Peronal notes
/index.xml
Recent content on Peronal notesHugo -- gohugo.ioen-usTue, 27 Nov 2018 00:00:00 +0530Notes for the book Gravity by Hartle
/post/gravity/
Tue, 27 Nov 2018 00:00:00 +0530/post/gravity/
<h2 id="questions">Questions</h2>
<ul>
<li>What are pulsars and quasars?</li>
</ul>
<h2 id="chapter-1">Chapter 1</h2>
English grammar
/post/grammar/
Mon, 13 Aug 2018 00:00:00 +0530/post/grammar/
<h2 id="present-continuous--i-am-doing">Present continuous (I am doing)</h2>
<p><strong>Usage</strong>: in the middle of ‘action’, started doing and haven’t finished yet; talk about things/changes happening in a period around now (for example, today / this week / this evening etc.).</p>
<p>Often the action is happening at the time of speaking, but not necessarily.</p>
<p><strong>Standard structure</strong>:</p>
<table>
<thead>
<tr>
<th>Subject</th>
<th>Present tense of <em>be</em></th>
<th>Present participle verb</th>
</tr>
</thead>
<tbody>
<tr>
<td>I</td>
<td>am (’m) / am not (’m not, ain’t)</td>
<td>-ing form of verb</td>
</tr>
<tr>
<td>he/she/it/name of a person</td>
<td>is (’s) / is not (’s not, isn’t)</td>
<td>-ing form of verb</td>
</tr>
<tr>
<td>we/you/they/name of group</td>
<td>are (’re) / are not (’re not,aren’t)</td>
<td>-ing form of verb</td>
</tr>
</tbody>
</table>
<p>For framing a question, reverse the first 2 columns.</p>
<p>Continuous is not used with stative verbs, use simple instead.</p>
<h2 id="present-simle--i-do">Present simle (I do)</h2>
Theory of groups for physics applications
/post/Groups/
Mon, 13 Aug 2018 00:00:00 +0530/post/Groups/
<h2 id="week-1">Week 1</h2>
<h3 id="introduction">Introduction</h3>
<ul>
<li>The origins of Group theory is in Premutations (the algebra obeyed) and Geometry (rotations) corresponding to Discrete and Continuous groups respectively.</li>
<li>Geometric rotations, in general do not commute.</li>
<li>Continuous groups is essentially Trignomentry.</li>
<li>In quantum mechanics, symmetry group substitutes for the geometry of shape and size.</li>
</ul>
<h3 id="algebraic-preliminaries">Algebraic preliminaries</h3>
<h4 id="sets-and-maps">Sets and maps</h4>
<p>Mathematicians usually classify maps as:</p>
<ul>
<li>Surjective/onto: Range is completely covered.</li>
<li>Injective/into: One to One, but need not exhaust range.</li>
<li>Bijective: Both one-to-one and covers the whole range.</li>
</ul>
<p>Algebraic Structure: There exists a binary relation with properties:</p>
<ul>
<li>commutatitive: \(a \circ b\) = \(b \circ a\)</li>
<li>associative: \(a \circ (b \circ c) = (a \circ b) \circ c\)
<ul>
<li>When not associative, Jacobi identity is used as an alternative.</li>
</ul></li>
<li>identity element: \(a \circ e = a = e \circ a\)</li>
<li>inverse: \(a \circ a^{-1} = e = a^{-1} \circ a\)</li>
</ul>
<p><strong>Homomorphism</strong>: Maps which preserves algebraic structure.</p>
<p><strong>Isomorphism</strong>: Maps which are bijective and preserves the algrbraic structure i.e., a map M: \(S_1 \rightarrow S_2\) with algrbraic structures \(a \circ_1 b \in S_1\) and \(a’ \circ_2 b’ \in S_2\) has the property - \((a \circ_1 b) \rightarrow (a’ \circ_2 b’)\) if \(a \rightarrow a’\) and \(b \rightarrow b’\).</p>
<h4 id="groups">Groups</h4>
<p><strong>Group</strong>: A set which satisfies:</p>
<ol>
<li>Closure: \(\forall a, b \in S\ \exists\ a \circ b \in S\)</li>
<li>Associative: \(a \circ (b \circ c) = (a \circ b) \circ c\)</li>
<li>Identity: \(\exists\ e \in S\) such that \(a \circ e = a = e \circ a\)</li>
