## Spin Precession in a Magnetic Field, Schrödinger Picture, Heisenberg Picture, Particle in a Potential, Example: Simple Harmonic Oscillator

Last time, we began discussing the classic example of precession of a spin- $\frac{1}{2}$ particle in a magnetic field. The Hamiltonian of this system is
$$H=-\frac{g e}{2 m} \boldsymbol{S} \cdot \boldsymbol{B}$$

With $\boldsymbol{B}=B \hat{\boldsymbol{z}}$, this becomes

$$H=-\frac{g e}{2 m} S^{z} B$$
The energy levels are
$$E_{\pm}=\mp \frac{g e \hbar B}{4 m},$$
giving a level splitting of
$$\hbar \omega=\left|\frac{\operatorname{ge\hbar} B}{2 m}\right| .$$
We define
$$\omega:=\frac{g|e B|}{2 m},$$
so that the Hamiltonian can be written simply as
$$H=\omega S^{z}$$

## Schrodinger Picture

The time-evolution operator can be expressed in the energy eigenbasis (which coincides with the $S^{z}$ eigenbasis) as
$$U(t, 0)=e^{-i H t / \hbar}=e^{-i \omega S^{z} t / \hbar} .$$
Suppose that we have an arbitrary initial state
$$|\psi\rangle=c_{+}|+\rangle+c_{-}|-\rangle \text {. }$$
Using the time-evolution operator, we find that
\begin{aligned} |\psi(t)\rangle &=U(t, 0)|\psi\rangle \ &=e^{-i \omega S^{z} t / \hbar}|\psi\rangle \ &=e^{-i \omega t / 2} c_{+}|+\rangle+e^{i \omega t / 2} c_{-}|-\rangle . \end{aligned}
We now know the state of the system at all times. For example, if we initially have $|\psi\rangle=|+\rangle$, then
$$|\psi(t)\rangle=e^{-i \omega t / 2}|+\rangle,$$
which has
$$\operatorname{Prob}\left(S^{z}=\frac{\hbar}{2}\right)=1$$
for all times. This is why energy eigenstates are often called stationary states. Because time evolution is generated by the Hamiltonian, energy eigenstates do not change under time evolution (up to an unphysical overall phase).

Consider instead the case where the initial state is the spin-up eigenstate of $S^{x}$, i.e.,
$$|\psi\rangle=\frac{1}{\sqrt{2}}(|+\rangle+|-\rangle) .$$
Then we have
$$|\psi(t)\rangle=\frac{1}{\sqrt{2}}\left(e^{-i \omega t / 2}|+\rangle+e^{i \omega t / 2}|-\rangle\right) .$$
Thus, after a time $t$ we have
\begin{aligned} \operatorname{Prob}\left(S^{x}=\frac{\hbar}{2} \text { at time } t\right) &=\left|\left\langle S_{+}^{x} \mid \psi(t)\right\rangle\right|^{2} \ &=\frac{1}{2} \mid\left(\left.\langle+|+\langle-|) \mid \psi(t)\rangle\right|^{2}\right.\ &=\cos ^{2}\left(\frac{\omega t}{2}\right) \end{aligned}
Similarly,
$$\operatorname{Prob}\left(S^{x}=-\frac{\hbar}{2} \text { at time } t\right)=\sin ^{2}\left(\frac{\omega t}{2}\right) \text { . }$$
We can check that this is true, but we know it must be true by conservation of probability. We then have
$$\left\langle S^{x}\right\rangle=\frac{\hbar}{2}\left[\cos ^{2}\left(\frac{\omega t}{2}\right)-\sin ^{2}\left(\frac{\omega t}{2}\right)\right]=\frac{\hbar}{2} \cos (\omega t) .$$
We see that this oscillates as a function of time, with angular frequency $\omega$, as we expect from our classical intuition. What can we learn from this calculation? If we have a general initial state represented by $\hat{n}$ on the Bloch sphere, then $\hat{n}$ will precess around $\boldsymbol{B}$ with angular frequency $\omega$. You will show this on the homework.

