# Physics 505 Homework No. 5 Solutions S Angular momentum uncertainty relations. A system is in the lm eigenstate of L 2, L z.

Save this PDF as:

Size: px
Start display at page:

Download "Physics 505 Homework No. 5 Solutions S5-1. 1. Angular momentum uncertainty relations. A system is in the lm eigenstate of L 2, L z."

## Transcription

1 Physics 55 Homewok No. 5 s S5-. Angula momentum uncetainty elations. A system is in the lm eigenstate of L 2, L z. a Show that the expectation values of L ± = L x ± il y, L x, and L y all vanish. ψ lm L ± ψ lm = h ll + mm ± ψ lm ψ l,m± =, since ψ lm ψ l,m± ae othogonal. Note in the special cases m = ±l fo L ±, the squae oot kills the expectation value! L x = L + + L /2 and L y = il + L /2, so these expectations must be zeo as well. b Detemine L i = L 2 i L i 2. Veify the genealized uncetainty elation holds fo all pais of angula momentum components. Comment on L x L y fo the cases m = and m = l. L z = m h, Lz 2 = m2 h 2, L z =. The state ψ lm is symmetic in x and y, so Lx 2 = L 2 y = L 2 L 2 z /2 = h 2 ll + m 2 /2. L x = L y = h ll + m 2 /2. L x L z = [L x, L z ] /2 = i hl y =. This is not vey exciting! Similaly, L y L z =, also not exciting. Howeve, L x L y = h 2 ll + m 2 /2 [L x, L y ] /2 = i hl z = h 2 m/2. In the case m =, thee is no angula momentum about the z-axis. The paticle is found as close to the z-axis as possible. In this case the minimum uncetainty poduct fo the x and y components is and this is well satisfied since all the angula momentum is in the x and y components but with vanishing expectation value. The only case whee the equality sign would be used in the uncetainty elation is the case l = in which case all components and thei squaes ae zeo. The case m = l coesponds to the maximum angula momentum component along the z-axis. One might visualize the paticle in the xy-plane otating about the z-axis. Of couse, it can t be exactly in the xy-plane and its out of plane motion poduces some components of L x and L y which aveage to, but have some spead aound the aveage. The uncetainty elation becomes L x L y = h 2 ll + l 2 /2 = h 2 l/2 = [L x, L y ] /2 = i hl z = h 2 l/2. In this case, the angula momentum components in the x and y diections have the minimum possible uncetainty poduct.

2 Physics 55 Homewok No. 5 s S Fun with angula momentum commutatos. a Suppose the vecto opeatos A and B commute with each othe and L. Show that [A L, B L] = i ha B L. We use the summation convention and the completely anti-symmetic tenso and just gind it out! [A L, B L] = A i L i B j L j B j L j A i L i = A i B j L i L j L j L i = A i B j [L i, L j ] = A i B j i hǫ ijk L k = A i B j i hǫ ikj L k = +A i B j i hǫ kij L k = i ha B k L k = i ha B L. b Suppose V is a vecto opeato which might be a function of x and p, so it doesn t necessaily commute with L. Show that [L 2, V ] = 2i hv L i hv. You might need the elation deived in lectue fo a vecto opeato, [L i, V j ] = i hǫ ijk V k. Again, ginding it out woks: [L 2, V ] = L i jl j V i V i L j L j = L j V i L j + L j [L j, V i ] L j V i L j + [L j, V i ]L j = i h L j ǫ jik V k + ǫ jik V k L j = i h ǫ jik L j V k + ǫ jik V k L j = i h ǫ jik V k L j + ǫ jik [L j, V k ] + ǫ jik V k L j = i h 2ǫ jik V k L j + i hǫ jik ǫ jkl V l = i h 2ǫ ijk V k L j i hǫ ijk ǫ jkl V l = i h +2ǫ ikj V k L j + i hǫ ijk ǫ jlk V l = i h +2ǫ ikj V k L j i hǫ ijk ǫ ljk V l = i h +2ǫ ikj V k L j 2i hδ il V l = i h +2ǫ ikj V k L j 2i hv i = 2i h V L i hv i.

