{
"cells": [
{
"cell_type": "markdown",
"id": "d473f2a8-ce90-43f0-bf35-3f001fdcb92c",
"metadata": {
"tags": []
},
"source": [
"# Week 1\n",
"\n",
"## Week 1, Monday\n",
"\n",
"### Brief description of the coming course\n",
"\n",
"Link to video: Introduction (22 minutes)\n",
"\n",
"\n",
"**Why is quantum field theory (QFT) necessary in condensed matter physics?** Any condensed matter system will consist of huge number of strongly correlated particles, making the precise quantum description impossible. Landau: do not try to describe all the particles, but *rather excitations (quasiparticles) on top of the ground state.* But number of excitations (quasiparticles) can change $\\rightarrow$ particle creation and annihilation processes $\\rightarrow$ need for a theory that can handle changing number of (quasi)particles $\\rightarrow$ second quantization (=QFT, more or less).\n",
"\n",
"In the QFT-coma(p) course, we will discuss various quantum field theoretical topics that are often encountered in high energy physics context. However, the goal of this course is to discuss the related, or analogous, phenomena in condensed matter context:\n",
"- particle-antiparticle creation/annihilation\n",
"- vacuum fluctuations\n",
"- long-range mediated interactions\n",
"- Feynman diagrams\n",
"- Dyson equation and self-energies\n",
"- renormalization\n",
"- spontaneous symmetry breaking, Goldstone modes and Anderson-Higgs mechanism. \n",
"\n",
"At the end of the course the students will have heard of these topics in high-energy context, know them in the condensed-matter context, and have an idea of how the microscopic theory of the condensed matter physics is underlying the more familiar mean-field theories from other courses. High-energy physics *is* an effective theory, with an unknown microscopic theory underlying it. In condensed matter physics, we know *both the effective theories and the underlying microscopic theories*. Students will also see examples of how condensed matter systems have been utilized in simulating hard-to-realize high-energy physics topics.\n",
"\n",
"The course will consist of three two-hour learning sessions per week for 12 weeks. The three sessions will consist of short lectures (~20 minutes), guided exercises, and various multiform teaching projects. All exercises and projects will be done in small groups. Solutions will not be collected nor graded, but course points are based on attendance (1 point per session, with total of 3x12 = 36 points). At the beginning of each week will be prelecture assigment, and at the end of the course will be a quantum field theory escape room that everyone is expected to pass in small groups. The course will be graded pass/fail and the passing grade is obtained by doing all the prelecture assigments, passing the escape room and by 70% attendance (25 attendance points).\n",
"\n",
"The weekly schedule is the following:\n",
"- prelecture assignment (deadline Monday 2 pm)\n",
"- Monday's learning session 2pm-4pm\n",
"- Wednesday's learning session 2pm-4pm\n",
"- Thursday's learning session 2pm-4pm"
]
},
{
"cell_type": "markdown",
"id": "5fcbda81-85e2-44d2-9e9f-f088335b41bf",
"metadata": {
"tags": []
},
"source": [
"### Course Topics\n",
"\n",
"**The preliminary schedule of the course is the following**\n",
"- Week 1: relativistic quantum (wave-)mechanics\n",
"- Week 2: relativistic hydrogen atom\n",
"- Week 3: linear response theory\n",
"- Week 4: basic scattering theory\n",
"- Week 5: Rayleigh and Thomson scattering\n",
"- Week 6: Green's functions and Feynman diagrams\n",
"- Week 7: Dyson equation and self-energies\n",
"- Week 8: Vacuum fluctuations\n",
"- Week 9: Collective modes\n",
"- Week 10: Gauge transformations\n",
"- Week 11: Spontaneous symmetry breaking and Higgs physics\n",
"- Week 12: TBD"
]
},
{
"cell_type": "markdown",
"id": "7573ff79-17b4-4e76-931b-633731a53c2f",
"metadata": {
"tags": []
},
"source": [
"### Radius of an electron?\n",
"\n",
"Link to video: Compton wavelength (6 minutes)\n",
"\n",
"**The electron is a point-like elementary particle with charge $-e$ and intrinsic spin angular momentum of $\\hbar/2$. Right?**\n",
"\n",
"Let’s imagine an experimental setup that would decide once and for all what is the radius of a single electron. By shining light, we try to take an image of the electron. The spatial resolution is determined by the wavelength, so we repeat the experiment with shorter and shorter wavelengths, halving the wavelength in each iteration $\\lambda$, $\\lambda/2$, $\\lambda/4$, .... Continuing this we eventually reach half of the Compton wavelength $\\lambda_\\mathrm{C} = h/(m_{\\rm e}c)$, where $m_{\\rm e}$ is the mass of the electron. What happens?\n",
