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1088 Chapter 22 Calculus of Variations 1 2 y x FIGURE 22.3 Circular optical path in medium. Irrespective of the values of C1 and C2, this light path is the arc of a circle whose center is on the line y = 0, namely the event horizon. The actual path of light passing from (x1, y1) to (x2, y2) will be on the circle throug... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,089 | 7th_Mathematical_Methods_for_Physicists | 0 |
22.1 Euler Equation 1089 (x1, y1) (x2, y2) y ds y x FIGURE 22.4 Surface of rotation, soap-film problem. See Fig. 22.4. The curve is required to pass through fixed endpoints (x1, y1) and (x2, y2). The variational problem is to choose the curve y(x) so that the area of the resulting surface will be a minimum. A physical ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,090 | 7th_Mathematical_Methods_for_Physicists | 0 |
1090 Chapter 22 Calculus of Variations which rearranges to (yx)−1 = dx dy = c1 q y2 −c2 1 . (22.23) We note in passing that c1 had better have a value that causes dy/dx to be real. Equa- tion (22.23) may be integrated to give x = c1 cosh−1 y c1 + c2, and, solving for y, we have y = c1 cosh x −c2 c1 . (22.24) Finally, c... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,091 | 7th_Mathematical_Methods_for_Physicists | 0 |
22.1 Euler Equation 1091 Before answering this question, consider the physical situation with the rings moved apart so that x0 = 1. Then Eq. (22.26) becomes 1 = c1 cosh 1 c1 , (22.28) which has no real solutions. The physical significance is that as the unit-radius rings were moved out from the origin, a point was reac... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,092 | 7th_Mathematical_Methods_for_Physicists | 0 |
1092 Chapter 22 Calculus of Variations member of the first line below, we have A = 4π x0 Z 0 y(1 + y2 x)1/2 dx = 4π c1 x0 Z 0 y2 dx = 4πc1 x0 Z 0 cosh x c1 2 dx = πc2 1 sinh 2x0 c1 + 2x0 c1 . (22.30) For x0 = 1 2, Eq. (22.30) leads to c1 = 0.2350 → A = 6.8456, c1 = 0.8483 → A = 5.9917, showing that the former can at mo... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,093 | 7th_Mathematical_Methods_for_Physicists | 0 |
22.1 Euler Equation 1093 For an excellent discussion of both the mathematical problems and experiments with soap films, we refer to Courant and Robbins in Additional Readings. The larger message of this subsection is the extent to which one must use caution in accepting solutions of the Euler equations. Exercises 22.1.... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,094 | 7th_Mathematical_Methods_for_Physicists | 0 |
the condition that J = Z f (x, y)dx has a stationary value (a) leads to f (x, y) independent of y and (b) yields no information about any x-dependence. We get no (continuous, differentiable) solution. To be a meaningful variational problem, dependence on y or higher derivatives is essential. | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,094 | 7th_Mathematical_Methods_for_Physicists | 0 |
1094 Chapter 22 Calculus of Variations Note. The situation will change when constraints are introduced (compare to Exer- cise 22.4.6). 22.1.6 A soap film stretched between two rings of unit radius centered at ±x0 will have its closest approach to the x-axis at x = 0, with the distance from the axis given by c1, with x0... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,095 | 7th_Mathematical_Methods_for_Physicists | 0 |
, yx, x)dx, with f = y(x), has no extreme values. (b) If f (y, yx, x) = y2(x), find a discontinuous solution similar to the Goldschmidt solution for the soap-film problem. 22.1.10 Fermat’s principle of optics states that a light ray in a medium for which n is the (position-dependent) index of refraction will follow the... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,095 | 7th_Mathematical_Methods_for_Physicists | 0 |
22.1 Euler Equation 1095 equation satisfied by the path if the transit time is to be a minimum. Assume the Earth to be a nonrotating sphere of uniform density. Hint. The potential energy of a particle of mass m a distance r < R from the center of the Earth, with R the Earth’s radius, is 1 2mg(R2 −r2)/R, where g is the ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,096 | 7th_Mathematical_Methods_for_Physicists | 0 |
Note. This is not an Euler equation problem, because the light path is not differentiable at x0. n1 θ1 θ2 (x0, 0) (x2, y2) (x1, y1) n2 x FIGURE 22.7 Snell’s law. | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,096 | 7th_Mathematical_Methods_for_Physicists | 0 |
1096 Chapter 22 Calculus of Variations 22.1.13 A second soap-film configuration for the unit-radius rings at x = ±x0 consists of a circular disk, radius a, in the x = 0 plane and two catenoids of revolution, one joining the disk and each ring. One catenoid may be described by y = c1 cosh x c1 + c3 . (a) Impose boundary... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,097 | 7th_Mathematical_Methods_for_Physicists | 0 |
/c2y, where c is an integration constant. It is helpful to make the substitution c2y = sin2 φ/2 and take (x0, y0) = (π/2c2,1/c2). 22.2 MORE GENERAL VARIATIONS Several Dependent Variables To apply variational methods to classical mechanics, we need to generalize the Euler equa- tion to situations in which there is more ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,097 | 7th_Mathematical_Methods_for_Physicists | 0 |