3 Physics 55 Homewok No. 5 s S5- Note that we used the fact that ǫ ijk ǫ ljk = 2δ il. This is easy to see. Pick an i, say. Then the only non-zeo values of ǫ ijk ae those with j, k = 2, o j, k =, 2. In the second tenso, the only non-zeo values will occu fo l =, the sign will be the same as the fist, and thee ae two contibutions.. Classically, a paticle moving in a spheically symmetic potential has the Hamiltonian H = p2 2m + L2 2m 2 + V, whee p = p/. Fo quantum mechanics, we must define with the Hemitian opeato p = 2 p + p p = h i +,. 2 Show the opeato defined in equation is the same as that in equation 2. Show that p in equation 2 is Hemitian conside ψ, θ, φ and ϕ, θ, φ and that when used in the Hamilton, p of equation 2 gives the coect Schoedinge equation. Also show that the opeato h/i/ is not Hemitian! = e. The momentum opeato in spheical coodinates is h i = h i e + e θ θ + e φ sin θ φ The fist tem in equation becomes h/2i/. The second tem is a bit moe wok. It s most easily evaluated in a mix of Catesian and spheical coodinates. Note that the momentum opeato in the second tem opeates on, /, and whateve might be to the ight. When it opeates on whateve is to the ight, we get a tem that s the same as the fist tem. So let s just evaluate the second tem when the p opeates on and /. 2 p = h 2i x x + y y + x z = h 2i x2 + y2 + z2 = h 2i x2 + y 2 + z 2 = h i,.

4 Physics 55 Homewok No. 5 s S5-4 and putting this togethe with the two othe pieces, we see that indeed, the opeato in equation is the same as that in equation 2. The adjoint of p is We conside p ϕ ψ = 2 d π sin θ dθ p = h i 2π + dφ h i. + ϕ, θ, φ ψ, θ, φ. The angula integations can be done independently of the adial integation, so we just assume they have been done o will be done and dop the angula vaiables fom the discussion. We ll use an integation by pats to move the deivative fom the ϕ to ψ. The deivative also acts on 2. p ϕ ψ = h i + ϕ ψ 2 d = h ϕ i ψ 2 d + ϕ ψ 2 d = h ϕ ψ 2 ϕ ψ i 2 d ϕ ψ 2 d + ϕ ψ 2 d = h + ϕ ψ i 2 d + ϕ 2 ψ 2 d ϕ ψ 2 d h = ϕ i + ψ 2 d = ϕ p ψ, and p is Hemitian. Note that the evaluation of the limits gives since the integand must convege at both ends! If we do the same calculation with p bogus = h/i/ we find p bogus ϕ ψ ϕ p bogus ψ = h i which is not zeo in geneal, so p bogus is not Hemitian. Finally, the Laplacian, in spheical coodinates, is 2 = sin θ ϕ 2 ψ θ sinθ θ + 2 sin 2 θ, 2 φ 2.

5 Physics 55 Homewok No. 5 s S5-5 If we multiply this by h 2, we get p 2 fo the Schoedinge equation expessed in spheical coodinates. We ve aleady seen in lectue that the angula pat is L 2 / 2. So it emains to show that the fist tem times h 2 is the same as p 2. The fist tem is h = h Squaing p, p 2 = h2 + + = h = h What if we like x instead of z? a Find the eigenfunction, ψ, of L 2 and L x with eigenvalues 2 h 2 and h, espectively. Since the eigenvalue of L 2 is 2 h 2, the eigenfunction has l =. The eigenfunctions of L z with l = ae Y,+ = sin θ e+iφ = Y, = + 4π cos θ = + 4π Y, = + sin θ e iφ = + x + iy z x iy. If we otate the coodinate system about the y-axis so the new x-axis is along the old z-axis then thee will be one unit of angula momentum along the new x-axis. This means eplacing z by x, y by y, and x by z. With these eplacements, the eigenfunction Y,+ will have one unit of angula momentum along the x-axis and of couse, it will still be an eigenfunction of L 2 with eigenvalue 2 h 2. The desied eigenfunction is then ψ = z + iy = cos θ + i sin θ sin φ.