"\n",
"The energy of the photon at half of the Compton wavelength is $E = \\frac{hc}{\\lambda} = \\frac{2hc m_{\\rm e} c}{h} = 2m_{\\rm e} c^2$, which is enough for spontaneous creation of an electron-positron pair. That is, the initial assumption of a single electron breaks down as another electron is created. \n",
"\n",
"Alternative idea from Bohr: enclose a single electron in a box. Shrink the size of the box. What happens?\n",
"\n",
"As the box shrinks, the position of the electron becomes better defined, leading to increase in the uncertainty of the momentum $p > h/L$, where $L$ is the size of the box.\n",
"When the size of the box reaches half of the Compton wavelength, the momentum uncertainty becomes $p > 2m_{\\rm e}c$. The energy of the electron becomes\n",
"$$\n",
"E = \\frac{p^2}{2m_{\\rm e}} = 2m_{\\rm e} c^2,\n",
"$$\n",
"which is again twice the electron rest mass energy and enough for creation of more particles in the box. Of course, the speed of the electron at that point is already relativistic, so one might want to use the relativistic energy-momentum relation\n",
"$$\n",
"E = \\sqrt{c^2 p^2 + m_{\\rm e}^2 c^4} = \\sqrt{4m_{\\rm e}^2 c^4 + m_{\\rm e}^2 c^4} = \\sqrt{5} m_{\\rm e}c^2,\n",
"$$\n",
"which has slightly different prefactor but the main picture remains unchanged.\n",
"\n",
"To quote Sidney Coleman:\n",
"*So we can localize something, but what we’re localizing is not a single particle. Because of\n",
"the phenomena of pair production, not only is momentum complementary to position, but\n",
"particle number is complementary to position. If we make a very precise measurement of\n",
"position, we’ll have a very big spread in momentum and therefore, because pair production takes\n",
"place, we do not know how many particles we have.*\n",
"\n",
"We have now found the first important length scale associated with the electron and also other elementary particles. The Compton wavelength $\\lambda_\\mathrm{C} = h/(m_{\\rm e} c)$ is the wavelength of a photon with the same energy as the electron rest energy (mass) $m_{\\rm e}c^2$. "
]
},
{
"cell_type": "markdown",
"id": "b0fafee3-b395-4926-92ce-1932a5a8e657",
"metadata": {
"tags": []
},
"source": [
"### Relativistic quantum mechanics\n",
"\n",
"Link to video: Relativistic quantum mechanics (9 minutes)\n",
"\n",
"**What is, and why do we require, Lorentz invariance from our theories?** (Although in condensed matter physics we mostly don't care...)\n",
"\n",
"The choice of the coordinate system is an important and often neglected step in description of a physical system. \n",
"In condensed matter physics, we usually choose implicitly the center-of-mass, or laboratory, coordinates, but in principle there is no reason why we couldn't use\n",
"some other choice of coordinates. This is the Galileo's principle of relativity and at low speeds it implies invariance to Galilei transformations (which in turn gives us various conservation laws by Noether's theorem). At speeds approaching the speed of light, the Galilei transformation needs to be replaced by the more general Lorentz transformation.\n",
"The key of the Lorentz transformation is that translations in space and time (i.e. 'movement') are connected so that the *spacetime distance*\n",
"$$\n",
"\\delta s = \\sqrt{(c\\delta t)^2 - (\\delta x)^2 - (\\delta y)^2 - (\\delta z)^2}\n",
"$$\n",
"is constant, i.e. does not depend on the choice of the inertial coordinate system. A Lorentz invariant theory has to satisfy this Lorentz invariance\n",
"and thus it has to have certain symmetry regarding temporal and spatial translations.\n",
"\n",
"The usual Schrödinger equation for a free particle\n",
"$$\n",
"i\\hbar \\frac{\\partial}{\\partial t}\\psi(r,t) = \\frac{p^2}{2m} \\psi(r,t)\n",
"$$\n",
"is not Lorentz invariant, since the time derivative is of the first order while the spatial derivative is of the second order.\n",
"Schrödinger actually started with a Lorentz invariant wave equation for his quantum theory\n",
"$$\n",
"-\\hbar^2 \\frac{\\partial^2}{\\partial t^2} \\psi = E^2 \\psi = \\left(c^2 p^2 + m^2 c^4\\right) \\psi,\n",
"$$\n",
"where the right hand side operator is the square of the relativistic energy-momentum-dispersion relation for free particle\n",
"$$\n",
"E_p = \\sqrt{c^2 p^2 + m^2 c^4}.\n",
"$$\n",
"However, the relativistic wave equation turned out to be problematic, with strange negative energy eigenstates and especially it failed to explain observed hydrogen spectrum's fine structure. Therefore Schrödinger opted for the non-Lorentz-invariant nonrelativistic limit of the equation. That is, the usual Schrödinger equation."