22.2 More General Variations 1097 Eq. (22.2). The generalization corresponds to functionals J of the form J = x2 Z x1 f u1(x),u2(x),...,un(x),u1x(x),u2x(x),...,unx(x), x dx. (22.31) We are now calling the dependent variables ui to be consistent with notations we will shortly introduce, and as before we use the subscrip... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,098 | 7th_Mathematical_Methods_for_Physicists | 0 |
when the integrand f is taken to be a Lagrangian L. The Langrangian (for nonrelativistic systems; see Exercise 22.2.5 for a relativistic particle) is defined as the difference of kinetic and potential energies of a system: L ≡T −V. (22.36) Using time as an independent variable instead of x and xi(t) as the dependent va... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,098 | 7th_Mathematical_Methods_for_Physicists | 0 |
1098 Chapter 22 Calculus of Variations xi(t) is the position and ̇xi = dxi/dt is the velocity of particle i as a function of time. The equation δJ = 0 is then a mathematical statement of Hamilton’s principle of classical mechanics, δ t2 Z t1 L(x1, x2,..., xn, ̇x1, ̇x2,..., ̇xn;t) dt = 0. (22.37) In words, Hamilton’s pr... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,099 | 7th_Mathematical_Methods_for_Physicists | 0 |
exercises. The result is a unification of otherwise sep- arate areas of physics. In the development of new areas, the quantization of Lagrangian particle mechanics provided a model for the quantization of electromagnetic fields and led to the gauge theory of quantum electrodynamics. One of the most valuable advantages ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,099 | 7th_Mathematical_Methods_for_Physicists | 0 |
22.2 More General Variations 1099 We will need ∂L ∂ ̇x = m ̇x, ∂L ∂x = −dV (x) dx = F(x). (22.39) We have identified the force F as the negative gradient of the potential. Inserting the results from Eq. (22.39) into the Lagrangian equation of motion, Eq. (22.38), we get d dt (m ̇x) −F(x) = 0, which is Newton’s second l... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,100 | 7th_Mathematical_Methods_for_Physicists | 0 |
is a real force. It is of some interest that this interpretation of centrifugal force as a real force is supported by the general theory of relativity. ■ Hamilton’s Equations Hamilton was the first to show that Euler’s equation for the Lagrangian enabled the equa- tions of motion to be reduced to the set of coupled fir... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,100 | 7th_Mathematical_Methods_for_Physicists | 0 |
1100 Chapter 22 Calculus of Variations This definition is consistent with the elementary definition of momentum in Cartesian coordinates, where (in one dimension) T = m ̇q2/2, p = m ̇q. From Eq. (22.41) and the Lagrangian equations of motion, Eq. (22.38), we have by direct substitution ̇pi = ∂L ∂qi , (22.42) and this p... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,101 | 7th_Mathematical_Methods_for_Physicists | 0 |
a function of several independent variables, u = u(x, y, z). In the three- dimensional case, for example, that equation becomes J = Z Z Z f <unk>u,ux,uy,uz, x, y, z dx dy dz, (22.48) where ux = ∂u/∂x, uy = ∂u/∂y, uz = ∂u/∂z, and u is assumed to have specified values on the boundary of the region of integration. General... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,101 | 7th_Mathematical_Methods_for_Physicists | 0 |
22.2 More General Variations 1101 where η is arbitrary except that it must vanish on the boundary. Our integral J is now, as in Section 22.1, a function of α, and our variational problem is to make J stationary with respect to α. Differentiating the integral Eq. (22.48) with respect to the parameter α and then setting ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,102 | 7th_Mathematical_Methods_for_Physicists | 0 |
u −∂ ∂x ∂f ∂ux −∂ ∂y ∂f ∂uy −∂ ∂z ∂f ∂uz = 0. (22.50) Remember that the derivative ∂/∂x operates on both the explicit and implicit x dependence of ∂f/∂ux; similar remarks apply to ∂/∂y and ∂/∂z. Example 22.2.3 LAPLACE’S EQUATION A variational problem with several independent variables is provided by electrostatics. An ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,102 | 7th_Mathematical_Methods_for_Physicists | 0 |
1102 Chapter 22 Calculus of Variations where φx stands for ∂φ/∂x. Thus, f (φ,φx,φy,φz, x, y, z) = φ2 x + φ2 y + φ2 z , so Euler’s equation, Eq. (22.50), yields (with u in that equation replaced by φ) −2(φxx + φyy + φzz) = 0, which in the usual vector notation is equivalent to ∇2φ(x, y, z) = 0. This is Laplace’s equatio... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,103 | 7th_Mathematical_Methods_for_Physicists | 0 |
for functions q and r. Replacing p, q, r, ... with yi and x, y, z, ... with xi, we can put Eq. (22.52) in a more compact form: ∂f ∂yi − X j ∂ ∂xj ∂f ∂yi j = 0, i = 1,2,..., (22.53) in which yi j ≡∂yi ∂xj . An application of Eq. (22.53) appears in Exercise 22.2.10. | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,103 | 7th_Mathematical_Methods_for_Physicists | 0 |
