Paper 12

Core XII: Quantum Mechanics and Applications

CBCS 2019: Core 11

Syllabus

Unit I

Schrodinger equation: Time dependent Schrodinger equation, Properties of Wave Function, Physical interpretation of wave function, Wave function of a free particle, Normalization, Probability current and probability current densities in three dimensions, Linearity and Superposition Principle, Wave Packet, Fourier Transform Theorem, Momentum space wave function and its significance, Representation of position vector in momentum space. Schrodinger equation in momentum space.

Unit II

Time Independent Schrodinger equation in 1-D, 2-D and 3-D, Hamiltonian, stationary states and energy Eigen values, expansion of an arbitrary wave function as a linear combination of energy Eigen functions, General solution of the time dependent Schrodinger equation in terms of linear combinations of stationary states. General Discussion of Bound states in an arbitrary potential: Continuity of wave function, Boundary condition and emergence of discrete energy levels.

Unit III

Operators: Operators, Commutator Algebra, Position, Momentum, Angular Momentum and Energy operators, Hermitian Operators, Expectation Value, Expectation values of position and momentum, Ehrenfest Theorem, Eigenvalues and Eigenfunctions of Hermitian Operator, Energy Eigen Spectrum, Degeneracy, Orthonormality of Eigen functions, Linear Dependance, Orthogonalisation,
Uncertainty Relation- Uncertainty product, minimum uncertainty wave packet- Gaussian Wave Packet.

Unit IV

Application to one dimensional problem- One dimensional infinitely rigid Box- Energy Eigen values and Eigen functions, normalization, quantum dot as an example, Quantum mechanical scattering and tunneling in one dimension across a Potential Step and Rectangular Potential Barrier, Finite Square well potential, Quantum mechanics of simple Harmonic Oscillator-Energy Levels and Energy Eigen
functions, ground state, zero point energy.

Text Books:
  • Introduction to Quantum Theory David Park (DoverPublications)
  • Introduction to Quantum Theory, D. J. Griffiths(Pearson)
  • Quantum Mechanics: Concepts and applications, N.Zettili, Wiley

Reference Books :

  • Quantum Mechanics, Theory and applications A.Ghatak and S.Lokanathan (McMillanIndia)
  • Quantum Mechanics- G.Aruldhas (PrinticeHallofIndia)
  • Quantum Physics–S. Gasiorowicz (Wiley)
  • Quantum Mechanics- G.R.Chatwal and S.K.Anand
  • Quantum Mechanics -J.L. Powell and B. Craseman(Narosa)
  • Introduction to Quantum Mechanics M.Das and P.K.Jena (Shri Krishna Publication).

Questions

2025 | +3-V-S-CBCS(MS)-Sc(H)-Core-XI-Phy | Full Marks: 60

PART-I | Answer all Questions. | 1x8
  1. Heisenberg's uncertainty principle is given by ______________
  2. The de Broglie wavelength is given by ________________
  3. Momentum operator ($\hat{p}$) is given by ___________________
  4. Space derivative is Hermitian (True/False). ________
  5. Schrodinger equation is _____ order in space derivative.
  6. Momentum of a free particle in a one-dimensional box of length L is given by ________
  7. Value of orbital angular momentum for a d-electron is ________
  8. One Bohr magneton is given by _______________
PART-II | Answer any eight within two to three sentences | 1.5x8
  1. Is $\psi (x) = x^3$ an acceptable wave function ? Give justification.
  2. What is the physical significance of wave function?
  3. Define adjoint of an operator.
  4. Find de Broglie wavelength of an electron accelerated through a potential 100 eV.
  5. Find the uncertainty in velocity of a particle if the uncertainty in its position is equal to its Compton wavelength.
  6. The width of an infinite deep potential well is halved. What will happen to the energy of a particle trapped inside it.
  7. Plot ground state and first excited state wave functions for a free particle in a box.
  8. What is Larmor precession ?
  9. What is Paschen-Back effect ?
  10. Define Lande g-factor.
PART-III | Answer any eight of the following (in maximum 75 words.) | 2x8
  1. If $\psi_1$ and $\psi_2$ are two solutions of time dependent Schrodinger equation, then prove that any linear combination of these two solutions is also another solution of time dependent Schrodinger equation.
  2. Show that $\left[ x^3, x\frac{d}{dx}\right] =3x^3$.
  3. Find probability current density for a plane wave $\psi = A\exp{i\vec{k}.\vec{r}}$.
  4. Show that group velocity $v_g$ and phase velocity $v_p$ of a relativistic particle satisfy the relation $v_g.v_p=c^2$.
  5. Determine the normalization constant A for given wave function: $$\psi =\begin{cases} A(a^2-x^2) &\text{ if } -a\le x\le a \ = 0 &\text{ otherwise} \end{cases}.$$
  6. What is tunnelling? Give one example.
  7. What is penetration depth? On what factors does it depend?
  8. Distinguish normal and anomalous Zeeman effect.
  9. Find the magnetic moment in Bohr magneton of an atom in $^3P_2$ state.
  10. Find value of total angular momentum and spin magnetic moment for a free electron.
PART-IV | Answer within 500 words each. | 6x4