]
},
{
"cell_type": "markdown",
"id": "50d434f9-ddea-4d6e-bcd5-fa0217011739",
"metadata": {
"tags": []
},
"source": [
"### Dirac equation\n",
"\n",
"Link to video: Dirac equation (8 minutes)\n",
"\n",
"**Dirac decided to try an ansatz which was of the first order in time and space derivatives**:\n",
"$$\n",
"i\\hbar \\frac{\\partial}{\\partial t} \\psi = H\\psi,\n",
"$$\n",
"where $H=c{\\bf \\alpha} \\cdot {\\bf p} + \\beta mc^2$.\n",
"\n",
"What kind of properties Dirac equation should have?\n",
"- Hamiltonian $H$ should be hermitian\n",
"- the prefactors $\\alpha$ and $\\beta$ should not depend on time or position (to preserve necessary symmetries), nor on momentum or energy. That is, they should be constants.\n",
"- in free space it should produce the energy-momentum dispersion relation $E_p = \\sqrt{c^2 p^2 + m^2 c^4}$\n",
"- in the nonrelativistic limit it should produce the Schrödinger equation\n",
"\n",
"and hopefully it will then yield correct hydrogen atom spectrum (more on this in week 2).\n",
"It turns out that these requirements can be satisfied if $\\alpha$ and $\\beta$ are 4x4-matrices (or larger) and the correct result can be already seen by demanding that the dispersion relation is linear in momentum as in:\n",
"$$\n",
"E_p = \\sqrt{c^2 p^2 + m^2 c^4} = c{\\bf \\alpha} \\cdot {\\bf p} + \\beta mc^2.\n",
"$$\n",
"Show that this relation is satisfied if $\\alpha$ and $\\beta$ are\n",
"- matrices\n",
"- traceless\n",
"- have eigenvalues $\\pm 1$\n",
"- even dimensional, for example 4x4 matrices\n",
"\n",
"We can use for example the Pauli-Dirac representation:\n",
"$$\n",
"{\\bf \\alpha} = \\left(\\begin{matrix} \n",
"0 & {\\bf \\sigma} \\\\\n",
"{\\bf \\sigma} & 0 \\\\\n",
"\\end{matrix}\\right),\n",
"$$\n",
"where the components of $\\sigma = (\\sigma_x,\\sigma_y,\\sigma_z)$ are the Pauli spin-matrices, and\n",
"$$\n",
"{\\beta} = \\left(\\begin{matrix} \n",
"\\mathcal{1}_2 & 0 \\\\\n",
"0 & -\\mathcal{1}_2 \\\\\n",
"\\end{matrix}\\right),\n",
"$$\n",
"and $\\mathcal{1}_2$ is the 2x2-identity matrix.\n",
"\n",
"However, if $\\alpha$ and $\\beta$ are matrices, then wavefunctions $\\psi({\\bf r})$ must be 4-dimensional vectors (in addition to what ever\n",
"description in the position space) $\\rightarrow$ wavefunctions are 4-dimensional spinors:\n",
"$$\n",
" \\psi({\\bf r},t) = \\left(\\begin{matrix}\n",
"\\phi_1({\\bf r},t)\\\\\n",
"\\phi_2({\\bf r},t)\\\\\n",
"\\phi_3({\\bf r},t)\\\\\n",
"\\phi_4({\\bf r},t)\n",
"\\end{matrix}\\right).\n",
"$$\n",
"What do all these different components of the spinor describe?"
]
},
{
"cell_type": "markdown",
"id": "bfd2905c-1004-4796-aecd-ad503706aee4",
"metadata": {
"jp-MarkdownHeadingCollapsed": true,
"tags": []
},
"source": [
"### Condensed-matter connection: Graphene\n",
"\n",
"Link to video: Graphene and two-dimensional Dirac equation (11 minutes)"
]
},
{
"cell_type": "markdown",
"id": "0895cca1-5beb-410e-8453-cf29134527e5",
"metadata": {
"tags": []
},
"source": [
"## Week 1, Wednesday\n",
"\n",
"### Free-particle solution of the Dirac equation\n",
"\n",
"Link to video: Free-particle solutions of the Dirac equation (17 minutes)\n",
"\n",
"The free particle solutions (no external potential) are of the plane-wave type as usual\n",
"$$\n",
"\\psi({\\bf r},t) = \\psi({\\bf r},t) = e^{i{\\bf p \\cdot r}/\\hbar - i\\omega_p t} u_{\\bf p}.\n",
"$$ \n",
"Using this as an ansatz and inserting in the Dirac equation yields the vectors $u_{\\bf p}$ which actually\n",
"gives four different solutions. First two solutions are\n",
"$$\n",
"u_{\\uparrow,{\\bf p}} = \\left(\\begin{matrix}\n",
"1\\\\\n",
"0\\\\\n",
"cp_z/(E_p + mc^2)\\\\\n",
"c(p_x+ip_y)/(E_p+mc^2)\n",
"\\end{matrix}\\right)\n",
"$$\n",
"$$\n",
"u_{\\downarrow,{\\bf p}} = \\left(\\begin{matrix}\n",
"0\\\\\n",
"1\\\\\n",
"c(p_x-ip_y)/(E_p+mc^2)\\\\\n",
"-cp_z/(E_p + mc^2)\\\\\n",
"\\end{matrix}\\right),\n",
"$$\n",
"and these correspond to positive energy solutions $E_p^+ = \\hbar \\omega_p^+ = E_p = \\sqrt{c^2 p^2 + m^2 c^4}$.\n",
"The other two solutions are\n",
"$$\n",
"v_{\\uparrow,{\\bf p}} = \\left(\\begin{matrix}\n",
"-cp_z/(E_p + mc^2)\\\\\n",
"-c(p_x+ip_y)/(E_p+mc^2)\\\\\n",
"1\\\\\n",
"0\n",
"\\end{matrix}\\right)\n",
"$$\n",
"$$\n",
"v_{\\downarrow,{\\bf p}} = \\left(\\begin{matrix}\n",
"-c(p_x-ip_y)/(E_p+mc^2)\\\\\n",
"cp_z/(E_p + mc^2)\\\\\n",
"0\\\\\n",
"1\n",
"\\end{matrix}\\right),\n",
"$$\n",
"and these correspond to negative energy solutions $E_p^- = \\hbar \\omega_p^- = -E_p = -\\sqrt{c^2 p^2 + m^2 c^4}$."