22.2 More General Variations 1103 Geodesics Particularly in general relativity, it is of interest to identify the shortest path between two points in a “curved space,” i.e., a space characterized by a metric tensor more general than that of Euclidean or even Minkoswki space. A path that is a “local minimum” (calculated... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,104 | 7th_Mathematical_Methods_for_Physicists | 0 |
s2 contains cross terms dqidq j with i <unk>= j. A path in our curved space can be described parametrically by giving the qi as functions of an independent variable that we will call u, and the distance between two points A and B can then be represented as J = B Z A ds du du = B Z A q gi j dqi dq j du du = B Z A s gi j... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,104 | 7th_Mathematical_Methods_for_Physicists | 0 |
1104 Chapter 22 Calculus of Variations Here the dot notation refers to derivatives with respect to the proper time τ (or to any other variable related thereto by an affine transformation (meaning the new variable, e.g., u, is related to τ by a transformation of the form u = aτ + b). This means that we can replace the m... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,105 | 7th_Mathematical_Methods_for_Physicists | 0 |
klgik = δl i, reaching (in a more expanded notation) the geodesic equation d2ql du2 + dqi du dq j du 1 2 gklh ∂gkj ∂qi + ∂gik ∂q j −∂gi j ∂qk i = 0. (22.61) Comparing with the formula for the Christoffel symbol, Eq. (4.63), we can rewrite Eq. (22.61) as d2ql du2 + dqi du dq j du 0l i j = 0. (22.62) Note that although E... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,105 | 7th_Mathematical_Methods_for_Physicists | 0 |
22.2 More General Variations 1105 Relation to Physics The calculus of variations as developed so far provides an elegant description of a wide variety of physical phenomena. The physics includes classical mechanics, as in Ex- amples 22.2.1 and 22.2.2; relativistic mechanics, Exercise 22.2.5; electrostatics, Exam- ple 2... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,106 | 7th_Mathematical_Methods_for_Physicists | 0 |
1106 Chapter 22 Calculus of Variations 22.2.5 Show that the Lagrangian L = m0c2 <unk> <unk>1 − s 1 −v2 c2 <unk> <unk>−V (r) leads to a relativistic form of Newton’s second law of motion, d dt m0vi p 1 −v2/c2 ! = Fi, in which the force components are Fi = −∂V/∂xi. 22.2.6 The Lagrangian for a particle with charge q in an... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,107 | 7th_Mathematical_Methods_for_Physicists | 0 |
22.3 Constrained Minima/Maxima 1107 principle to the Lagrangian density (the integrand), now with two independent vari- ables, leads to the classical wave equation, ∂2u ∂x2 = ρ τ ∂2u ∂t2 . 22.2.9 Show that the stationary value of the total energy of the electrostatic field of Example 22.2.3 is a minimum. Hint. Investig... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,108 | 7th_Mathematical_Methods_for_Physicists | 0 |
, z are actually independent, and in principle one could solve the constraint equation to obtain z as a function of x and y: z = z(x, y), after which one could obtain the desired minimum by setting to zero the derivatives ∂ ∂x f x, y, z(x, y) and ∂ ∂y f x, y, z(x, y) . However, it may be cumbersome, or in some cases ne... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,108 | 7th_Mathematical_Methods_for_Physicists | 0 |
1108 Chapter 22 Calculus of Variations Lagrangian Multipliers Continuing with our three-dimensional illustration in which we seek to minimize f (x, y, z) subject to the constraint g(x, y, z) = C, our starting point is that the constraint equation implies dg = ∂g ∂x yz dx + ∂g ∂y xz dy + ∂g ∂z xy dz = 0, where (as indic... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,109 | 7th_Mathematical_Methods_for_Physicists | 0 |
22.3 Constrained Minima/Maxima 1109 The generalization of Eqs. (22.66) to n variables and k constraints is ∂f ∂xi − k X j=1 λ j ∂g j ∂xi = 0, i = 1,2,...,n. (22.67) The n equations, Eqs. (22.67), contain n + k unknowns (the n xi and the k λ j), and they are to be solved subject also to the k constraint equations. In so... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,110 | 7th_Mathematical_Methods_for_Physicists | 0 |
rh) = 0, ∂S ∂h −λ∂V ∂h = 2πr −λπr2 = 0. Eliminating λ from these equations, we find h/r = 2. Because we have not also used the constraint equation, we get only the ratio of the two variables h and r (which is the information that is relevant for the present problem). However, if we specify the volume V (i.e., use the c... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,110 | 7th_Mathematical_Methods_for_Physicists | 0 |
1110 Chapter 22 Calculus of Variations Exercises 22.3.0 The following problems are to be solved by using Lagrangian multipliers. 22.3.1 The ground-state energy of a quantum particle of mass m in a pillbox (right-circular cylinder) is given by E = ̄h2 2m (2.4048)2 R2 + π2 H2 , in which R is the radius and H is the heigh... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,111 | 7th_Mathematical_Methods_for_Physicists | 0 |