## Heisenberg Picture

We can carry out this same calculation in the Heisenberg picture. In this picture, the states do not change as a function of time, but rather the operators do. The spin operator in the Heisenberg picture evolves according to the equation
$$\boldsymbol{S}(t)=e^{i \omega S^{z} t / \hbar} \boldsymbol{S} e^{-i \omega S^{z} t / \hbar} .$$
Taking the time derivative of this expression, we find
$$\frac{\mathrm{d} \boldsymbol{S}}{\mathrm{d} t}=\frac{i \omega}{\hbar} e^{i \omega S^{z} t / \hbar}\left[S^{z}, \boldsymbol{S}\right] e^{-i \omega S^{z} t / \hbar}$$
The $z$ -component of this equation of motion is simple, because $\left[S^{z}, S^{z}\right]=0$, so we have
$$\frac{\mathrm{d} S^{z}}{\mathrm{~d} t}=0 .$$
The $x$ -component is found using the fact that $\left[S^{z}, S^{x}\right]=i \hbar S^{y}$, which gives
$$\frac{\mathrm{d} S^{x}}{\mathrm{~d} t}=\frac{i \omega}{\hbar} e^{i \omega S^{z} t / \hbar} i \hbar S^{y} e^{-i \omega S^{z} t / \hbar}=-\omega S^{y}(t)$$
Similarly, using $\left[S^{z}, S^{y}\right]=-i \hbar S^{x}$, we find
$$\frac{\mathrm{d} S^{y}}{\mathrm{~d} t}=\omega S^{x}(t)$$

We can write these three expression compactly in vector notation as
$$\frac{\mathrm{d} \boldsymbol{S}}{\mathrm{d} t}=-\omega \boldsymbol{S} \times \hat{\boldsymbol{z}}$$
This is the same equation as for classical spins, but the interpretation is entirely different because $S$ is now an operator. If we take the expectation value of each side of this expression, we will get the same answers that we found in the Schrödinger picture.

## Particle in a Potential

We have said, on general grounds, that time evolution is given by a unitary operator. Furthermore, we have said that the infinitesimal form of this operator corresponds to a Hermitian operator, which we have called the Hamiltonian. However, we must ask: how do we specify a quantum mechanical system? There are two main ingredients we need: we have to first specify the Hilbert space that the states live in, and then we have to specify the Hamiltonian. Either of these is insufficient without the other.

For a particle in a potential, the Hilbert space is the space of square-integrable functions (modulo magnitude and phase). In order to specify the Hamiltonian, we will simply take the classical Hamiltonian and replace the variables $x$ and $p$ by the corresponding operators $x$ and $p .$ We must be careful, because although the regular variables $x$ and $p$ commute, the corresponding operators do not. Suppose we have the Hamiltonian
$$H=\frac{p^{2}}{2 m}+V(x) .$$
How do we proceed? In the Schrödinger picture, we first find the energy eigenkets $|j\rangle$ and eigenvalues $E_{j}$, which satisfy
$$H|j\rangle=E_{j}|j\rangle$$
Once we have found these eigenkets, we can expand an arbitrary state as
$$|\psi\rangle=\sum_{j} c_{j}|j\rangle$$
Time evolution is carried out as
$$|\psi(t)\rangle=\sum_{j} c_{j} e^{-i E_{j} t / \hbar}|j\rangle$$
In principle, this is a complete procedure to determine the dynamics of any quantum system. However, in practice this can often be very difficult. In the Heisenberg picture, we simply care about the operators,
$$\begin{array}{l} x_{\mathrm{H}}(t)=e^{i H t / \hbar} x e^{-i H t / \hbar}, \ p_{\mathrm{H}}(t)=e^{i H t / \hbar} p e^{-i H t / \hbar} . \end{array}$$
Using the Heisenberg picture equation of motion, we see that the operator $x_{H}$ evolves according to
$$\frac{\mathrm{d} x_{\mathrm{H}}}{\mathrm{d} t}=\frac{1}{i \hbar}\left[x_{\mathrm{H}}, H\right]=\frac{1}{i \hbar}\left[x_{\mathrm{H}}, \frac{p_{\mathrm{H}}^{2}}{2 m}+V\left(x_{\mathrm{H}}\right)\right]=\frac{1}{i \hbar}\left[x_{\mathrm{H}}, \frac{p_{\mathrm{H}}^{2}}{2 m}\right]$$

From this point forward, we will drop the subscript $H$ on the Heisenberg picture operators. Using a commutator identity, we then have
\begin{aligned} \frac{\mathrm{d} x}{\mathrm{~d} t} &=\frac{1}{i \hbar}\left[x, \frac{p^{2}}{2 m}\right] \ &=\frac{1}{i \hbar}\left(\left[x, \frac{p}{2 m}\right] p+p\left[x, \frac{p}{2 m}\right]\right) \ &=\frac{p}{m} \end{aligned}
This is the expected result, but it is now an operator equation. Similarly, we find
\begin{aligned} \frac{\mathrm{d} p}{\mathrm{~d} t} &=\frac{1}{i \hbar}[p, V(x)] \ &=\frac{1}{i \hbar}\left(-i \hbar \frac{\mathrm{d}}{\mathrm{d} x}(V(x) \cdot)+V(x) i \hbar \frac{\mathrm{d}}{\mathrm{d} x} \cdot\right) \ &=-\frac{\mathrm{d} V}{\mathrm{~d} x} . \end{aligned}
Here, we have made clear the action of the derivative operator by using $\cdot$ to denote an arbitrary position-space wavefunction on which these operators could act. Once again, this resembles the classical expression, but is now an operator equation. If we take the expectation values of Eqs. (7.29) and $(7.30)$, we find
$$\begin{array}{l} \frac{\mathrm{d}\langle x\rangle}{\mathrm{d} t}=\frac{\langle p\rangle}{m} \ \frac{\mathrm{d}\langle p\rangle}{\mathrm{d} t}=-\left\langle\frac{\mathrm{d} V}{\mathrm{~d} x}\right\rangle \end{array}$$
This result is called Ehrenfest’s theorem. It is important, when using the second equation, to remember to differentiate the potential first before taking the expectation value, because reversing the order of these operators will often change the result.