  1. Show that a wave packet corresponding to minimum uncertainty product for position and momentum is a Gaussian wave packet.

    OR

    Explain probability current and probability density. Show that change in probability density in a region of space is equal to net change in probability current into that region.

  2. Define Hermitian operator. Prove that all eigen values of a Hermitian operator are real and eigen vectors of a Hermitian operator with different eigen values are orthogonal.

    OR

    Derive the expression for square of angular momentum operator in spherical polar co-ordinates.

  3. Solve Schrodinger equation for a particle moving in a one-dimensional potential given by
    $$V(x) = \begin{cases} 0 &\text{for } x \le 0 \
    V_0 &\text{for } x > 0 \end{cases}$$
    Calculate the reflection and transmission coefficients, assuming total energy E is less than barrier height $V_0$.

    OR

    Solve Schrodinger equation for a linear harmonic oscillator to obtain eigen values and eigen functions.

  4. Describe Stern-Gerlach experiment. How does it explain the space quantization and electron spin?

    OR

    Explain space quantization taking account of orbital angular momenta of electrons in atomic orbitals.

2024 | +3-V-S-CBCS(MS)-Sc(H)-Core-XI-Phy

PART-I | Answer the following questions. | 1x8
  • For stationary states, divergence of probability current is ____.
  • Dimension of Planck's constant is ____.
  • The Hamiltonian operator is the operator associated with ____ .
  • According to Ehrenfest, the average motion of a wave packed must obey ____ Laws.
  • Uncertainty in momentum for a single monochromative wave is ____.
  • Energy of nth state of a simple harmonic oscillator in quantum mechanics is ____.
  • The value of Lande's g-factor for an s-electron is ____.
  • The first experiment that confirmed the presence of electron spin was ____ experiment.
PART-II | within two to three sentences. | 1.5x8
  • What is stationary state ?
  • What are the conditions for a wave function to be acceptable?
  • Is cosine a linear operator ? Justify your answer.
  • Is A and B are Hermitian operators, then i[A,B] Hermitian ?
  • A particle is in the first excited state of a one-dimensional box of length 1m. What is the minimum value of its momentum?
  • A Proton and a deuteron having the same energy penetrate a given rectangular barrier. Which particle has greater depth of penetration ?
  • Explain quantum dot energy ?
  • Explain why normal Zeeman effect occurs only in atoms with even number of electrons.
  • Calculate the maximum spin and spin magnetic moment in terms of Bohr Magneton for an atom having 6 unpaired electrons.
  • Explain spin gyromagnetic ratio.
PART-III | in maximum 75 words | 2×8
  • Show that momentum space wave function $\phi(\vec{p},t)$ is normalized to unity, if $\psi(\vec{r},t)$ is normalized to unity.
  • Prove that the vector momentum operator $\vec{p}$ is Hermitian?
  • Write the various properties of wave function.
  • Explain the expectation value of Hamiltonian H.
  • Explain energy eigen values of a particle in one dimensional box.
  • Explain Magnetic quantum number.
  • Calculate possible angles between $\vec{L}$ and $\vec{S}$ for a "d" electron in one electron atom.
  • Explain space quantisation.
  • Show that the zero point energy of Harmonic oscillator is a consequence of the uncertainty principle.
  • Find the position of nodes for a wave function $\psi_n(\chi)$ of in one dimensional box.
PART-IV | Answer within 500 words each. | 6x4

4) Derive the Schrodinger time dependent wave equation in one dimensional momentum space.