6 Physics 55 Homewok No. 5 s S5-6 We can check this by opeating with L x in pola coodinates. L x ψ = h i sin φ cos φ cotθ θ φ cos θ + i sin θ sin φ = h sin φ sin θ i sin 2 φ cos θ i cos θ cos 2 φ i = h cos θ + i sin θ sin φ = hψ. b Expess the ψ just found as a linea combination of eigenfunctions of L 2 and L z. Of couse, if we want to wite ψθ, φ = +l l= m= l c lm Y lm θ, φ, we find the c lm by pojecting ψ onto the Y lm, c lm = π sin θ dθ 2π dφ Ylm θ, φ ψθ, φ, but in this case, it s easie just to look at ψ and the Y,+, Y,, and Y, and decide what we need: ψ = 2 Y,+ + 2 Y, + 2 Y,. The sum of the squaes of the coefficients is, as it should be. 5. Algeba all the way. We used algebaic techniques in lectue to deduce that L 2 and L z could be simultaneously diagonalized that is, eigenfunctions could be eigenfunctions of L 2 and L z at the same time and that the eigenvalues ae ll+ h 2 and m h, espectively, with l and m integes o half integes and with l and m = l, l +,...l, l. We then abandoned the algebaic technique and solved a diffeential equation to find the obital angula momentum eigenfunctions. Hee we will outline the use of algebaic techniques to deduce the obital angula momentum eigenfunctions. We stat by intoducing x ± = x±iy.

7 Physics 55 Homewok No. 5 s S5-7 a Show that the following commutation elations hold you may use elations aleady deived in lectue: [L z, x ± ] = ± hx ± [L ±, x ± ] = [L ±, x ] = ±2 hz [L 2, x + ] = 2 hx + L z + 2 h 2 x + 2 hzl +. [L z, x] = i hy and [L z, iy] = ii hx = hx, so [L z, x ± iy] = h±x + iy = ± hx ±. [L ±, x ± ] = [L x ± il y, x ± iy] = ±i[l x, y] ± i[l y, x] = ± hz + ± hz =. [L ±, x ] = [L x ± il y, x iy] = i[l x, y] ± i[l y, x] = +± hz + ± hz = ±2 hz. Fo the final commutato, we use the following, [AB, C] = ABC BAC = ACB + A[B, C] [C, A]B ACB = A[B, C] + [A, C]B. Also, fom lectue L 2 = L L + + hl z + L 2 z. So [L L +, x + ] = L [L +, x + ] + [L, x + ]L + = 2 hzl +, h[l z, x + ] = h 2 x +, [L 2 z, x +] = L z [L z, x + ] + [L z, x + ]L z = hl z x + + hx + L z = hx + L z + h 2 x + + hx + L z, and adding it all up, we have [L 2, x + ] = 2 hx + L z + 2 h 2 x + 2 hzl +. b Show that L z x + l, l = x + L z l, l + hx + l, l = hl + x + l, l, and L 2 x + l, l = h 2 ll + x + l, l + 2 h 2 l + x + l, l = h 2 l + l + 2x + l, l. This means that x + is the ladde o aising opeato fo states in which m = l.

