Dr. Aniruddha Chakraborty

School of Basic Sciences

Indian Institute of Technology Mandi

Mandi, Himachal Pradesh 175001

India

ph: +(91)1905-237930

achakrab

**Course Details:**

Studying atomic and molecular world in more detail will help the students to learn to see more than what is obvious to others.

**Module 1: Molecular properties using Quantum Mechanics **

Time-dependent and time-independent Schrödinger equation, stationary & non-stationary states, Quantum Mechanical Tunneling.

* Tutorial 1:* Free Electron Model, Bands of Orbitals in Solids and elementary idea of wave packet dynamics.

**Module 2: Understanding the motions of the electrons in atoms and in molecules**

Atomic Orbitals, molecular orbitals, shape of molecules, Elementary ideas for Treating Electron Correlation, Current Challenges.

*Tutorial 2:* Approximate methods in Quantum Mechanics: simple examples of perturbation and variational method, recent applications.

**Module 3: Dance of Molecules **

The Born-Oppenheimer Approximation, Vibrational and Rotational motion, Electronic Absorption Spectroscopy, Raman Spectroscopy, current status.

*Tutorial 3:* Examples of tri-atomic molecules.

**Module 4: Treating Large Numbers of Molecules Together **

Elementary Statistical Mechanics, Distribution of Energy Among Levels, Partition Functions and Thermodynamic Properties, Equilibrium Constants in Terms of Partition Functions.

*Tutorial 4:* Simulation of Properties: simple examples for understanding the idea behind Monte Carlo and Molecular Dynamics Method.

**Module 5: Understanding Motion of the atoms within the Molecule **

Elementary reaction rate theories, experimental probes, current challenges.

*Tutorial 5:* Elementary Rate calculations: classical methods with quantum corrections.

**Suggested Books: **

- R. S. Berry, S. A. Rice and J. Ross, Physical Chemistry, 2nd edition, Oxford Univ. Press, 2000.
- D. A. McQuarrie and J. D. Simon, Physical Chemistry, University science books, 1999.
- R. J. Silbey and R. A. Alberty, Physical Chemistry, 3rd edition, John Wiley, 2002.

**Marks Distribution:**

End Sem Exam: 50%, Quizzes: 20 %, Regular Assignments 20%, Interactive Assignment: 10%.

**Lectures:**

Lecture 1: What is Theoretical Chemistry?

Lecture 2: Thermodynamics, Quantum, Classical & Statistical Mechanics.

Lecture 3: Principle of Least action.& Feynman's Path Integral Formulation.

Lecture 4: Generalized or Good Coordinates.

Lecture 5: Energy, Hamiltonian, and Momentum.

Lecture 6: The state of a system.

Lecture 7: Properties of one dimensional potential functions.

Lecture 8: The de Broglie Hypothesis. Work of Planck, Einstein & Bohr.

Lecture 9: Wave properties, Uncertainty principle, work of Born.

Lecture 10: Postulates of Quantum Mechanics.

Lecture 11: Hilbert space: delta function orthogonality.

Lecture 12: Superposition principle, commutator relations & uncertainty.

Lecture 13: Time development of the state function.

Lecture 14: Time development of expectation values.

Lecture 15: Conservation of energy, momemtum.

Lecture 16: Conservation of parity.

Lecture 17: Initial value problem in qunatum Mechanics.

Lecture 18: General solutions of time independent Schrodinger equation.

Lecture 19: Bound states & unbound states.

Lecture 20: One particle in one dimentional box.

Lecture 21: Two particle in one dimesional box & two beads on a wire.

Lecture 22: N particles in one dimensional box.

Lecture 23: Particles in n-dimensional box.

Lecture 24: Step potential & rectangular barrier.

Lecture 25: Dirac delta potential and arbitrary potentials.

Lecture 26: Suddent Perturbations : examples.

Lecture 27: Adiabatic perturbations: examples.

Lecture 28: Harmonic Oscillator in position space.

Lecture 29: Harmonic Oscillator in momentum space.

Lecture 30: Morse Oscillator.

Lecture 31: Diract delta potential well : At least one bound state.

Lecture 32: Finite Potential wells.

Lecture 33: Principles for periodic lattice and Band Gap.

Lecture 34: Standing waves at the band. Conduction in solid.

Lecture 35: Variational method.

Lecture 36: Linear Combinations of Atomic Orbitals.

Lecture 37: Time independent Perturbation Theory.

Lecture 38: Coupled Oscillators : Good quantum number ?