volume of the ellipsoid is 2/π √ 3 ≈0.367. 22.3.7 Find the maximum value of the directional derivative of φ(x, y, z), dφ ds = ∂φ ∂x cosα + ∂φ ∂y cosβ + ∂φ ∂z cosγ, subject to the constraint, cos2 α + cos2 β + cos2 γ = 1. | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,111 | 7th_Mathematical_Methods_for_Physicists | 0 |
22.4 Variation with Constraints 1111 22.4 VARIATION WITH CONSTRAINTS As in earlier sections, we seek the path that will make the integral J = Z f yi, ∂yi ∂xj , xj dxj (22.68) stationary. This is the general case in which xj represents a set of independent variables and yi a set of dependent variables. Now, however, we ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,112 | 7th_Mathematical_Methods_for_Physicists | 0 |
Z φk yi, ∂yi ∂xj , xj dxj = constant. (22.72) The effect of this constraint can be brought to a form consistent with Eq. (22.71) by writing δ Z λk φk yi, ∂yi ∂xj , xj dxj = 0. (22.73) Note that in this equation λk does not depend on the xj but is simply a constant, as it is only the integral of φk that is required to b... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,112 | 7th_Mathematical_Methods_for_Physicists | 0 |
1112 Chapter 22 Calculus of Variations Section 22.3, and may use a formula analogous to Eq. (22.67). In our present notation, we obtain δ Z " f yi, ∂yi ∂xj , xj + X k λkφk yi, ∂yi ∂xj , xj # dxj = 0. (22.74) Remember that the Lagrangian multiplier λk may depend on the xj when φ(yi, xj) is given in the form of Eq. (22.6... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,113 | 7th_Mathematical_Methods_for_Physicists | 0 |
t variables. Usually the generalized coordinates qi are chosen to eliminate the forces of constraint, but this is not necessary and not always desirable. In the presence of holo- nomic constraints (those that can be expressed via mathematical expressions, e.g., φk = 0), Hamilton’s principle is δ Z " L(qi, ̇qi,t) + X k ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,113 | 7th_Mathematical_Methods_for_Physicists | 0 |
22.4 Variation with Constraints 1113 Usually the constraint is of the form φk = φk(qi,t), independent of the generalized veloci- ties ̇qi. In this case the coefficient aik is given by aik = ∂φk ∂qi . (22.78) Then aikλk (no summation) represents the force of the kth constraint in the ˆqi-direction, appearing in Eq. (22.... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,114 | 7th_Mathematical_Methods_for_Physicists | 0 |
1114 Chapter 22 Calculus of Variations Substituting from the equation of constraint (r = l, ̇r = 0), these equations become ml ̇θ2 + mg cosθ = −λ1, ml2 ̈θ + mgl sinθ = 0. (22.82) The second equation may be solved for θ(t) to yield simple harmonic motion if the am- plitude is small (sinθ ≈θ), whereas the first equation ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,115 | 7th_Mathematical_Methods_for_Physicists | 0 |
= 1, aθ1 = ∂φ1 ∂θ = 0, we reach m ̈r −mr ̇θ2 + mg cosθ = λ1(θ), mr2 ̈θ + 2mr ̇r ̇θ −mgr sinθ = 0. θ r FIGURE 22.10 A particle sliding on a cylindrical surface. | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,115 | 7th_Mathematical_Methods_for_Physicists | 0 |
22.4 Variation with Constraints 1115 We have chosen to identify the constraining force λ1 as a function of the angle θ, a valid choice since θ is a single-valued function of the independent variable t. Inserting the constrained values r = l, ̈r = ̇r = 0, these equations reduce to −ml ̇θ2 + mg cosθ = λ1(θ), (22.85) ml2 ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,116 | 7th_Mathematical_Methods_for_Physicists | 0 |
1116 Chapter 22 Calculus of Variations Example 22.4.3 SCHRÖDINGER WAVE EQUATION As a final illustration of a constrained minimum, let us find the Euler equations for the quantum mechanical problem of a particle of mass m subject to a potential V, δJ = Z ψ∗(r)Hψ(r)d3r, (22.90) with the constraint that ψ is the normalize... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,117 | 7th_Mathematical_Methods_for_Physicists | 0 |
ψ∗−∂ ∂x ∂g ∂ψ∗x −∂ ∂y ∂g ∂ψ∗y −∂ ∂z ∂g ∂ψ∗z = 0. This yields V ψ −λψ − ̄h2 2m (ψxx + ψyy + ψzz) = 0, or − ̄h2 2m ∇2ψ + V ψ = λψ. (22.95) | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,117 | 7th_Mathematical_Methods_for_Physicists | 0 |
22.4 Variation with Constraints 1117 The Euler equation for yi = ψ gives the complex conjugate of Eq. (22.95), and therefore provides no further information. Reference to Eq. (22.92) enables us to identify λ physi- cally as the energy of the quantum mechanical system. With this interpretation, Eq. (22.95) is the celebr... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,118 | 7th_Mathematical_Methods_for_Physicists | 0 |
.100) is equivalent to the earlier formulation because py2 x + qy2 is homoge- neous in y and the denominator normalizes y without otherwise changing its functional form. The J satisfying Eq. (22.100) evaluates to the eigenvalue λ. | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,118 | 7th_Mathematical_Methods_for_Physicists | 0 |
1118 Chapter 22 Calculus of Variations In the frequently occurring case that p(x) is actually independent of x, we can manipu- late the y2 x term in the integrand of J, causing Eqs. (22.96), (22.97), and (22.100) to assume the useful forms, δJ = δ b Z a <unk>−p yxx + q(x)y2 dx = 0, (constrained minimum), (22.101) p d2y... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,119 | 7th_Mathematical_Methods_for_Physicists | 0 |