<li>Inverse: \(\forall a \in S\ \exists\ a^{-1} \in S\) such that \(a \circ a^{-1} = e = a^{-1} \circ a\)</li>
</ol>
<p>Examples of groups: set of matrices with determinant \(\ne\) 0; set of all posible rotational configurations of a rigid object.</p>
<p><strong>Abelian group</strong>: A group which satisfies an additional property - Commutativiy: \(a \circ b\) = \(b \circ a\). Example: Rotations in 2D plane - SO(2).</p>
<h4 id="linear-vector-space">Linear vector space</h4>
<p>A set \(V\) with + (addition) and \(\bullet\) (scalar multiplication) and an auxiliary set of scalars (s) which satisfies:</p>
<ul>
<li>Abelian group under +</li>
<li>Under \(\bullet\):
<ul>
<li>\(a \bullet v \in V\) when \(a \in s\)</li>
<li>\(a \bullet (v_1 + v_2) = a \bullet v_1 + a \bullet v_2\): multiplication is distributive</li>
<li>Additionally, scalars have their own abelian structure with + so that \(a_1+a_2 \in s\) etc.</li>
<li>\((a_1+a_2) \bullet v = a_1 \bullet v + a_2 \bullet v\)</li>
</ul></li>
</ul>
<h4 id="permutations">Permutations</h4>
<p>Permutation group is the group of all possible ways of rearranging n objects. The “possible ways” are elements of the group.</p>
<p>Any discrete group is a sub-group of some permutation group.</p>
<p>Can be represented as matrices.</p>
<h4 id="equivalence-realtion">Equivalence realtion</h4>
<p>For a set \(s\) a “relation” \(R\) is a <em>conditional</em> about a,b etc. \(\in s\) such that:</p>
<ul>
<li>\(a R a\) : a is always related to a: Reflexivity</li>
<li>\(a R b \Rightarrow b R a\): Symmetry</li>
<li>\(a R b\) and \(b R c \Rightarrow a R c\): Tansitivity</li>
</ul>
<p>Any relation \(R\) with above properties is called an <em>Equivalence relation</em>. Example: “parallel” relation of st. lines in a plane is an equivalence relation ; “perpendicular” is not an equivalence relation (does not satisfy first req.)</p>
<p><strong>Theorem</strong>: An equivalence relation divides a set into disjoint subsets whose union makes up the whole set \(s\).</p>
<p><strong>Proof</strong>: Let $s_1, s_2, …$ be some subsets.
Let \(s_1\) be such that all elements in it are related by R. Similarly, consider \(s_2\). Now, \(s_1 \cap s_2\) because \(a \in s_1\) and \(b \in s_2\) and hypothesis \(a R b\) forces \(s_1\) and \(s_2\) to be same subsets due to transitivity and symmetry.</p>
<h2 id="week-2">Week 2</h2>
<h3 id="lagrange-s-theorem">Lagrange’s Theorem:</h3>
<p>If H is a subgroup</p>
<h2 id="reference-books">Reference books:</h2>
<ul>
<li>Morton Hammermesh</li>
<li>Sadri Hassani Ch. 23 and 24</li>
<li>Brian C. Hall</li>
<li>Ramadevi’s draft book for applications</li>
</ul>
Notes for GRE Math Review
/post/math-review/
Tue, 07 Aug 2018 00:00:00 +0530/post/math-review/
<h2 id="arithematic">Arithematic</h2>
<ul>
<li>r = c - qd, r is the reminder and q is the quotient when an integer c is divided by a positive integer d</li>
<li>The first 10 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. 1 is not a prime number. 2 is the only prime number that is even.</li>
<li>The fraction part’s value of a mixed number has to be between 0 and 1.</li>
<li>Numers of form c/d, where either c or d is not an integer and d!=0, are called fractional expressions.</li>
<li>The exponent 0^0 is undefined.</li>
<li>The expression consisting of the square root symbol \sqrt placed over a nonnegative number denotes the nonnegative square root (or the positive square root if the number is greater than 0) of that nonnegative number. Square roots of negative numbers are not defined in real number system.</li>