Lecture 39: Coupled Oscillators : classical dynamics.

Lecture 40: Vibrational Dynamics: normal modes vs. local modes.

Lecture 41: Rigin rotor: ro-vibrational spectra.

Lecture 42: Molecular Spectroscopy & Dynamics.

**Assignment 1 Questions:**

Find the Reflection and the Transmission probability for the following two step potentials (a) if x <0, V(x)=0, otherwise V(x) = - V and (b) if x <0, V(x)= - V, otherwise V(x) = 0.

**Assignment 2 Questions:**

Find the Reflection and the Transmission probability for the following two step potentials. (a) For t < 0, if x < 0, V(x) = 0, otherwise V(x) = V. At t = 0, the potential is suddenly changed to the following if x<0, V(x) = 0, otherwise V(x ) = - V. (b) For t < 0, if x < 0, V(x) = 0, otherwise V(x) = V. At t = 0, the potential is suddenly changed to the following if x < a, V(x) = 0, otherwise V(x) = V, where 'a' is a positive number. (c) For t < 0, if x < 0, V(x) = 0, otherwise V(x) = V. At t = 0, the potential is suddenly changed to the following if x < a, V(x) = 0, otherwise V(x) = - V, where 'a' is a positive number.

**Assignment 3 Questions:**

1. Find the Transmission and Reflection probability for for the Dirac delta potential problem.

2. Find the Transmission and reflection probability for the oscillating (stregth) Dirac delta potential problem.

**Assignment 4 Questions:**

In a rectangular potential barrier problem, derive analytical expressions for transmission and reflection probabilities at E = V ?

2. In a particle in a box (of length L) problem, suppose the system is in ground state at time t=0. Now if the length of the box is suddenly changed to 2L at time t= t', what is the probability of finding the system at the ground state of the new Hamiltonian at time t (t > t') ?

**Literature Survey:**

Do a literature survey any research area of your choice related to 'Theoretical Chemistry' and submit a brief report which should include (a) reference of at least 10 research papers & (b) information of at least 10 research groups.

**Reading Assignment:**

Study & understand and then submit a brief report on the following topic - “Can temperature be defined when the system is not in thermodynamic equilibrium”.

**Interactive Assignment:**

Topic: Quantum Mechanics of Hydrogen Atom - Numerically. Time Duration: One Month.Students are expected to solve this assignment with the help of Instructor, they are allowed to interact with the instructor by phone,e-mail, chat, tweitt and face to face interactions. Instructor will supply all study materials, help students to understand the concept, derivation,interpretation etc. (that is the reason for calling it 'interactive assignment'). Students will submit an elaborate solution of the problem within one month.

**Short Project:**

Choose a problem related to 'Theoretical Chemistry', solve it using Mathematica and submit a brief report along with the Mathematica notebook.

**Quiz 1 Questions:**

1. Suppose we have a potential of the form V(x) = - K_{0} δ(x), where K_{0 }is a positive number. Search for the bound states (E < 0).

2. Find the reflection coefficient through the assymteric rectangular potential barrier_{.}

3. The operator **Ã** has only non degenerate eigenvectors {φ_{n}} and eigenvalues {a_{n}}. What are the eigenvectors and eigenvalues of the inverse operator **Ã**^{-1}? Is your answer consistent with the commutator theorem?

4. A particle moving in one dimension has the wave function Ψ(x, t) = N e^{[i (a x – b t)]}, where ‘a’ and ‘b’ are real constants, (a) what is the potential field V(x) in which the particle is moving ? (b) if the momentum is measured, what value is found (in terms of ‘a’ and ‘b’) ? (c)if the energy is measured, what value is found ?

5. What is the eigenfunction of the operator **x** corresponding to the eigenvalue x_{0}? (a) In the momentum representation. (b) In the position representation.

6. Consider time independent Schrödinger equation with a periodic potential. Are the eigenfunctions of this potential necessarily periodic? Justify.

7. Over a very long interval of the x-axis, a uniform distribution of 10^{4} electrons is moving to the right with velocity 10^{8} cm/sec and 10^{4} electrons are moving to the left with velocity 10^{8} cm/sec. Assuming that electrons do not interact with one another, construct a state function that yields the preceding properties for the combined beam. Calculate <**p**> for this state.