]
},
{
"cell_type": "markdown",
"id": "4c52c21f-fbad-4570-bcb9-d36adf24e0ad",
"metadata": {
"tags": []
},
"source": [
"### Discussion\n",
"\n",
"Link to video: Dirac sea/Fermi sea (15 minutes)\n",
"\n",
"\n",
"How to interpret negative energy solutions? \n",
"Dirac: negative energy states are occupied. Creating a hole in this Dirac sea corresponds to an antiparticle: a positron.\n",
"- Positron properties: positive energy, electron mass, opposite charge\n",
"- Energy density"
]
},
{
"cell_type": "markdown",
"id": "0f972344-800e-4ef7-a0ef-e41eaabe7eec",
"metadata": {
"tags": []
},
"source": [
"### Nonrelativistic limit of the Dirac equation\n",
"\n",
"Link to video: Nonrelativistic limit of the Dirac equation (27 minutes)\n",
"\n",
"\n",
"**How to interpret the four different spinor solutions of the Dirac equation?**\n",
"For this, it is instructive to have a look at the nonrelativistic limit of the Dirac equation and what kind of \n",
"solutions these correspond to in that limit.\n",
"\n",
"To get a bit better handle on what kind of solutions these are, we can slightly generalize our Dirac equation by adding \n",
"coupling with the electromagnetic field by the minimal coupling. That is, replace the momentum operator $p$ by $p-\\frac{e}{c}{\\bf A}$, where $A$ is the vector potential and add scalar potential $eA_0$ as a simple energy shift. \n",
"The Dirac equation with the electromagnetic field is now\n",
"$$\n",
"i\\hbar \\frac{\\partial}{\\partial t} \\psi({\\bf r},t) = \\left[ c \\boldsymbol{\\alpha} \\cdot \\left( {\\bf p} - \\frac{e}{c} {\\bf A}\\right) + eA_0 + \\beta mc^2\\right] \\psi({\\bf r},t),\n",
"$$\n",
"Recall that the matrices $\\boldsymbol{\\alpha} = (\\alpha_x, \\alpha_y, \\alpha_z)$ and $\\beta$ are given by\n",
"$$ \n",
"\\alpha_k = \\begin{pmatrix} 0 & \\sigma_k \\\\ \\sigma_k & 0 \\end{pmatrix}\\,\\qquad \\beta = \\begin{pmatrix} \\sigma_0 & 0 \\\\ 0 & -\\sigma_0 \\end{pmatrix} \n",
"$$\n",
"where $\\sigma_{k = x,y,z}$ are the standard Pauli matrices and $\\sigma_0$ is the $2\\times 2$ identity matrix.\n",
"\n",
"The stationary solutions of the Dirac equation are four-component spinors, which can be expressed in two-component representation as\n",
"$$\n",
"\\psi({\\bf r},t) = \\begin{pmatrix} \\phi ({\\bf r},t) \\\\ \\xi({\\bf r},t) \\end{pmatrix},\n",
"$$\n",
"where $\\phi({\\bf r},t)$ and $\\xi({\\bf r},t)$ are both two-component spinors themselves.\n",
"\n",
"In the nonrelativistic limit, the rest energy of the particle $mc^2$ is much larger that all other energy scales. This implies that the time-evolution of the states involves rapidly oscillating parts (due to rest energy) and slowly rotating parts (due to everything else). For positive energy solutions we can separate the rapid and slow parts with an ansatz\n",
"$$\n",
"\\begin{pmatrix} \\phi ({\\bf r},t) \\\\ \\xi({\\bf r},t) \\end{pmatrix} = \\begin{pmatrix} \\phi_0 ({\\bf r},t) \\\\ \\xi_0({\\bf r},t) \\end{pmatrix} e^{-imc^2t/\\hbar},\n",
"$$\n",
"where the fields $\\phi_0({\\bf r},t)$ and $\\xi_0({\\bf r},t)$ are only slowly varying in time.\n",
"(For negative energy solutions use factor $e^{imc^2t/\\hbar}$.)\n",
"\n",
"Substitute these into the Dirac equation and take advantage of the slow variation of the fields\n",
"$$\n",
"|i\\hbar \\frac{\\partial}{\\partial t} \\xi_0| \\ll |mc^2 \\xi_0|\n",
"$$\n",
"and assume a weak electrostatic potential\n",
"$$\n",
" |eA_0 \\xi_0| \\ll |mc^2 \\xi_0|.\n",
"$$\n",
"What you should get in the end is for the lower component of the Dirac equation:\n",
"$$\n",
" \\xi_0 = \\frac{\\sigma \\cdot \\left( {\\bf p} - \\frac{e}{c} {\\bf A}\\right)}{2mc} \\phi_0,\n",
"$$ \n",
"which in turn gives for the upper component \n",
"$$\n",