eigenfunc- tions can be chosen to form a complete orthogonal set, that we can write the expansion y = c0y0 + ∞ X i=1 ci yi. (22.104) We shall assume that we picked y sensibly enough that it is not orthogonal to the ground state, so c0 <unk>= 0. Invoking the orthogonality property, Ey, the expectation value of the 9This... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,119 | 7th_Mathematical_Methods_for_Physicists | 0 |
22.4 Variation with Constraints 1119 energy for wave function y, is Ey = ⟨y|H|y⟩ ⟨y|y⟩ = ∞ X i=0 |ci|2λi ∞ X i=0 |ci|2 , (22.105) where H, the operator defining the Schrödinger equation, typically has the form H = − ̄h2 2m d2 dx2 + V (x). The Schrödinger equation and its approximate solution Ey are then seen to corresp... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,120 | 7th_Mathematical_Methods_for_Physicists | 0 |
verges to λ0 as our approximate eigenfunction y improves (ci →0). In practical problems in quantum mechanics, y often depends on parameters that may be varied to minimize Ey and thereby improve the estimate of the ground-state energy λ0. This is the “variational method” discussed in quantum mechanics texts. It was illu... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,120 | 7th_Mathematical_Methods_for_Physicists | 0 |
1120 Chapter 22 Calculus of Variations ̄h = 1) by the lowest-eigenvalue eigenstate of the Schrödinger equation, −1 2m d2ψ dx2 + kx2 2 ψ = Eψ, (22.107) subject to the boundary conditions ψ(0) = ψ(∞) = 0. A guessed wave function consis- tent with the boundary conditions is y(x) = xe−αx. Let’s find the value of α making t... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,121 | 7th_Mathematical_Methods_for_Physicists | 0 |
k/m . The approximate wave functions of this example are compared with the exact wave func- tion in Fig. 22.11. Note that the second approximation yields an eigenvalue that is in error 2 Exact Exact Approx Approx 4 6 2 4 6 FIGURE 22.11 Exact and approximate ground-state wave functions for quantum oscillator, Example 22... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,121 | 7th_Mathematical_Methods_for_Physicists | 0 |
22.4 Variation with Constraints 1121 by less than 3%, even though the approximate wave function exhibits considerably larger relative errors. ■ Example 22.4.5 VARIATION OF LINEAR PARAMETERS A frequent use of the Rayleigh-Ritz technique is the approximation of an eigenfunction of a Schrödinger equation, Hψ(x) = Eψ(x), a... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,122 | 7th_Mathematical_Methods_for_Physicists | 0 |
by a finite matrix equation. ■ Exercises 22.4.1 A particle of mass m is on a frictionless horizontal surface. In terms of plane polar coordinates (r,θ), it is constrained to move so that θ = ωt (accomplished by pushing it with a rotating radial arm against which it can slide frictionlessly). With the initial conditions... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,122 | 7th_Mathematical_Methods_for_Physicists | 0 |
1122 Chapter 22 Calculus of Variations (a) Find the radial position as a function of time. ANS. r(t) = r0 coshωt. (b) Find the force exerted on the particle by the constraint. ANS. F(c) = 2m ̇rω = 2mr0ω2 sinhωt. 22.4.2 A point mass m is moving on a flat, horizontal, frictionless plane. The mass is con- strained by a st... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,123 | 7th_Mathematical_Methods_for_Physicists | 0 |
θθ −2r2 θ −r2 . Note. The problems of this section, variation subject to constraints, are often called isoperimetric. The term arose from problems of maximizing area subject to a fixed perimeter, as in part (a) of this problem. 22.4.6 Show that requiring J, given by J = b Z a b Z a K(x,t)φ(x)φ(t)dx dt, | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,123 | 7th_Mathematical_Methods_for_Physicists | 0 |
22.4 Variation with Constraints 1123 to have a stationary value subject to the normalizing condition b Z a φ2(x)dx = 1 leads to a Hilbert-Schmidt integral equation of the form given in Eq. (20.64). Note. The kernel K(x,t) is symmetric. 22.4.7 An unknown function satisfies the differential equation y′′ + π 2 2 y = 0 and... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,124 | 7th_Mathematical_Methods_for_Physicists | 0 |
1124 Chapter 22 Calculus of Variations with λ = 1 for the ground state; see Eq. (18.17). Take ψtrial = <unk> <unk> <unk> 1 −x2 a2 , x2 ≤a2 0, x2 > a2 for the ground-state wave function (with a2 an adjustable parameter) and calculate the corresponding ground-state energy. How much error do you have? Note. Your parabola ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,125 | 7th_Mathematical_Methods_for_Physicists | 0 |
Includes a discussion of sufficiency conditions for solutions of variational problems. Lanczos, C., The Variational Principles of Mechanics, 4th ed. Toronto: University of Toronto Press (1970), reprinted, Dover (1986). This book is a very complete treatment of variational principles and their applications to the develo... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,125 | 7th_Mathematical_Methods_for_Physicists | 0 |