<li>For odd order roots, there is exactly one root for every number n, even when n is negative.</li>
<li>For even order roots, there are exactly 2 roots for every positive number n and no roots for any negative number n.</li>
<li>Every fraction is equivalent to a decimal that either terminates or repeats. COnverse is also true: every terminating or repeating decimal represents as rational number.</li>
<li>Triangle inequality |r+s| <= |r|+|s| is satisfied by all real numbers.</li>
<li>Amount of change is calculated as percent of the initial amount.</li>
</ul>
<h2 id="algebra">Algebra</h2>
<ul>
<li>Like terms: terms with same variable and same corresponding exponents.</li>
<li>A polynomial is the sum of a finite number of terms in which each term is either a constant term or a product of a coefficient and one or more variables with <strong>positive integer exponents</strong>.</li>
<li>When a factor is canceled out (in both numerator and denominator) the expression is equivalent only for the values of x (variables) for which the original is defined.</li>
<li>In linear equations: none of the variables are multiplied together or raised to power greater than 1.</li>
<li>It is possible for a linear equation to have no solutions, or turn out to be an identity.</li>
<li>When both sides of the inequality are multiplied or divided by the same nonzero constant, the direction of the inequality is preserved if the constant is positive but the direction is reversed if the constant is negative.</li>
<li>Without an explicit restriction, the domain is assumed to be the set of all values of x for which f(x) is a real number.</li>
<li>In Simple interest, value V = P (1 + (rt)/100) where r is the annual interest rate expressed in percent and t is the time in years.</li>
<li>In Compund interest, value V = P (1 + r/100)^(nt) where n is the nuber of times interest is compounded (1 if annualy, 4 is quaterly) and t is expressed in years.</li>
<li>Points satisfying y < mx+c are those either on the line or below the line. Similarly for y > mx+c are on or above the line.</li>
<li>Interchanging x and y in the equation of any graph yields another graph that is the reflection of the original graph about the line y = x.</li>
<li>The graph of a quadratic equation of the form y = ax^2 + bx + c, where a,b,and c are constants and a != 0, is a parabola. If a is positive, the parabola opens upward and the vertex is its lowest point. If a is negative, the parabola opens downward and the vertex is its highest point. Every parabola is symmetric with itself about the vertical line that passes through its vertex.</li>
<li>The expression f(x) = \(a x^2 + b x + c\) can be expressed as \(a \left(x + \frac{b}{2a} \right)^2 + \frac{4 a c - b^2}{4a}\). So, the axis is given by the straight line x = \(- \frac{b}{2a}\) and the vertex by V = \(\left( - \frac{b}{2a}, \frac{4 a c - b^2}{4a} \right)\).</li>
<li>Graph of \sqrt{x} represents only the upper half of a parabola lying on its side.</li>
<li>When the variable (x) is replaced by x+c then the graph is shifted c units left.</li>
<li>The graph of ch(x) is the graph of h(x) streched vertically by a factor of c if c > 1, or shrunk verically by a factor of c if 0 < c < 1.</li>
</ul>
<h2 id="geometry">Geometry</h2>
<ul>
<li>Line segments that have equal lengths are called congruent line segments.</li>
<li>Opposite/vertical angles have equal measure, and angles that have equal measure are calledcongruent angles.