"i\\hbar \\frac{\\partial}{\\partial t} \\phi_0 = \\frac{{\\bf \\sigma}\\cdot \\left( {\\bf p} - \\frac{e}{c} {\\bf A}\\right) {\\bf \\sigma} \\cdot \\left( {\\bf p} - \\frac{e}{c} {\\bf A}\\right)}{2m} \\phi_0 + eA_0 \\phi_0.\n",
"$$\n",
"This is almost like a nonrelativistic Schrödinger equation for an electron in an electromagnetic field. With some $\\nabla$-algebra you can write it in the final form\n",
"$$\n",
"i\\hbar \\frac{\\partial}{\\partial t} \\phi_0 = \\left[ \\frac{\\left( {\\bf p} - \\frac{e}{c} {\\bf A}\\right)^2}{2m} - \\frac{e\\hbar}{2mc} {\\bf \\sigma} \\cdot {\\bf B} + eA_0 \\right]\\phi_0.\n",
"$$\n",
"Here's the solution:\n",
"https://sites.ualberta.ca/~gingrich/courses/phys512/node46.html\n",
"\n",
"\n",
"An alternative approach to the $\\nabla$-algebra is a brute force approach here:\n",
"$$\n",
"\\begin{split}\n",
"\\left[{\\bf \\sigma}\\cdot \\left( {\\bf p} - \\frac{e}{c} {\\bf A}\\right)\\right]^2 &=\n",
"\\sum_{kl} \\sigma_k \\left(p_k - \\frac{e}{c} A_k\\right) \\sigma_l \\left( p_l - \\frac{e}{c} A_l\\right) \\\\\n",
"&= \\sum_{k} \\sigma_k^2 \\left(p_k - \\frac{e}{c} A_k\\right) \\left( p_k - \\frac{e}{c} A_k\\right) + \\sum_{k\\neq l} \\sigma_k \\sigma_l \\left(p_k - \\frac{e}{c} A_k\\right) \\left( p_l - \\frac{e}{c} A_l\\right).\n",
"\\end{split}\n",
"$$\n",
"Noticing that $\\sigma_k^2 = 1$ and $\\sigma_k \\sigma_l = \\sum_j i \\epsilon_{klj} \\sigma_j$ (this relation can be easily derived by combining the commutation- and anticommutation relations of the Pauli spin matrices) we get\n",
"$$\n",
"\\left({\\bf p} - \\frac{e}{c} {\\bf A}\\right)^2 + \\sum_{k\\neq l} i \\sigma_j \\epsilon_{klj} \\left(p_k - \\frac{e}{c} A_k\\right) \\left( p_l - \\frac{e}{c} A_l\\right).\n",
"$$\n",
"On the other hand, the cross-product is \n",
"$$\n",
"A \\times B = \\sum_{ijk} \\epsilon_{ijk} A_i B_j \\hat e_k,\n",
"$$\n",
"where $\\hat e_k$ is the unit vector. We can identify this structure in the above equation and write\n",
"$$\n",
"\\left({\\bf p} - \\frac{e}{c} {\\bf A}\\right)^2 + i {\\bf \\sigma} \\cdot \\left({\\bf p} - \\frac{e}{c} {\\bf A}\\right) \\times \\left( {\\bf p} - \\frac{e}{c} {\\bf A}\\right),\n",
"$$\n",
"which can be simplified\n",
"$$\n",
"\\left({\\bf p} - \\frac{e}{c} {\\bf A}\\right)^2 - i {\\bf \\sigma} \\cdot \\left({\\bf p} \\times \\frac{e}{c} {\\bf A} + \\frac{e}{c} {\\bf A} \\times {\\bf p} \\right)\n",
"$$\n",
"Notice that the operator $p = -i\\hbar \\nabla$ may operate both on the vector potential ${\\bf A}$ (if it is on the right hand side of $p$) and the state, which is not shown here. \n",
"We get\n",
"$$\n",
"\\left({\\bf p} - \\frac{e}{c} {\\bf A}\\right)^2 - \\frac{e\\hbar}{c} {\\bf \\sigma} \\cdot \\left(\\nabla \\times {\\bf A}\\right) = \\left({\\bf p} - \\frac{e}{c} {\\bf A}\\right)^2 - \\frac{e\\hbar}{c} {\\bf \\sigma} \\cdot {\\bf B}\n",
"$$\n",
"\n",
"\n",
"And thats it! As you can see, the nonrelativistic limit of Dirac equation is the Schrödinger equation for a particle of mass $m$, charge $-e$ and intrinsic magnetic moment of $e\\hbar/(2mc)$, which is the Bohr magneton.\n",
"Also, it is in particular the upper two components of the four-component spinor that correspond to this particle state, and the two solutions have opposite signs of the magnetic moment.\n",
"The negative energy solutions have same properties except for the strange fact that their energies are negative.\n",
"\n",
"The nonrelativistic limit calculated above is the lowest order result from the Dirac equation. The first order corrections are also interesting and we will use them later when studying relativistic hydrogen atom spectrum. The first order correction provides the following perturbation to the energy\n",
"$$\n",
" \\hat V \\phi_0 = \\left[-\\frac{{\\bf p}^4}{8m^3c^2} + \\frac{q\\hbar}{4m^2 c^2}\\frac{1}{r} \\frac{\\partial A_0}{\\partial r} \\sigma \\cdot L + \\frac{q\\hbar^2}{8m^2 c^2} \\nabla^2 A_0\\right] \\phi_0.\n",