CHAPTER 23 PROBABILITY AND STATISTICS Probabilities arise in many problems dealing with random events or large numbers of parti- cles defining random variables. An event is called random if it is practically impossible to predict from the initial state. This includes cases where we have merely incomplete infor- mation ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,126 | 7th_Mathematical_Methods_for_Physicists | 0 |
be presented. The material found here may be adequate to provide a conceptual basis for statistical mechanics, but can at best be an elementary introduction to the ideas needed to 1125 Mathematical Methods for Physicists. © 2013 Elsevier Inc. All rights reserved. | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,126 | 7th_Mathematical_Methods_for_Physicists | 0 |
1126 Chapter 23 Probability and Statistics gain maximum information from data-intensive experimental studies such as those arising from the study of cosmic rays or the data from high-energy particle accelerators. A more complete picture of the role of statistics in physics and engineering can be obtained from a number ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,127 | 7th_Mathematical_Methods_for_Physicists | 0 |
three possible values of x, which we designate x0, x1, x2, where we are now letting xi stand for the occurrence of i heads in the two tosses. Obviously, the only possible values of x are 0, 1, and 2. But we also know that the four possible results of two successive tosses are (heads, then heads), (heads, then tails), (... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,127 | 7th_Mathematical_Methods_for_Physicists | 0 |
23.1 Probability: Definitions, Simple Properties 1127 the smaller the spread in probability about 1/2 will be. Moreover, the more trials we run, the closer the average of all the individual trial probabilities will be to 1/2. We could even pick single grains and check if the probability 1/4 of picking two black grains ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,128 | 7th_Mathematical_Methods_for_Physicists | 0 |
points, then events meeting certain specifica- tions can be identified as subsets A, B,... of S, denoted as A ⊂S, etc. Two sets A, B are equal if A is contained in B, denoted A ⊂B, and B is contained in A, denoted B ⊂A. The union A ∪B consists of all points (events) that are in A or B or both (see Fig. 23.1). The inter... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,128 | 7th_Mathematical_Methods_for_Physicists | 0 |
1128 Chapter 23 Probability and Statistics A B FIGURE 23.1 The shaded area gives the intersection A ∩B, corresponding to the A and B event sets; the dashed line encloses A ∪B, corresponding to the event set A or B. The entire sample space has P(S) = 1. The probability of the union A ∪B of mutually exclusive events is t... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,129 | 7th_Mathematical_Methods_for_Physicists | 0 |
CONDITIONAL PROBABILITY Consider a box of 10 identical red and 20 identical blue pens, from which we remove pens successively in a random order without putting them back. Suppose we draw a red pen first, event R, followed by the draw of a blue pen, event B. One way to compute P(R, B) is to note that our sample space co... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,129 | 7th_Mathematical_Methods_for_Physicists | 0 |
23.1 Probability: Definitions, Simple Properties 1129 now 29 pens of which 20 are blue, we easily compute P(B|R) = 20/29, and the probability of the sequence “red, then blue” is P(R, B) = 10 30 20 29 = 20 87, (23.5) equal to the result we obtained previously. ■ The generalization of the result in Eq. (23.5) is the very... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,130 | 7th_Mathematical_Methods_for_Physicists | 0 |
who graduate, 97% have an SAT score of at least 1400 points, while 80% of those who drop out have an SAT score below 1400. Suppose a student has an SAT score below 1400. What is his/her probability of graduating? Let A represent all students with an SAT test score below 1400, and let B represent those with scores ≥1400... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,130 | 7th_Mathematical_Methods_for_Physicists | 0 |
1130 Chapter 23 Probability and Statistics Among the 5% of students who do not graduate, 80% are in set A and 20% are in set B, so P(A ∩ ̃C) = (0.05)(0.80) = 0.0400, P(B ∩ ̃C) = (0.05)(0.20) = 0.0100. Since P(A) = P(A ∩C) + P(A ∩ ̃C), and likewise for P(B), we have P(A) = 0.0285 + 0.0400 = 0.0685, P(B) = 0.9215 + 0.010... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,131 | 7th_Mathematical_Methods_for_Physicists | 0 |
components. This relation follows from the obvious decom- position B = ∪i(B ∩Ai) (this notation indicates the union of all the quantities B ∩Ai, see Fig. 23.2), which implies P(B) = P i P(B ∩Ai) because the components B ∩Ai are mu- tually exclusive. For each i, we know from Eq. (23.9) that P(B ∩Ai) = P(Ai)P(B|Ai), whic... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,131 | 7th_Mathematical_Methods_for_Physicists | 0 |