8 Physics 55 Homewok No. 5 s S5-8 This is a piece of cake afte pat a. L z x + l, l = x + L z l, l + [L z, x + ] l, l = x + L z l, l + hx + l, l = x + l + l, l. Similaly, L 2 x + l, l = x + L 2 l, l + [L 2, x + ] l, l = x + L 2 l, l + 2 hx + L z + 2 h 2 x + 2 hzl + l, l = x + h 2 ll + + 2l l, l = h 2 l + l + 2x + l, l. So, any state l, m can be found by applying the opeato x + to the state, l times and then applying L l m times. l, m = CL l m xl +,, whee C is a nomalization constant. We can show that, is independent of angle. L, =, so otating the state with U δϕ intoduced in lectue just gives back,. This means,, must be a constant. c Detemine l, l equivalently, Y ll θ, φ up to a phase using the x ±. Hint: commutes with L, L 2, and x, so it is just a constant as fa as all these opeatos ae concened. l, l = Cx l + = Cx + iyl = C sin θ cos φ + i sin θ sin φ l = C l sin l θe ilφ, whee C is a nomalization constant to be detemined and we have eplaced the constant, by which will be omitted in the subsequent discussion. 2π l, l l, l = = C 2 2l dφ π + sin 2l θ sin θ dθ = 2π C 2 2l 2 l dx.

9 Physics 55 Homewok No. 5 s S5-9 One can look up the integal o integate by pats l times. + 2 l dx = 2 l x + = 2l + + 2l 2 l x 2 dx 2 + l x + 22 ll + 2 l 2 x 5 = 22 ll 5 = We deduce that, up to a phase, and + 2 ll l l l! 5 2l 2 l l! = 2 5 2l + 2 l l! = 2 2l +!!. + + x 2l dx C = l 2l +!! 4π 2 l l! 2l +!! l, l = sin l θ e ilφ. 4π 2 l l! + 2 l x 6 dx, 2 l 2 x 4 dx Note that this is almost Y ll θ, φ. It s missing l, but this phase is detemined by convention! Fom hee one could go on to use L to detemine up to a phase all the angula momentum eigenfunctions fo intege l. The nomalization constants wee given in lectue. Howeve, it s unlikely that this will lead to new insights, so this poblem ends hee!

10 Physics 55 Homewok No. 5 s S5- Appendix. Since we wee somewhat ushed with the coveage of P lm s and Y lm s, I include a few items hee. Much moe can be found in any efeence on mathematical functions such as Abamowitz and Stegun o any quantum text. Associated Legende equation. Afte sepaation of vaiables θ and φ so the solutions fo φ ae exp±imφ the equation fo θ becomes sin θ + ll + m2 sin θ θ θ sin 2 fθ =. θ This is the associated Legende equation and it s customay to change the vaiable to µ = cos θ. µ 2 d2 dµ 2 2µ d m2 + ll + dµ µ 2 P lm µ =, whee the non-singula at µ = ± solution has been witten as P lm µ which is known as an associated Legende function. When m =, the equation is known as the Legende equation with egula solutions P l µ = P l µ which ae Legende polynomials. P l µ = l 2 l l! l d µ 2 l. dµ P l µ is an l th ode polynomial in µ and is eithe even o odd depending on whethe l is even o odd. The nomalization be caeful when consulting efeences, not eveyone uses the same is P l =. The associated Legende functions ae given by m, m l+m d P lm µ = µ 2 m/2 P l µ = l d µ 2 m/2 µ 2 l. dµ 2 l l! dµ Fo m = l, the associated Legende function is paticulaly simple. The fathest ight facto is a polynomial in l in which the highest powe is µ 2 l Since the polynomial is diffeentiated 2l times, only this tem suvives. The l cancels the l in font. The l deivatives poduce a facto 2l! which combined with the othe factos in font poduces 2l 2l 2l 5 which is often abbeviated 2l!! whee!! is ead double factoial. So P ll µ = 2l!! µ 2 l/2 = 2l!! sin l θ. The associated Legende polynomials with diffeent l, but the same m, ae othogonal on the inteval to +, + dµ P lm µp l mµ = 2 2l + l + m! l m! δ ll. The nomalized, othogonal, complete eigenfunctions of L 2 and L z fo obital angula momentum ae the spheical hamonics which ae defined in tems of the associated Legende functions fo the pola angle and the azimuthal wave fo the azimuthal angle. Fo m, these ae Y lm θ, φ = m 2l + 4π l m! l + m! P lmcos θe imφ,