"$$\n",
"The various terms in this perturbation operator can be interpreted as: the relativistic correction to the kinetic energy, the spin-orbit coupling and so-called Darwin term. We will analyze these further next week."
]
},
{
"cell_type": "markdown",
"id": "b8b75cce-abd7-4a71-bab3-5134997fbc9f",
"metadata": {
"tags": []
},
"source": [
"### Condensed matter connection\n",
"\n",
"Link to video: Solar panels and photosynthesis (8 minutes)\n",
"\n",
"\n",
"Band filling diagram of materials. Nanite, Creative Commons.\n",
" \n",
"\n",
"Band structures of silicon and copper. Ville Havu.\n",
"\n",
"- discuss particle-hole excitations in semiconductors, band structure\n",
"- demonstrate solar cells, LEDs\n",
"- discuss photosynthesis"
]
},
{
"cell_type": "markdown",
"id": "2823f294-aaa9-481e-b09f-7fb99e7c764d",
"metadata": {
"tags": []
},
"source": [
"## Week 1, Thursday\n",
"\n",
"### One-photon box\n",
"\n",
"Link to video: One-photon box (12 minutes)\n",
"\n",
"Begin the session by watching together this PBS Space Time video. Until 3:33.\n",
"\n",
"In the video, Matt says that a weightless box containing a single photon of energy $E=hf$ has inertial mass equal to $m=E/c^2$. Derive that.\n",
"\n",
"Optional hints:\n",
"Consider a box of dimensions $(L_x, L_y, L_z)$ with perfectly reflecting walls. For the sake of\n",
"simplicity, lets assume that the photon is moving in the x-direction.\n",
"1. Assume initially that the box is stationary. What is the pressure exerted on the left and right\n",
"walls by the single photon?\n",
"2. Lets push the box so that it's acceleration is ${\\bf a}$ to the right (along the x-axis). After the photon gets reflected from the left wall, and before it traverses through the box to the right wall, the\n",
"box and the right wall have gained more speed. The right wall thus feels the photon as red-\n",
"shifted. Similarly, the left wall sees the photon reflected from the right wall as blue-shifted.\n",
"3. An observer moving with velocity ${\\bf v}$ away from a light source emitting photon with\n",
"frequency $f_S$ sees the photon frequency shifted according to the relativistic longitudinal\n",
"Doppler shift, i.e. $f_O = \\sqrt{(1-\\beta)/(1+\\beta)} f_S$, where $\\beta = v/c$.\n",
"4. Now determine the pressure on the left wall and on the right wall, assuming that $v \\ll c$\n",
"(linear term in the relativistic Doppler shift is enough).\n",
"5. Finally, determine the force ${\\bf F}$ required to have the box accelerating with magnitude $a$ and\n",
"show that the inertial mass of the box is $m=E/c^2$, where $E$ is the momentary energy of the\n",
"photon. Does the acceleration affect the energy and the mass of the box?\n",
"\n",
"\n",
"For a stationary box and a photon with wavelength $\\lambda$, the pressure at each wall is due to the radiation pressure of the photon. At each reflection, the impulse from the photon reflection is $I = 2p = 2\\hbar k = 2h f/c$, where $f$ is the frequency and $c$ the speed of light.\n",
"\n",
"If the box is moving, the photon frequency is shifted by the Doppler effect. Therefore the frequency of the photon in the box is constantly changing: frequency decreases when it is reflected from the right wall (that is 'running away', or red-shifted) and increases when it is reflected from the left wall ('running towards', or blue-shifted). For small enough velocities of the box, we linearize the relativistic Doppler shift (here $\\beta = v/c$, which is a small number)\n",
"$$\n",
"\\begin{split}\n",
" f_O &= \\sqrt{(1-\\beta)/(1+\\beta)} f_S \\\\\n",
" &\\approx (1-\\beta) f_S.\n",
"\\end{split} \n",
"$$\n",
"So, if for the back-and-forth trip from the left wall to right wall and back the frequency of the photon is initially $f_L$, the impulse upon reflection from the left wall is\n",
