin{displaymath} {\widehat T}= \frac{{\widehat p}_x^2 + {\widehat p}_y^2 + {\widehat p}_z^2}{2 m} \end{displaymath}(3.7)

Its shadow is the New­ton­ian no­tion that the ki­netic en­ergy equals: 

\begin{displaymath} T = \frac12 m \left( u^2 + v^2 + w^2 \right) = \frac{(mu)^2 + (mv)^2 + (mw)^2}{2m} \end{displaymath}

This is an ex­am­ple of the “New­ton­ian anal­ogy”: the re­la­tion­ships be­tween the dif­fer­ent op­er­a­tors in quan­tum me­chan­ics are in gen­eral the same as those be­tween the cor­re­spond­ing nu­mer­i­cal val­ues in New­ton­ian physics. But since the mo­men­tum op­er­a­tors are gra­di­ents, the ac­tual ki­netic en­ergy op­er­a­tor is, from the mo­men­tum op­er­a­tors above: 
\begin{displaymath} {\widehat T}= - \frac{\hbar^2}{2m} \left( \frac{\partial^... ...^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \right). \end{displaymath}(3.8)

Math­e­mati­cians call the set of sec­ond or­der de­riv­a­tive op­er­a­tors in the ki­netic en­ergy op­er­a­tor the Lapla­cian, and in­di­cate it by $\nabla^2$: 

\begin{displaymath} \fbox{$\displaystyle \nabla^2 \equiv \frac{\partial^2}{\p... ...artial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} $} \end{displaymath}(3.9)

In those terms, the ki­netic en­ergy op­er­a­tor can be writ­ten more con­cisely as: 
\begin{displaymath} \fbox{$\displaystyle {\widehat T}= - \frac{\hbar^2}{2m} \nabla^2 $} \end{displaymath}(3.10)

Fol­low­ing the New­ton­ian anal­ogy once more, the to­tal en­ergy op­er­a­tor, in­di­cated by $H$, is the the sum of the ki­netic en­ergy op­er­a­tor above and the po­ten­tial en­ergy op­er­a­tor $V(x,y,z,t)$: 

\begin{displaymath} \fbox{$\displaystyle H = -\frac{\hbar^2}{2m} \nabla^2 + V $} \end{displaymath}(3.11)

This to­tal en­ergy op­er­a­tor $H$ is called the Hamil­ton­ian and it is very im­por­tant. Its eigen­val­ues are in­di­cated by $E$ (for en­ergy), for ex­am­ple $E_1$, $E_2$, $E_3$, ...with: 

\begin{displaymath} H \psi_n = E_n \psi_n \quad\mbox{for } n = 1, 2, 3, ... \end{displaymath}(3.12)

where $\psi_n$ is eigen­func­tion num­ber $n$ of the Hamil­ton­ian.

It is seen later that in many cases a more elab­o­rate num­ber­ing of the eigen­val­ues and eigen­vec­tors of the Hamil­ton­ian is de­sir­able in­stead of us­ing a sin­gle counter $n$. For ex­am­ple, for the elec­tron of the hy­dro­gen atom, there is more than one eigen­func­tion for each dif­fer­ent eigen­value $E_n$, and ad­di­tional coun­ters $l$and $m$ are used to dis­tin­guish them. It is usu­ally best to solve the eigen­value prob­lem first and de­cide on how to num­ber the so­lu­tions af­ter­wards.

(It is also im­por­tant to re­mem­ber that in the lit­er­a­ture, the Hamil­ton­ian eigen­value prob­lem is com­monly re­ferred to as the time-in­de­pen­dent Schrö­din­ger equa­tion. How­ever, this book prefers to re­serve the term Schrö­din­ger equa­tion for the un­steady evo­lu­tion of the wave func­tion.)

Hamiltonian operator:

  1. It is the summation of the kinetic and potential energy operator. 
  2. The operator of kinetic energy should be the same for all the models but potential energy changes as per the parameters. 
  3. It is denoted the H^ as an operator.

The description of the Hamiltonian operator is shown below:

  1. It is used to calculate the system energy in terms of the wave function of quantum mechanics.
  2. The input of this operator is wavefunction and the output is energy in the form of eigenvalues.
  3. The wavefunction is the smallest amount of energy in the form of vibrational energy, also known as zero-point energy.
  4. It is expressed in terms of momenta and position coordinates. 
  5. The formula for the Hamiltonian operator is: H^=p22m+V(x)
    \begin{displaymath} \fbox{$\displaystyle H = -\frac{\hbar^2}{2m} \nabla^2 + V $} \end{displaymath}
  6. where the first term denotes the kinetic energy and the second term denotes the potential energy. In this equation, p is momentum, m is mass, and V is potential.
  7. The operator is calculated in a particular direction of time.

CONSTRUCTION of OPERATOR (Click here)


Orthogonality, orthogonality and normalization of wave functions, Kronecker’s delta. 


Commuting operators, Hermitian operators and their significances. Postulates of quantum mechanics, Eigen functions and Eigen values. Dirac bracket notations. Approximate solution of Schrodinger equation: Perturbation method; Variation method and their applications.

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