-</li>
</ul>
<figure >
<img src="/ox-hugo/lines.png" />
</figure>
<ul>
<li>Convex polygon (are the only ones considered here): polygon in which each interior angle is less than 180^0.</li>
<li>If a polygon has n sides, it can be divided into n-2 triangles. It follows that the sum of the interior angles of an n-sided polygon is (n-2)(180^0).</li>
<li>Regular polygon: polygon in which all sides and all interior angles are congruent.</li>
<li>In a triangle, length of each side must be less than the sum of lengths of the other two sides.</li>
<li>Legs: the two sides other than hypotenuse of a right triangle.</li>
<li>The length of median of an equilateral triangle is \(\frac{\sqrt{3}}{2}\) of length of the side.</li>
<li>While calculating the area of a triangle, base can be any side and height is the perpendicular line segment from the opposite vertex to base.</li>
<li>Two triangles are congruent if their vertices can be matched up so that the corresponding angles and the corresponding sides are congruent.</li>
<li>Side-Side-Side (SSS) congruence: if three sides of both triangle are congruent.</li>
<li>Side-Angle-Side (SAS) congruence: if two sides and the included angle of two triangles are congruent.</li>
<li>ASA congruence: if two angles and included side are congruent. AAS follows from this (since congruence of two angles implies for third angle).</li>
<li>Two triangle that have same shape but not necessarily the smae size are called similar triangles. scale factor of similarity: ratio of lengths of corresponding sides.</li>
<li>A quadrilateral in which at least one pair of opposite sides is parallel is called a trapezoid. Those sides are called bases.</li>
<li>In parallelograms, opposite angles are congruent.</li>
<li>All squares are rectangles. All rectangles are parallelograms. All parallelograms are trapezoids.</li>
<li>The area of a trapezoid is given by \(A = \frac{1}{2} (b_1+b_2)h\), where height is measured as the length of perpendicular line segment between bases.</li>
<li>Since any two points on a circle always repesents two different arcs, arc is frequently identified by 3 points.</li>
<li>A central angleof a circle is an angle with its vertex at the center of the circle.</li>
<li>The measure of an arc is the measure of its central angle, which is the angle formed by two radii that connect the center of the circle to the two endpoints of the arc.</li>
<li>The ratio of the length of an arc to the circumference is equal to the ratio of the degree measure of the arc to 360^0.</li>
<li>A sector of a circle is a region bounded by an arc of the circle and two radii.</li>
<li>The ratio of the area of a sector of a circle to the area of the entire circle is equal tothe ratio of the degree measure of its arc to 360^0.</li>
<li>It is also possible for the center of the circle to be outside the inscribed triangle, or on one of the sides of inscribed triangle.</li>
<li>If one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle. And converse is true.</li>
<li>A polygon is circumscribed about a circle if each side of the polygon is tangent to the circle, or equivalently, the circle is inscribed in the polygon.</li>
<li>A circular cylinder consists of two bases that are congruent circles lying in parallel planes and a <strong>lateral surface</strong> made of all line segments that join points on the two circles and that are parallel to the line segment joining the centers (axis) of the two circles.</li>
<li>A right circular cylinder is the one with axis perpendicular to its bases.</li>
</ul>
<h2 id="data-analysis">Data Analysis</h2>
<ul>
<li>The relative frequency of a category or a numerical value is the corresponding frequency divided by the total number of data.</li>
<li>A segmented bar graph, or stacked bar graph, is similar to a regular bar graph except that in a segmented bar graph, each rectangular bar is divided, or segmented, into smaller rectangles that show how the variable is “separated” into other related variables.</li>
<li>Histograms are graphs of frequency distributions that are similar to bar graphs,but they must havea number line for the horizontal axis, which represents the numerical variable. Also there are no regular spaces between the bars because we have divided the data into intervals of equal length.</li>
<li>The median is a measure of central tendency that is fairly unaffected by unusually high or low values relative to the rest of the data, unlike mean.</li>
<li>Like the median M, quartiles and percentiles are numbers that divide the data into roughly equal groups after the data have been ordered from the least value L to the greatest value G.</li>
<li>After the data are listed in increasing order, Q_1 is the median of the first half of the data in the ordered list and Q_3 is the median of the second half of the data.</li>
<li>Sometimesa data value isunusually small or unusually large in comparison with the rest of the data. They are called outliers and directly affect the range.</li>
<li>Interquartile range: differnce between the third quartile and first quartile.</li>
<li><strong>Boxplot</strong> or <strong>box-and-whisker plots</strong> summarize a group of numerical data and illustrate its center and spread.