"$$\n",
" -2\\frac{hf_L}{c},\n",
"$$\n",
"where the minus sign means impulse to the left.\n",
"The Doppler shifted frequency upon reflection from the right wall will be $\\left(1-\\frac{at}{c}\\right)f_L$, where $a$ is the acceleration of the box and $t$ is the time it took to travel across the box, $t = L/c$. The impulse to the right is thus\n",
"$$\n",
" 2\\frac{hf_L}{c} \\left(1-\\frac{at}{c}\\right).\n",
"$$\n",
"The force required to accelerate the box at acceleration $a$ is equal to the total impulse divided by unit time. With unit time equal to the back-and-forth travel time, $2t$, the force is\n",
"$$\n",
"\\begin{split}\n",
" F &= \\frac{I}{2t} = \\frac{1}{2t} \\left[2\\frac{hf_L}{c} \\left(1-\\frac{at}{c}\\right)-2\\frac{hf_L}{c}\\right]\\\\\n",
" &= -\\frac{hf_L}{c} \\frac{a}{c}\\\\\n",
" &= -\\frac{E}{c^2}a \\\\\n",
" &= -ma,\n",
" \\end{split}\n",
"$$\n",
"where the mass can be identified as the energy of the photon divided by $c^2$: $m = E/c^2$.\n",
"Thus the photon in the massless box appears as if it has inertial mass given by the relativistic mass-energy relation.\n",
""
]
},
{
"cell_type": "markdown",
"id": "181e0221-d9dd-4ea8-9d4e-fbfdc2c6b8b1",
"metadata": {},
"source": [
"### Energy of an electron\n",
" \n",
"Link to video: Classical electron radius (8 minutes)\n",
"\n",
"Using classical electromagnetism, determine the total energy of the static electric field produced by a single electron. For the purpose of the calculation, assume that the electron is a small metallic sphere of radius $r_{\\rm e}$. Compute first the electric field inside and outside the sphere (you should know this from the electromagnetism course) and then the total energy using that the energy density of the electric field is $u_E = \\frac{1}{2}\\varepsilon_0 \\mathbf{E}^2$ where $\\mathbf{E}$ is the electric field and $\\varepsilon_0$ the vacuum permittivity. What is the total energy in the limit $r_{\\rm e} \\to 0$?\n",
"\n",
" \n",
"Lets assume that electron is a spherical shell with uniform charge density. The electric field 'inside' the electron is then zero and outside the electron it's magnitude is \n",
"$$\n",
"E(r) = \\frac{1}{4\\pi \\epsilon_0} \\frac{e}{r^2}.\n",
"$$\n",
"The energy density is $u_\\mathrm{E}(r) =\\frac{1}{2}\\epsilon_0 |E(r)|^2$. The total energy of this field is thus\n",
"$$\n",
"\\begin{split}\n",
" E &= \\int d{\\bf r} u_\\mathrm{E}(r) \\\\\n",
" &= 4\\pi \\int_{r_\\mathrm{e}}^\\infty dr\\, \\frac{1}{2} \\epsilon_0 |E(r)|^2 \\\\\n",
" &= 4\\pi \\int_{r_\\mathrm{e}}^\\infty dr\\, r^2 \\frac{1}{2} \\epsilon_0 \\frac{1}{16\\pi^2 \\epsilon_0^2} \\frac{e^2}{r^4} \\\\\n",
" &= \\frac{e^2}{8\\pi \\epsilon_0} \\int_{r_\\mathrm{e}}^\\infty dr\\, \\frac{1}{r^2}\\\\\n",
" &= \\frac{e^2}{8\\pi \\epsilon_0} \\frac{1}{r_\\mathrm{e}}\n",
"\\end{split}.\n",
"$$\n",
"In the limit $r_\\mathrm{e} \\rightarrow 0$, the energy goes to infinity. \n",
"\n",
"\n",
"The *classical electron radius* is the value of $r_{\\rm e}$ obtained by postulating that all of the electron mass originates from the energy associated to the electric field (remember that mass-energy equivalence principle of relativity: $E = mc^2$ where $c$ is the speed of light). Compute the classical electron radius. \n",
"\n",
"\n",
"So, we got the condition\n",
"$$\n",
" mc^2 = \\frac{e^2}{8\\pi \\epsilon_0} \\frac{1}{r_\\mathrm{e}}\n",
"$$\n",
"or\n",
"$$ \n",
"r_\\mathrm{e} = \\frac{1}{8\\pi \\epsilon_0} \\frac{e^2}{mc^2}.\n",
"$$\n",
"For a charge distribution that is not a hollow shell, the prefactor is different, but this was easiest to calculate.\n",
"\n",
"\n",
"You should obtain a result that differs from the conventional definition of the classical electron radius by a dimensionless prefactor that depends on what it is assumed for the charge distribution inside the electron. The conventional value of the classical electron radius is\n",