**Quiz 2 Questions:**

1. What are the energy eigenfunctions and eigenvalues for the one dimensional box problem discussed in the class if the ends of the box are at x= –L/2 and x= +L/2. What will happen to those eigenfunctions and eigenvalues, if we add one repulsive Dirac Delta Potential at x = 0 to the above mentioned (particle in a one dimensional box) problem ?

2. (a) Show that for a particle in a one dimensional box, in an arbitrary state Ψ(x,0)_{, < E > > E1}, (b) Under what condition does the equality maintain?

3. A particle in the one-dimensional box with walls at x = 0 and x = L, is in the ground state. One of the wall of the box is moved to the position x = 2L, in a time short compared to the natural period 2π/ω_{1}, where ħω_{1 = }E_{1}. If the energy of the particle is measured soon after this expansion, what value of energy is most likely to be found ? How does this energy compare to the particle’s initial energy (E_{1})?

4. At t = 0 it is known that of 1000 neutrons in a one dimensional box of width 10^{-5} cm, 100 have energy 4E_{1}, and 900 have energy 225 E_{1} (a) Construct a state function that has these properties. (b) How many neutrons are in the left half of the “box”?

5. What is the expectation value of momentum for a particle in the state Ψ(x,t)=A e ^{i} ^{ω t} Cos[x], where A is a real constant.

6. For what values of the real angle θ will be constant C = ½(e^{iθ}-1) have no effect in calculation involving the modulus |CΨ|?

7. Give an argument in support of the statement that one cannot measure the momentum of a particle in a one dimensional box, with absolute accuracy.

**Final Exam. Questions:**

1. Show that, if a particle is in a stationary state at a given time, it will always remain in that stationary state.

2. Discuss the advantages and limitations of using Harmonic Oscillator model for understanding vibrational motion of molecules.

3. What are the units, if any, for the wave function of a particle in a one dimensional box ?

4. Discuss the degeneracies of the first few energy levels of a particle in a three-dimensional box when all three sides have a different length.

5. Show that the expectation value of an observable whose operator does not depend on time explicitly, is a constant with zero uncertainty.

6. “The Schrodinger equation for the Helium atom cannot be solved exactly” – Justify.

7. What is solvation coordinate? Explain.

8. In the Born-Oppenheimer approximation, we assume that because the nuclei are so much more massive than the electrons, the electron can adjust essentially instantaneously to any nuclear motion, and hence we have a unique and well defined energy, E(R), at each inter-nuclear separation R. Under the same approximation, E(R) is the inter-nuclear potential and so is the potential field in which the nuclei vibrate. Argue, then, that under the Born-Oppenheimer approximation, the force constant is independent of isotopic substitution.

9. A particle of mass m is confined to one dimensional infinite square well of side 0 < x < L. At t = 0, the wave function of the system is ψ (x, 0) = c_{1} Sin (πx/L)+c_{2} Sin(2πx/L), where c_{1} and c_{2} are the normalization constants for the respective states. (a) What is the wave function at time t ? Is it a stationary state ? Why ? (b) What is the average energy of the system at time t ?

9. Two Hamiltonians H_{1} and H_{2} differ from one another by a constant potential V (independent of x and t). Compare the eigenvalues and eigenfunctions of H_{1} and H_{2}.

10. Consider a one dimensional box of length L. Four electrons are to be put in it. What is the lowest energy state available for the system (neglect electron- electron repulsion effect).

11. If you were to use a trial function of the form φ(x)=(A+Bx)Exp[- α x^{2} ], where A and α are constants. B is a variational parameter to calculate the ground state energy of a harmonic oscillator, what do you think the value of B will come out to be ? Why ?

12. A particle is free to move from - ∞ to + ∞ along the x-axis. Prove that well behaved solutions exist for all values of the energy E > 0. Can there be any solution for E < 0? Explain.

13. What is meant by the parity of a wave function? Prove that the parity operator can have only two eigenvalues +1/- 1.

14. Using Harmonic oscillator as the unperturbed problem, calculate the first order correction to the ground state energy of a Morse oscillator.

15. Evaluate the integration of (A +B x^{2}) Exp[- α x^{2} ], where x varies from - ∞ to + ∞.

16. Discuss the basic idea behind single molecule spectroscopy ?

17. “The stability of a chemical bond is a quantum mechanical effect” – Justify.

18. “Most molecules are, in the ground vibrational state and excited rotational states at room temperature” –Justify.

Dr. Aniruddha Chakraborty

School of Basic Sciences

Indian Institute of Technology Mandi

Mandi, Himachal Pradesh 175001

India

ph: +(91)1905-237930

achakrab