<img src="/ox-hugo/boxplot.png" alt="" /></li>
<li>The usual standard deviation (also called population standard deviation) is different from sample standard deviation which is computed by dividing the sum of squared differences by n-1 instead of n. The latter is preferred for for samples from larger population of data.</li>
<li>Standardization: Any point p is \(\frac{p - mean}{sd}\) standard deviations away (above or below) from the mean.</li>
<li>Fact: In any group of data, most of the data are within 3 standard deviations of the mean.</li>
<li>A <strong>list</strong> is like a set but different in two ways: order matters, elements can be repeated and it matters.</li>
<li>Permutations: The number of ways n objects can be ordered is n!.</li>
<li>Permutations of n objects taken k at a time: The number of ways to select and order k out of n objects is given by \(\frac{n!}{(n-k)!}\).</li>
<li>Combinations: number of ways to select without order = (number of ways to select with order) / (number of ways to order).</li>
<li>P(E or F) = P(E) + P(F) - P(E and F). P(E and F) = 0 if E and F are mutually exclusive. P(E and F) = P(E)P(F) if E and F are independent.</li>
<li>If P(E) != 0 amd P(F) != 0 then events E and F cannot be both mutually exclusive and independent.</li>
<li>Distribution/density/frequency curve: A model curve that us close to tops of the bars in a histogram representing large data sets. Vertical scale is usually adjusted such that the area under the curve is 1.</li>
<li>A probability is essentially relative frequency represented as decimal instead of percent.</li>
<li>Approximately normally distributed data have following properties:
<ul>
<li>The mean, median, and mode are all nearly equal. (for normal - exaclty equal)</li>
<li>The data are grouped fairly symmetrically about the mean. (for normal - perfectly symmetrical)</li>
<li>About <sup>2</sup>⁄<sub>3</sub> of the date are within 1 standard deviation of the mean.</li>
<li>Almost all the data are within 2 standard deviation of the mean.</li>
</ul></li>
</ul>
Notes for the course PH5211
/post/HEP/
Tue, 07 Aug 2018 00:00:00 +0530/post/HEP/
<h2 id="nuclear-physics">Nuclear Physics</h2>
<h3 id="stable-nuclei-nomenclature-and-units">Stable nuclei, Nomenclature and units</h3>
<ul>
<li>Atomic scale is 10<sup>-10</sup>m, nuclear scale is 10<sup>-15</sup>m. For convenience we use the unit <strong>Fermi</strong> (fm) = 10<sup>-15</sup>m = 1 femtometer. Nuclear sizes range from 1 fm to 7 fm. Particle physics usually happens at an even smaller scale << 10<sup>-15</sup>m.</li>
<li>A nuclear species or <em>nuclide</em> is denoted by \(^A_ZX_N\). Here, X is the chemical symbol. Z is the <strong>atomic number</strong>: the number of protons. A is the <strong>mass number</strong>: the integer nearest to the ratio between the nuclear mass and the fundamental mass unit (<sup>1</sup>⁄<sub>12</sub> th of mass of \(^{12}_6C\)). N = A-Z represents the number of neutrons.</li>
<li>Before the discovery of neutrons, it was believed that nucleus contains A protons along with A-Z electrons to justify the charge. But it is unsatisfactory because: A force stronger than Coulomb is required between protons and electrons; Confining electrons in 10<sup>-14</sup>m requires momentum of range 20 MeV/c (acc. to Uncertainty principle), but β rays usually have energies less than 1 MeV; total intrinsic angular momentum (spin) of nuclei (A-Z) would disagree with spin addition of A protons and A-Z electrons; nuclei containing unpaired electrons would be expected to have magnetic dipole moments greater than observed.</li>
<li><strong>Isotopes</strong>: Nuclides with same Z but different N (and A). Like \(^{35}Cl\) and \(^{37}Cl\).</li>
<li>Radioisotopes/radioactive isotopes: unstable isotopes artificially produced in nuclear reactions.</li>
