# FYSA2031 Quantum Mechanics, part A (4–6 cr)

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## Description

Schrödinger equation, Hamiltonian, wave function, statistical interpretation.

Expectation value and its time evolution, variance.

Time-independent Schrödinger equation and stationary states.

One-dimensional potentials: Potential well, harmonic oscillator, free particle and the wave packet, the bound and scattering states of the delta potential and square well. Reflection and transmission coefficients. New mathematical methods: Fourier transform and the Dirac delta.

Hilbert space, the Dirac notation, inner product.

Observables and hermitian operators.

Generalized statistical interpretation

Generalized uncertainty principle and its energy-time version. Compatible observables. The time evolution of expectation values and its relation to conservation laws.

## Learning outcomes

After completing the course the student

Can explain what means the generalized statistical interpretation of the wave function and can extract from it probabilities of observables and compute expectation values and variances of observables. Can apply the properties of an orthonormal basis set in the above-mentioned contexts.

Can relate an observable and the corresponding hermitian operator to a property of a physical system.

Can tell what the compatibility of observables means, e.g., from the point of view of measurements.

Can relate the incompatibility of two operators to the corresponding uncertainty principle.

Can explain the main features of stationary states and apply them in the description of the time evolution of a wave function.

Is able to handle continuity conditions for bound and scattering states of one-dimensional potential problems. Can identify physical situations that enable quantum-mechanical reflection and transmission, and can handle the plane wave and the wave packet of a free particle.

Is able to handle ladder operators and apply them in the context of the harmonic oscillator.

Is able to handle the inner product and utilize it in the context of operator hermiticity and hermitian conjugation.

Can explain the meaning of the generalized uncertainty principle and its energy-time version. Can relate the time evolution of expectation values to conservation laws.

## Additional information

Nanotieteen koulutusohjelmassa opintojakso suoritetaan 4 op laajuisena ilman laboratoriotöitä, jos se on osa kandidaatintutkinnon pääaineopintoja.

## Description of prerequisites

- Basic physics courses (in particular the wave motion)
- FYSA2001-FYSA2002 Modern Physics, parts A&B
- 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)

## 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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### Method 3

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### Method 4

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**Parts of the completion methods**

### Participation in teaching (1.5 cr)

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##### 9/5–10/12/2022 Lectures

### Exam (2.5 cr)

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##### 10/14–10/14/2022 Final exam

##### 11/11–11/11/2022 Exam

##### 2/10–2/10/2023 Exam

##### 5/12–5/12/2023 Exam

### Participation in teaching (2 cr)

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##### 8/29–12/16/2022 Laboratory work

### Exam (4 cr)

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