"$$\n",
" r_\\mathrm{e} = \\frac{1}{4\\pi \\varepsilon_0} \\frac{e^2}{m_\\mathrm{e} c^2}.\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "14254d66-aef1-4185-989e-9655b5a53b78",
"metadata": {},
"source": [
"### Fine structure constant\n",
"\n",
"Link to video: Bohr radius and fine structure constant (7 minutes)\n",
"\n",
"We have now encountered two important length scales associated to the electron and other elementary particles. The first was the Compton wavelength $\\lambda_\\mathrm{C} = h/(m_{\\rm e} c)$, which is the wavelength of a photon with the same energy as the electron rest energy (mass) $m_{\\rm e}c^2$. By replacing the Planck constant $h$ with the reduced Planck constant $\\hbar = h/(2\\pi)$, one obtains the reduced Compton wavelength $\\bar{\\lambda}_{\\rm C} = \\hbar/(m_{\\rm e} c)$.\n",
"\n",
"The second length scale is the classical electron radius, whose expression you found out in the previous exercise.\n",
"\n",
"The third and final length scale is the Bohr radius \n",
"$$ \n",
"a_0 = \\frac{4\\pi \\varepsilon_0\\hbar^2}{e^2 m_{\\rm e}} \n",
"$$\n",
"which sets the size of the electronic orbitals of the hydrogen atom. Up to prefactors, the Bohr radius can be obtained by using the Heisenberg uncertainty principle and equating the potential and kinetic energy of the electron in the hydrogen atom. Complete this simple argument for obtaining the Bohr radius.\n",
"\n",
" \n",
"This is a slightly different reasoning: in a circular orbit, the centripetal force must be $m\\frac{v^2}{r}$, and this is provided by the Coulomb force\n",
"$$\n",
" m\\frac{v^2}{r} = \\frac{1}{4\\pi \\varepsilon_0} \\frac{e^2}{r^2}.\n",
"$$\n",
"On the other hand, the circumference of the orbit is a multiple of the de Broglie wavelength $\\lambda = h/p = h/(mv)$. In particular, for the ground state the circumference is equal to the deBroglie wavelength, $2\\pi r = \\lambda$. This gives $v = h/(m\\lambda) = \\hbar/(mr)$ and the equation of motion becomes\n",
"$$\n",
" m\\frac{\\hbar^2}{m^2r^3} = \\frac{1}{4\\pi \\varepsilon_0} \\frac{e^2}{r^2}.\n",
"$$\n",
"Solving for $r$ which we identify as the Bohr radius gives\n",
"$$\n",
"r = a_0 = \\frac{4\\pi \\varepsilon_0\\hbar^2}{e^2 m}.\n",
"$$\n",
"\n",
"\n",
"Check that the ratio between the reduced Compton wave length and the Bohr radius $\\bar{\\lambda}_{\\rm C} / a_0$ is equal to the ratio between the classical electron radius and the reduced Compton wave length $r_{\\rm e}/\\bar{\\lambda}_{\\rm C}$ and find the expression for this ratio in terms of fundamental constants. \n",
"\n",
" \n",
"Indeed\n",
"$$\n",
" \\frac{\\bar{\\lambda}_{\\rm C}}{a_0} = \\frac{\\hbar}{m_{\\rm e} c} \\frac{e^2 m}{4\\pi \\varepsilon_0\\hbar^2}= \\frac{e^2}{4\\pi \\varepsilon_0 c\\hbar }\n",
"$$\n",
"and\n",
"$$\n",
" r_{\\rm e}/\\bar{\\lambda}_{\\rm C} = \\frac{1}{4\\pi \\varepsilon_0} \\frac{e^2}{m_\\mathrm{e} c^2} \\frac{m_{\\rm e} c}{\\hbar} = \\frac{e^2}{4\\pi \\varepsilon_0 c \\hbar}.\n",
"$$\n",
"\n",
"\n",
"The ratio is called the fine structure costant $\\alpha$ and it is of fundamental importance in quantum electrodynamics, that is the quantum theory of the electromagnetic field, since it is a measure of the strength of the electromagnetic interaction between photons and charged particles. The fine structure costant is a small quantity\n",
"$$ \\alpha = 0.00729735\\dots \\approx \\frac{1}{137}. $$\n",
"Since the fine structure constant is small, a perturbation theory can be used for calculating very accurate results in quantum electrodynamics."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "10a16632-f182-405b-99c9-731cb219e1ea",
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.9.5"
}
},
"nbformat": 4,
"nbformat_minor": 5
}