<li><strong>Isotones</strong>: Nuclides with same N but different Z (so X,A) like \(^2H\) and \(^3He\).</li>
<li><strong>Isobars</strong>: Nuclides with same A like \(^3He\) and \(^3H\).</li>
<li>The properties of nuclides we measure include: mass, radius, relative abundance (for stable), decay modes and half-lives (for radioactive), reaction modes and cross sections, spin, magnetic dipole and electric quadrupole moments, and excited states.</li>
<li>Black = stable (radioactive lifetime is huge), Grey = Unstable
<img src="/ox-hugo/stable.jpeg" alt="" /></li>
<li>Typical β and γ decay energies are in the range of 1 MeV, and low-energy nuclear reactions take place with kinetic energies of order 10 MeV which are far smaller than nuclear rest energies. So non-relativistic formulation is justified for nucleons, but β -decay electrons must be treated relativistically.</li>
<li>Electromagnetic (γ) decays generally occur with lifetimes of order nanoseconds to picoseconds. α and β decays occur with longer lifetimes, often minutes or hours. Many nuclear reactions (\(^5He\) or \(^8Be\) breaking apart) take place in the order of 10<sup>-20</sup> s, which is roughly the time that the reacting nuclei are within range of each other’s nuclear force.</li>
</ul>
<h4 id="remember">Remember</h4>
<ul>
<li>Z=92 for U, 26 for Fe, 17 for Cl.</li>
<li>1 u = 931.502 MeV = 1.661 \(\times 10^{-27}\) Kg.</li>
<li>1 eV = \(1.602 \times 10^{-19}\) J: the energy gained by a single unit of electronic charge when accelerated through a potential difference of 1 Volt.</li>
</ul>
<h4 id="further-reading">Further Reading</h4>
<ul>
<li>How does smoke detector’s working depend on nuclear physics.</li>
</ul>
<h3 id="size-and-shape-of-nuclei">Size and shape of nuclei</h3>
<ul>
<li>Like the radius of an atom, the radius of a nucleus is not precisely defined. The density of nucleons and the nuclear potential have similar spatial dependence - relatively constant over short distances beyond which they drop rapidly to zero.</li>
<li>(Spherical) nuclear shape is characterized by: <strong>mean radius</strong>, where the density is half its central value, and the <strong>skin thickness</strong> over which the density drops from near its maximum to near its minimum.</li>
<li>In some experiments like high-energy electron scattering, muonic X rays, optical and X-ray isotope shifts, and energy differences of mirror nuclei, we measure Coulomb interaction of a charged particle with the nucleus determining the <em>distribution of nuclear charge</em> (primarily distribution of protons but also involving somewhat that of neutrons). In other experiments such as Rutherford scattering, α decay, and pionic X rays, we measure the strong nuclear interaction determining the <em>distribution of nuclear matter</em>.</li>
</ul>
<h4 id="the-distribution-of-nuclear-charge">The Distribution of Nuclear charge</h4>
<ul>
<li>Shape and size of an object is determined by examining the radiation scattered (elastically) from it for which we need wavelength smaller than the details of the object. For nuclei with diameter 10 fm, we require λ <= 10 fm i.e., p >= 100 MeV/c.</li>
<li>Rusults for such an experiment looks like diffraction pattern (only an approximation of the potential scattering in 3D) by a circular disk (2D) of diameter D i.e., the first minimum appears at \(\theta = sin^{-1}(1.22 \lamda / D)\).</li>
<li>Since the nucleus does not have a sharp boundary, the minima do not fall to zero.</li>
</ul>
<figure >
<img src="./../images/electron-Pb-elastic-scattering.png" />
</figure>
<h3 id="isospin-and-the-shell-model">Isospin and The Shell Model</h3>
<ul>
<li>In the absence of magnetic field, there is no distinction between “spin-up” and “spin-down” nucleons.</li>