OR

What is normalization of wave function ? Discuss about the Dirac-Delta Normalization.

5) State and prove Ehrentest's Theorem.

OR

Define expectation value of a dynamical variable. Give expression for expectation value of position and momentum.

6) Derive an expression for Boundary conditions using Schrodinger's quation to one dimensional square well potential.

OR

Derive an expression for quantum mechanical tunneling across a rectangular potential barrier.

7) Explain Normal Zeeman effect using classical theory and also obtain an expression for Zeeman shift.

OR

State and explain Larmour's Theorem. Then derive an expression for Larmor's frequency.

2022 | +3-V-S-CBCS(MS)-Sc(H)-Core-XI-Phy-R&B | Full Marks : 60

PART-I | Answer all questions | 1x8
  • Two observables are compatible, if their operators are _____.
  • The eigen values of Hamiltonian operator represents ______ of the system.
  • Write the Wave function represents a free particle moving along the X-axis.
  • Ouantum Mechanics deals with _____.
  • The splitting of a spectral line in the presence of external electric field is termed as ______ effect.
  • The value of spin quantum number of an electron in Hydrogen atom is ______.
  • According to quantum mechanics momentum in a 'particle in a box' is ______.
  • The energy of a one-dimensional Harmonic oscillator in first excited state is ______.
PART-II | Answer any eight within two to three sentences | 1.5x8
  • If a 5000 $\AA$ line exhibits normal zeeman splitting of $1.1 \times 10^{-3} \AA$. Find the field strength.
  • Determine the total Angular Momentum for a free electron.
  • What is the Zero point Energy ?
  • What is tunneling ? Give one example.
  • Why the minimum energy of a 'particle in a box' cannot be zero?
  • Calculate the Uncertainty in momentum of an electron if the Uncertainty in its position is 0.4 nm.
  • Find the eigen functions of the operator $d^2/dx^2$.
  • What is the Hermitian operator ?
  • What is importance of schrodinger's equation ?
  • Write the limitations of the ware function for a free particle.
PART-III | Answer any eight of the following (in maximum 75 words. | 2x8
  • Calculate the Lande's g-factor for the $2p_{3/2}$ state.
  • Distinguish between L-S and J-J coupling.
  • Explain space Quantisation.
  • Explain Quantum Dot.
  • Justify that "tunneling" is a Quantum mechanical phenomenon.
  • What are eigen values and eigen functions?
  • Find whether the operator $\hat A = 3x^2$ and $\hat B = d/dx$ commute or not.
  • Find the the probability of a particle in a box of width 'L' to be found between x=0 and x =L/n, when it is in the nth state.
  • Determine whether or not $\psi (x)=e^x$ is an acceptable ware function.
  • Comment on the statement, "Heisenberg's Uncertainty principle is valid for all kind of particles."
PART-IV | Answer within 500 words each | 6x4

  1. Derive the time dependent schrodinger quation for a non- relativistic particle.

    OR

    Explain normalization of a Wave function. How is normalisation expressed mathematically ? Prove that normalisation is independent of time.

  2. State and prove Ehrenfest's Theorem

    OR

    Define expectation value of a dynamical variable. How is it related to probability ? Give expression for expectation valuc of position and momentum.

  3. Derive an expression for quantum mechanical tunneling across a rectangular potential barrier

    OR

    Derive an expression for Boundary conditions using schrodinger's equation to one dimensional square Well potential.

  4. Explain the Normal Zeeman effect using classical theory and also obtain an expression for zeeman shift.

    OR

    State and explain Larmor's Theorem. Then derive an expression for Larmor's frequency

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