23.1 Probability: Definitions, Simple Properties 1131 B A1 A2 A3 FIGURE 23.2 The shaded area B is composed of mutually exclusive subsets of B belonging also to A1, A2, A3, where the Ai are mutually exclusive. Generalizing this, suppose there are n people but only k < n chairs to seat them. In how many ways can we seat ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,132 | 7th_Mathematical_Methods_for_Physicists | 0 |
shable (numbered) objects, and we place n1 of these into Box 1, n2 into Box 2, etc. We wish to know how many different ways this can be done (this is a combination problem because the objects in each box do not form ordered sets). A simple way to solve this problem is to identify each permutation of the n objects with ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,132 | 7th_Mathematical_Methods_for_Physicists | 0 |
1132 Chapter 23 Probability and Statistics of permutations) divided by n1!, n2!, etc. Thus, our overall formula is B(n1,n2,...) = n! n1!n2!.... (23.14) This quantity is sometimes referred to as a multinomial coefficient; if there were only two boxes it reduces to the binomial coefficient. For a related problem with rep... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,133 | 7th_Mathematical_Methods_for_Physicists | 0 |
al mechanics, we frequently need to know the number of ways in which it is possible to put n particles in k boxes subject to various additional specifications. If we are working in classical theory, our more complete specification includes the notion that the particles are distinguishable, and we refer to the probabili... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,133 | 7th_Mathematical_Methods_for_Physicists | 0 |
23.1 Probability: Definitions, Simple Properties 1133 any number of indistinguishable particles may be placed in the same box; in Fermi-Dirac statistics no box may contain more than one indistinguishable particle. Application of the various kinds of statistics in general problems is outside the scope of this text; howe... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,134 | 7th_Mathematical_Methods_for_Physicists | 0 |
(A ∪B), P(A ∪C), P(B ∪C), and P(A ∪B ∪C). 23.1.6 Determine directly or by mathematical induction (Section 1.4) the probability of a dis- tribution of N (Maxwell-Boltzmann) particles in k boxes with N1 in Box 1, N2 in Box 2,..., Nk in the kth box for any numbers N j ≥1 with N1 + N2 + ··· + Nk = N, k < N. Repeat this for... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,134 | 7th_Mathematical_Methods_for_Physicists | 0 |
1134 Chapter 23 Probability and Statistics 23.2 RANDOM VARIABLES In this section we define properties that characterize the probability distributions of random variables, by which we mean variables that will assume various numerical values with individual probabilities. Thus, the name of a color (e.g., “black” or “whit... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,135 | 7th_Mathematical_Methods_for_Physicists | 0 |
ity (1/6)2 +(1/6)2 = 1/18. Continuing, the value 4 is reached in three equally probable ways: 2 + 2, 3 + 1, and 1 + 3 with total probability 3(1/6)2 = 1/12; the values 5 and 6 are reached with the respective probabilities 4(1/6)2 = 1/9 and 5(1/6)2 = 5/36; and the value 7 occurs with the maximum probability, 6(1/6)2 = 1... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,135 | 7th_Mathematical_Methods_for_Physicists | 0 |
23.2 Random Variables 1135 P(x) x 6 36 5 36 4 36 3 36 2 36 1 36 0 1 2 3 4 5 6 7 8 9 10 11 12 FIGURE 23.3 Probability distribution P(x) of the sum of points when two dice are tossed. In summary, then, • If a discrete random variable X can assume the values xi, each value occurs by chance with a probability P(X = xi) = p... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,136 | 7th_Mathematical_Methods_for_Physicists | 0 |
1136 Chapter 23 Probability and Statistics The value of this integral can be checked by identifying it as a gamma function: ∞ Z 0 e−2r/ar2dr = a 2 3 ∞ Z 0 e−xx2dx = a3 8 0(3) = a3 4 . ■ Computing Discrete Probability Distributions In Example 23.2.1 the overall probability of a particular value of a discrete random vari... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,137 | 7th_Mathematical_Methods_for_Physicists | 0 |
nm) is just the multinomial coefficient encountered earlier; in the present context the numbered objects correspond to events numbered from 1 to N and each box corresponds to an individual-event outcome. Thus, our final formula for the probability of a distribution defined by n1, n2, etc., is P(n1,n2,...,nm) = N! n1!n2... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,137 | 7th_Mathematical_Methods_for_Physicists | 0 |
23.2 Random Variables 1137 is the key link of experimental data with probability theory. This observation and practical experience suggest defining the mean value for a discrete random variable X as ⟨X⟩≡ X i xi pi, (23.20) while defining the mean value for a continuous random variable x characterized by prob- ability d... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,138 | 7th_Mathematical_Methods_for_Physicists | 0 |