<li>The charge independence of strong interaction means that protons and neutrons can be grouped together (as nucleons) analogous to spin: called isospin. In the absence of EM fields, they are indistinguishable.</li>
<li>The isospin(T) obeys usual rules for angular momentum vectors: length of an isospin vector \(\sqrt{t(t+1}\hbar\); 3-axis projections \(T_3 = m_T \hbar\); and addition rules.</li>
<li>For any nucleus: \(T_3 = \frac{1}{2}(Z-N)\) expressed in terms of \(\hbar\). T can take any value at least as great as \(\left\| T_3 \right\|\).</li>
</ul>
<h2 id="clarifications">Clarifications</h2>
<ul>
<li>The moodle says 10 best out of 11 assignments. But 8 out of 11 is mentioned in class.</li>
<li>The ted link on the moodle is not working. What is it supposed to point to?</li>
</ul>
<p>size and shape of nuclei. Krane: Section 3.1</p>
<p>Lecture 3 (2/8/18): mass, abundance and binding energy of nuclei. Krane: Sections 3.2 and 3.3</p>
<p>Lecture 4 (7/8/18): semi-empirical mass formula. Krane 3.3.</p>
<p>Lecture 5 (9/8/18): spin, parity and magnetic moments. Krane 3.4, 3.5 and 16.1</p>
<p>Lecture 6 (13/8/18) [Wednesday timetable so class at 9am]: the deuteron. Krane 4.1 and 3.5.</p>
<p>Lecture 7 (14/8/18) : cross sections and charge independence. Krane 4.4 and 11.4, Perkins 2.10 and Martin and Shaw, Particle Physics (Wiley), Appendix B.</p>
<p>Lecture 8 (16/8/18): exchange nature of the nuclear force. Krane 4.5, Perkins 2.2.</p>
<p>Lecture 9 (20/8/18) [11am Monday to make-up for cancelled class]: isospin Krane 11.3, Perkins 13.12 and 3.13. The shell model evidence Krane 5.1</p>
<p>Lecture 10 (21/8/18): Basic principles, spin-orbit and shell model predictions; gamma decay Krane 5.1, 10.1-10.4.</p>
<p>[No classes on <sup>22</sup>⁄<sub>8</sub> and <sup>23</sup>⁄<sub>8</sub>]</p>
<p>Lecture 11 (28/8/18): decays in general and resonances. Krane 6.1-6.3, Perkins 2.11</p>
<p>Lecture 12 (29/8/18): alpha decay. Krane Chapter 8.1-8.5</p>
<p>Lecture 13 (30/8/18): beta decay. Krane Chapter 9.1-9.5</p>
<p>Lecture 14 (4/9/18): fission and fusion. Krane Chapters 13 and 14</p>
Solution manual for Quantum Computation and Quantum Information
/post/SolutionsforQCQI/
Tue, 07 Aug 2018 00:00:00 +0530/post/SolutionsforQCQI/
<div class="ox-hugo-toc toc">
<div></div>
<div class="heading">Table of Contents</div>
<ul>
<li><a href="#1-introduction-and-overview">1 Introduction and Overview</a></li>
<li><a href="#2-introduction-to-quantum-mechanics">2 Introduction to quantum mechanics</a></li>
</ul>
<p></div>
<!--endtoc--></p>
<h2 id="1-introduction-and-overview">1 Introduction and Overview</h2>
<div class="ox-hugo-toc toc local">
<div></div>
<ul>
<li><a href="#exercise-1">Exercise 1</a></li>
<li><a href="#exercise-2">Exercise 2</a></li>
<li><a href="#problem-1">Problem 1</a></li>
<li><a href="#problem-2">Problem 2</a></li>
</ul>
<p></div>
<!--endtoc--></p>
<h3 id="exercise-1">Exercise 1</h3>
<p>It is already shown that a deterministic classical computer would require \(2^n/2+1\) queries.</p>
<p>Instead, if we use a probabilistic classical computer i.e, \(f(x)\) is evaluated for randomly chosen \(x\), with just one execution we cannot determine whether \(f(x)\) is constant or balanced function (atleast not with probability of error ε < <sup>1</sup>⁄<sub>2</sub>). If the second evaluation gives a different result than first, we can say with certainity that \(f(x)\) is a balanced function. In the other case, the probaility that we get same result twice in a row if the function was balanced would be <sup>1</sup>⁄<sub>2</sub> for the first evaluation times \(\frac{2^n/2-1}{2^n-1}\) for the second which is less than <sup>1</sup>⁄<sub>2</sub> if:</p>
<p>\begin{equation}
\begin{split}
\frac{1}{2} \times \frac{2^n/2-1}{2^n-1} & < \frac{1}{2}\\<br />
2^n-2 & < 2(2^n-1)\\<br />
2^n & < 2^{n+1} \\<br />
n & < n+1