## Simple Harmonic Oscillator

Recall the simple harmonic oscillator Hamiltonian,
$$H=\frac{p^{2}}{2 m}+\frac{1}{2} m \omega^{2} x^{2}$$
We are likely all familiar with the approach in the Schrödinger picture. In the Heisenberg picture, we have
$$\frac{\mathrm{d} x}{\mathrm{~d} t}=\frac{p}{m}, \quad \frac{\mathrm{d} p}{\mathrm{~d} t}=-m \omega^{2} x$$
Solving these equations gives
$$\begin{array}{l} x(t)=x(0) \cos (\omega t)+\frac{p(0)}{m \omega} \sin (\omega t), \ p(t)=-m \omega x(0) \sin (\omega t)+p(0) \cos (\omega t) . \end{array}$$
Finding these equations in the Schrödinger picture is messy, though it can be done; in the Heisenberg picture, the result was immediate.

Keep in mind that these are operator equations. If, for example, we square the operators $x(t)$ or $p(t)$, we must be careful with the order of $x(0)$ and $p(0)$ in the cross terms, as these operators do not commute.

# Advanced Quantum Theory – Material for the year 2020-2021

## Primary tabs

We have updated our Undergraduate exams guidance in preparation for the Trinity Term examinations.

Please see our new webpages dedicated to TT exams.2020-2021Course URL: https://www-thphys.physics.ox.ac.uk/people/AndreiStarinets/C6-MASTER/C6-Homepage-2020-2021.htmlExternal Lecturer(s): Prof. Sid ParameswaranCourse Term: Michaelmas

### Assessment type:

• Invigilated written examination in TT

Course Overview:

20 lectures
Weight: 1.25 units
Areas: CMT, foundational course

Link to submit your homework (this is not a homework completion course but you still submit your work online)
Sirui Ning’s class: https://cloud.maths.ox.ac.uk/index.php/s/bYkYYjw3joiS7QS

Saraswat Bhattacharyya’s class: https://cloud.maths.ox.ac.uk/index.php/s/smDcxY382fCxCoE
Deadlines for Saraswat’s class: Monday 12 noon, in the week the class takes place (e.g the first deadline will be Monday 12 noon, 2nd Nov)

Michael Nee’s class: https://cloud.maths.ox.ac.uk/index.php/s/4fgt84cqWAMqXfj
Deadlines for Michael’s class: Monday 5pm, in the week of the tutorial.

Max McGinley’s class: https://cloud.maths.ox.ac.uk/index.php/s/zAwKRjWsc8JNdBg
Deadlines for Max’s class: 9 am on the Monday before each session.Course Synopsis:

Path integrals in Quantum Mechanics; the propagator.
Path Integrals in Quantum Statistical Mechanics; correlation functions; perturbation theory;
Feynman diagrams.
Path Integrals and Transfer Matrices.
Transfer matrix approach to the Ising Model.
Second quantisation. Ideal Fermi gas in second quantization.
Weakly interacting Bose gas: Bogoliubov theory; superfluidity.
Spinwaves in a ferromagnet.
Landau theory of phase transitions.

# 00691 – Quantum Mechanics

## Learning outcomes

At the end of the course, the student has the basic knowledge of the foundations, the theory and the main applications of quantum
mechanics. In particular he/she is able to solve problems through the Schroedinger equation and its resolution methods, knows the
algebraic formalism and its main applications, the theory and the
applications of angular momentum and spin, can discuss simple
problems of perturbation theory.

