# FYSA2032 Quantum Mechanics, part B (4 cr)

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## Description

Schrödinger equation and Hamiltonian in three dimensions.

Central-force problem, separation of the Schrödinger equation to the angular and radial parts.

Solution of the angular part of the central-force problem, spherical harmonics.

Solutions of the radial part for simple piecewise constant potentials (free particle, potential well, finite square well) and the hydrogen atom.

Eigenstates of angular momentum, orbital angular momentum and spin. General angular momentum and its matrix representation.

Coupling of angular momenta, singlet and triplet states, Clebsch-Gordan coefficients.

Two-particle systems, identical particles, bosons and fermions.

Time independent first- and second-order perturbation theory: The non-degenerate and degenerate cases.

## Learning outcomes

After completing the course the student

Is able to apply the tools of wave mechanics and formal quantum mechanics, acquired in part A, to the treatment of the three-dimensional Schrödinger equation.

Can explain the meaning of a central force field and its simple nature, and how it enables the separation of the radial and angular parts of the Schrödinger equation.

Is able to handle the spherical harmonics, the angular wave function of the central-force problem.

Can explain qualitatively the nature of the radial wave functions involved in central-force problems.

Can explain the properties of the eigenfunctions of angular momentum and can form the matrix representation of an angular-momentum operator.

Is able to couple two angular momenta (spin with spin, spin with orbital, two general angular momenta). Is able to relate the results of angular-momentum coupling to Clebsch-Gordan coefficients.

Knows how to treat two-particle systems and can construct the wave functions of two identical particles (bosons or fermions).

Is able to apply time-independent first- and second-order perturbation theory in the non-degenerate and degenerate cases.

## Description of prerequisites

- FYSA2031 Quantum mechanics, part A
- MATP211-213 Calculus 1-3 (in particular derivatives and integrals of basic functions, integration by parts, chain rule of derivatives)
- MATP121, MATA122 linear algebra 1 & 2 (in particular linear vector space, matrices and determinants, the eigenvalue problem and diagonalization)
- MATA114 differential equations (in particular separable and second-order linear differential equations with constant coefficients)
- MATA200 complex calculus (in particular complex conjugation, absolute value)

## Study materials

## Literature

- Griffiths: Introduction to Quantum Mechanics. 2nd Edition, Cambridge University Press, ISBN-10 1107179866, ISBN-13 9781107179868.
- Spiegel, Lipschutz, Liu: Mathematical Handbook of Formulas and Tables.

## Completion methods

### Method 1

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### Method 2

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### Teaching (4 cr)

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Exercises and exam.

#### Teaching

##### 10/25–12/17/2021 Lectures

##### 12/3–12/3/2021 Final exam

##### 1/14–1/14/2022 Final exam

### Independent study (4 cr)

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Self-study, exercises, exam.