\end{split}
\end{equation}</p>
<p>which is always true for all positive integer \(n\) . So if we get same evaluation twice, we can say that \(f(x)\) is a constant function with a probability of error ε < <sup>1</sup>⁄<sub>2</sub>. Therefore, the best classical algorithm (probabilistic) will require 2 evaluations, irrespective of size of the input.</p>
<h3 id="exercise-2">Exercise 2</h3>
<p>If a device, upon input of one of two non-orthogonal quantum states correctly identified the state without collapsing, then we can perform certain unitary transformation on an extra quantum state to create either of the quantum states, since we know its coefficients. Thus creating a clone of the input quantum state.</p>
<p>Conversely, if we have a device for cloning, we can in principle, generate multiple copies of the unknown quantum states and perform ensemble measurement to find it’s coefficients (hidden information - not accessible in single measurement) with enough precision to identify/distinguish them.</p>
<h3 id="problem-1">Problem 1</h3>
<p>As suggested, I will attempt this once I finish rest of the book. An example: <a href="https://www.douban.com/group/topic/22546986">https://www.douban.com/group/topic/22546986</a>.</p>
<h3 id="problem-2">Problem 2</h3>
<p>As suggested, I will attempt this once I finish rest of the book.</p>
<h2 id="2-introduction-to-quantum-mechanics">2 Introduction to quantum mechanics</h2>
<div class="ox-hugo-toc toc local">
<div></div>
<ul>
<li><a href="#exercise-1">Exercise 1</a></li>
<li><a href="#exercise-2">Exercise 2</a></li>
<li><a href="#exercise-3">Exercise 3</a></li>
</ul>
<p></div>
<!--endtoc--></p>
<h3 id="exercise-1-1">Exercise 1</h3>
<p>To prove that a set of vectors are linearly dependent, it is enough to show that there exists a set of complex numbers \((a_1,…,a_n)\), not all zero, such that
\(\newcommand{\ket}[1]{\lvert #1 \rangle}\)
\[a_1\ket{v_1}+a_2\ket{v_1}+…+a_n\ket{v_n} = 0.\]
Here,</p>
<p>\begin{equation}
\begin{split}
a_1
\begin{bmatrix}
1 \\<br />
-1
\end{bmatrix}
+a_2
\begin{bmatrix}
1 \\<br />
2
\end{bmatrix}
+a_3
\begin{bmatrix}
2 \\<br />
1</p>
<h1 id="end-bmatrix">\end{bmatrix}</h1>
<p>\begin{bmatrix}
0 \\<br />
0
\end{bmatrix}
\\<br />
\\<br />
\Rightarrow
a_1 + a_2 + 2 a_3 = 0,\\<br />
-a_1 + 2 a_2 +a_3 = 0.\\<br />
\end{split}
\end{equation}</p>
<p>This is a set of 2 simultaneous equations in 3 variables, so there exists \(\infty\) number of solutions like \(a_1=1, a_2=1,a_3=-1\) and any multiples of them.</p>
<h3 id="exercise-2-1">Exercise 2</h3>
<p>The matrix representation of an operator depends on the choise of basis for the underlying vector space (both input and output).</p>
<p>In this case, if we use \(\ket{0}\) and \(\ket{1}\) as the basis for V,</p>
<p>\begin{equation*}
\begin{split}
A\ket{v_1} &= A\ket{0} = A_{11}\ket{w_1}+A_{21}\ket{w_2} = A_{11}\ket{0}+A_{21}\ket{1}=\ket{1}
\Rightarrow A_{11} &= 0, A_{21} = 1\\<br />
A\ket{1} &= A_{12}\ket{0}+A_{22}\ket{1} = \ket{0} \Rightarrow A_{12} = 1,A_{22}=0\\<br />
\therefore A &=
\begin{bmatrix}
0 & 1 \\<br />
1 & 0
\end{bmatrix}
\end{split}
\end{equation*}</p>
<p>Instead, if we use \(\frac{\ket{0}+\ket{1}}{2}\) and \(\frac{\ket{0}-\ket{1}}{2}\) as the basis for V,</p>
<p>\begin{equation*}
\begin{split}
A\left(\frac{\ket{0}+\ket{1}}{2}\right) &= A_{11}\frac{\ket{0}+\ket{1}}{2}+A_{21}\frac{\ket{0}-\ket{1}}{2}= \frac{\ket{1}+\ket{0}}{2} \Rightarrow A_{11} = 1, A_{21} = 0\\<br />
A\left(\frac{\ket{0}-\ket{1}}{2}\right) &= A_{12}\frac{\ket{0}+\ket{1}}{2}+A_{22}\frac{\ket{0}-\ket{1}}{2}= \frac{\ket{1}-\ket{0}}{2} \Rightarrow A_{12} = 0, A_{22} = -1\\<br />
\therefore A &=
\begin{bmatrix}
1 & 0 \\<br />
0 & -1
\end{bmatrix}
\end{split}
\end{equation*}</p>
<h3 id="exercise-3">Exercise 3</h3>
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