## Course contents

Module 1 Theory (Prof. Roberto Zucchini)

1) From classical physics to quantum physicsUndulatory theory of light, interference and diffraction
Photoelectric effect and Compton effect
Corpuscular theory of light
Material waves and de Broglie theory
Wave particle duality
Experience of Davisson and Germer
Atomic spectra
Experience of Franck and Hertz
Bohr-Sommerfeld atomic model
Correspondence principle
Experience of Stern and Gerlach
Angular momentum and spin in quantum physics
Spatial quantisation
2) The Schroedinger equationThe wave equation and geometric optics
Hamilton-Jacobi equation and its relation to geometric optics
Quasiclassical limit
Derivation of the Schroedinger equation
Wave function and its probabilistic interpretation
Energy eigenfunctions and levels
Time evolution of the wave function
Schroedinger equation for a particle with spin
3) Solution of the Schroedinger equation

Schroedinger equation in one dimension
Energy eigenfunctions and levels
Potential boxes and wells
The one-dimensional harmonic oscillator
Schroedinger equation in three dimensions
Schroedinger equation for a central potential
Orbital angular momentum, parity and spherical harmonics
Spherical sotential boxes and wells
The hydrogen atom
Other examples and applications
4) Collision theoryCollision in quantum physics
Scattering in one dimension
Reflection and transmission coefficients
Potential barriers
Scattering in three dimensions
Differential and total scattering cross section
Scattering in a central potential
Born approximation
Partial waves expansion
Coulomb scattering
Examples and applications
5) Foundations of quantum physicsBasic quantum experiences
States, observables and measurement
Definition and eigenstates
Measurement and state reduction
Probabilistic nature of quantum physics
Spectrum of an observable
Superposition and completeness
Expectation values and uncertainty of an observable
Compatible observables and simultaneous eigenstates
Indetermination principle
6) Formalism of quantum mechanicsBras, kets and orthonormal bases
States and kets
Schroedinger, momentum and Heisenberg representations
Quantisation and canonical commutation relations
Ehrenfest theorem and quasiclassical limit
7) Elementary applications

Equazione di Schroedinger for a particle in an electromagnetic field
Two-state systems
The harmonic oscillator in the operator formalism
Other examples and applications
8) Angular momentum theoryAngular momentum commutation relations
Angular momentum spectral theory
Sum of angular momenta and Clebsh-Gordan coefficients
Wigner-Eckart theory
The hydrogen atom in the operator formalism
Pauli Theory of the spinning electron
9) Identical particlesIdentity and quantum indistinguishability
Spin and statistics, bosons and fermions
Pauli exclusion principle
10) Time independent perturbation theory

Perturbations and lift of degeneracy
Non degenerate and degenerate perturbation theory
Perturbative expansion
Examples and applications
11) Time dependent perturbation theory

Schroedinger equation and evolution operator
Time dependent perturbations
Schroedinger, Heisenberg and Dirac representation
Pulse perturbations
Periodic perturbations
Fermi golden rule
Examples and applications

No supplementary contents are envisaged for non-attending students.

Module 2 Problem solving  (Dr. Davide Vodola)

Problem solving in the following topics of the course

One-dimensional potentials
Harmonic oscillator
Central potentials
Hydrogenlike atoms
Angular momentum and spin
Time independent perturbation theory
Time dependent perturbation theory

For the preparation of the exam, we recommend reading the course notes:

R. Zucchini
Quantum mechanics: Lecture Notes
Available on the Insegnamenti OnLine website

The following texts can be consulted for further information on the course contents.

P. A.M. Dirac
The Principles of Quantum Mechanics
Oxford University Press
ISBN-13: 978-0198520115
ISBN-10: 0198520115C. Cohen-Tannoudji, B. Diu & F. Laloe
Quantum Mechanics I & II
Wiley-Interscience
ISBN 10: 047116433X
ISBN 13: 9780471164333J. J. Sakurai & J. Napolitano
Modern Quantum Mechanics
ISBN-13: 978-0805382914
ISBN-10: 0805382917A. Galindo & P. Pascual
Quantum Mechanics I & II
Springer-Verlag
ISBN 978-3-642-83856-9
ISBN 978-3-642-84131-6L. D. Landau, E. M. Lifshitz
Quantum Mechanics: Non-Relativistic Theory
Elsevier
ISBN: 9780080503486
ISBN: 9780750635394

## Teaching tools

The following educational material is available on the Insegnamenti OnLine web site

1) Lectute notes in English

2) Texts of the problems proposed in the problem solving classes.

3) Texts of past written exams

## Office hours

See the website of Roberto Zucchini

See the website of Davide Vodola

# Quantum mechanics II

Linnaeus University

7.5 credits

The course includes:•Dirac’s formalism and the general formulation of quantum mechanics•Quantum dynamics, semi-classical approximations, propagators and Feynman’s path integral•General theory of angular momentum•Symmetry in quantum mechanics•Time dependent problems: perturbation theory, the sudden approximation, the adiabatic approximation and Berry’s phase•Many-particle systems, identical particles•Scattering theory