1138 Chapter 23 Probability and Statistics setting the derivative equal to zero yields 2P i(x −xi) = 0, or x = 1 n X i xi ≡ ̄x, that is, the arithmetic mean. The arithmetic mean has another important property: If we denote by vi = xi − ̄x the deviations, then P i vi = 0, that is, the sum of positive deviations equals t... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,139 | 7th_Mathematical_Methods_for_Physicists | 0 |
, so the spread in the xi has caused ⟨x2 i ⟩to increase. Example 23.2.3 STANDARD DEVIATION OF MEASUREMENTS From the measurements x1 = 7, x2 = 9, x3 = 10, x4 = 11, x5 = 13, we extract ̄x = 10 for the mean value and, using Eq. (23.23), σ = s (−3)2 + (−1)2 + 02 + 12 + 32 5 = 2.2361 for the standard deviation, or spread. ■ | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,139 | 7th_Mathematical_Methods_for_Physicists | 0 |
23.2 Random Variables 1139 There is yet another interpretation of the standard deviation, in terms of the sum of squares of measurement differences: X i<k (xi −xk)2 = 1 2 n X i=1 n X k=1 x2 i + x2 k −2xi xk = 1 2 h 2n2⟨x2⟩−2n2⟨x⟩2i = n2σ 2. (23.25) The last step in the above equation made use of Eq. (23.24). Now we are... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,140 | 7th_Mathematical_Methods_for_Physicists | 0 |
σ 2(Y) = ∞ Z −∞ (ax + b −a⟨X⟩−b)2 f (x)dx = ∞ Z −∞ a2(x −⟨X⟩)2 f (x)dx = a2σ 2(X). | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,140 | 7th_Mathematical_Methods_for_Physicists | 0 |
1140 Chapter 23 Probability and Statistics 3. Probabilities of random variables satisfy the Chebyshev inequality, P(|x −⟨X⟩| ≥kσ) ≤1 k2 , (23.29) which demonstrates why the standard deviation serves as a measure of the spread of an arbitrary probability distribution from its mean value ⟨X⟩. We first derive the simpler ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,141 | 7th_Mathematical_Methods_for_Physicists | 0 |
23.2 Random Variables 1141 Moments of Probability Distributions It is straightforward to generalize the mean value to higher moments of probability distri- butions relative to the mean value ⟨X⟩: D (X −⟨X⟩)kE = X j <unk>x j −⟨X⟩ k p j, discrete distribution, D (X −⟨X⟩)kE = ∞ Z −∞ (x −⟨X⟩)k f (x)dx, continuous distribut... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,142 | 7th_Mathematical_Methods_for_Physicists | 0 |
1142 Chapter 23 Probability and Statistics Setting t = 0, we get ⟨X⟩= M′(0) = 5, ⟨X2⟩= M′′(0) = 80 3 . Thus, the mean of X is found to be 5, and its variance is given by σ 2 = ⟨X2⟩−⟨X⟩2 = 80 3 −25 = 5 3. In this example we see that the moment-generating function does (in a systematic way) the same thing as direct forma... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,143 | 7th_Mathematical_Methods_for_Physicists | 0 |
23.2 Random Variables 1143 is a measure of how much the random variables X and Y are correlated (or related): It is zero for independent random variables because cov(X,Y) = Z (x −⟨X⟩)(y −⟨Y⟩) f (x, y)dx dy = Z (x −⟨X⟩) f (x)dx Z (y −⟨Y⟩) g(y)dy = (⟨X⟩−⟨X⟩)(⟨Y⟩−⟨Y⟩) = 0. The normalized covariance cov(X,Y)/σ(X)σ(Y), whic... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,144 | 7th_Mathematical_Methods_for_Physicists | 0 |
that the relation between them is linear. Our first step in proving this theorem is to show that P(Y = aX + b) = 1 (meaning that Y = aX + b) implies that cov(X,Y)/σ(X)σ(Y) = ±1. For the mean ⟨Y⟩, we simply compute ⟨Y⟩= ⟨aX + b⟩= a⟨X⟩+ b. For the variance, σ(Y)2 = ⟨Y 2⟩−⟨Y⟩2 = ⟨(aX + b)2⟩−(a⟨X⟩+ b)2 = a2⟨X2⟩+ 2ab⟨X⟩+ b2... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,144 | 7th_Mathematical_Methods_for_Physicists | 0 |
1144 Chapter 23 Probability and Statistics which is equivalent to σ(Y) = ±aσ(X). We also need cov(X,Y), which is cov(X,Y) = ⟨(X −⟨X⟩)((aX + b) −(a⟨X⟩+ b))⟩ = a ⟨X2⟩−⟨X⟩2 = aσ 2(X) = ±σ(X)σ(Y), where the last equality was obtained by identifying aσ(X) as ±σ(Y). This result completes the first step in our proof of the th... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,145 | 7th_Mathematical_Methods_for_Physicists | 0 |
reach D σ(Y)2(X −⟨X⟩)2 + σ(X)2(Y −⟨Y⟩)2 <unk>2σ(X)σ(Y)(X −⟨X⟩)(Y −⟨Y⟩) E . Making now the substitutions (X −⟨X⟩)2 = σ(X)2, (Y −⟨Y⟩)2 = σ(Y)2, and (X −⟨X⟩) (Y −⟨Y⟩) = ±σ(X)σ(Y), our quadratic expectation value reduces to zero. Marginal Probability Distributions It is sometimes useful to integrate out (i.e., average over... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,145 | 7th_Mathematical_Methods_for_Physicists | 0 |
23.2 Random Variables 1145 motivated by the geometric aspects of projection. It is straightforward to show that these marginal distributions satisfy all the requirements of properly normalized probability dis- tributions. Here is a comprehensive example that illustrates the computation of probability distri- butions an... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,146 | 7th_Mathematical_Methods_for_Physicists | 0 |
to 2 and with the three individual-event probabilities pa, pb, and pc. The number of events a is x, the number of events b is y, and therefore the number of events c is 2 −x −y, and, by Eq. (23.18), P(x, y) = 2! x! y!(2 −x −y)!(pa)x(pb)y(pc)2−x−y = 2! x! y!(2 −x −y)! 5 13 x 2 13 y 6 13 2−x−y , (23.39) with 0 ≤x + y ≤2.... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,146 | 7th_Mathematical_Methods_for_Physicists | 0 |
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