11 Physics 55 Homewok No. 5 s S5- and fo negative m, we use The otho-nomality elation is Y l, m θ, φ = m Ylm θ, φ. π sin θ dθ 2π dφ Y lm θ, φy l m θ, φ = δ ll δ mm. The completeness elation is +l l= m= l Some of the low ode spheical hamonics ae, Y lm θ, φy lmθ, φ = sin θ δθ θ δφ φ. Y = + 4π Y = + 4π cos θ Y = sin θeiφ 5 Y 2 = + cos 2 θ 6π 5 Y 2 = sin θ cos θeiφ 5 Y 22 = + 2π sin2 θe 2iφ 7 Y = + 5 cos θ cosθ 6π 2 Y = 64π sinθ 5 cos 2 θ e iφ 5 Y 2 = + 2π sin2 θ cos θe 2iφ 5 Y = 64π sin θe iφ

### Hour Exam No.1. p 1 v. p = e 0 + v^b. Note that the probe is moving in the direction of the unit vector ^b so the velocity vector is just ~v = v^b and

Hou Exam No. Please attempt all of the following poblems befoe the due date. All poblems count the same even though some ae moe complex than othes. Assume that c units ae used thoughout. Poblem A photon

### Vector Calculus: Are you ready? Vectors in 2D and 3D Space: Review

Vecto Calculus: Ae you eady? Vectos in D and 3D Space: Review Pupose: Make cetain that you can define, and use in context, vecto tems, concepts and fomulas listed below: Section 7.-7. find the vecto defined

### LINES AND TANGENTS IN POLAR COORDINATES

LINES AND TANGENTS IN POLAR COORDINATES ROGER ALEXANDER DEPARTMENT OF MATHEMATICS 1. Pola-coodinate equations fo lines A pola coodinate system in the plane is detemined by a point P, called the pole, and

### So we ll start with Angular Measure. Consider a particle moving in a circular path. (p. 220, Figure 7.1)

Lectue 17 Cicula Motion (Chapte 7) Angula Measue Angula Speed and Velocity Angula Acceleation We ve aleady dealt with cicula motion somewhat. Recall we leaned about centipetal acceleation: when you swing

### 2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,

3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects

### Gravitation and Kepler s Laws Newton s Law of Universal Gravitation in vectorial. Gm 1 m 2. r 2

F Gm Gavitation and Keple s Laws Newton s Law of Univesal Gavitation in vectoial fom: F 12 21 Gm 1 m 2 12 2 ˆ 12 whee the hat (ˆ) denotes a unit vecto as usual. Gavity obeys the supeposition pinciple,

### Transformations in Homogeneous Coordinates

Tansfomations in Homogeneous Coodinates (Com S 4/ Notes) Yan-Bin Jia Aug, 6 Homogeneous Tansfomations A pojective tansfomation of the pojective plane is a mapping L : P P defined as u a b c u au + bv +

### Physics 235 Chapter 5. Chapter 5 Gravitation

Chapte 5 Gavitation In this Chapte we will eview the popeties of the gavitational foce. The gavitational foce has been discussed in geat detail in you intoductoy physics couses, and we will pimaily focus

### Solutions to Homework Set #5 Phys2414 Fall 2005

Solution Set #5 1 Solutions to Homewok Set #5 Phys414 Fall 005 Note: The numbes in the boxes coespond to those that ae geneated by WebAssign. The numbes on you individual assignment will vay. Any calculated

### Functions of a Random Variable: Density. Math 425 Intro to Probability Lecture 30. Definition Nice Transformations. Problem

Intoduction One Function of Random Vaiables Functions of a Random Vaiable: Density Math 45 Into to Pobability Lectue 30 Let gx) = y be a one-to-one function whose deiatie is nonzeo on some egion A of the

### Chapter 22. Outside a uniformly charged sphere, the field looks like that of a point charge at the center of the sphere.

Chapte.3 What is the magnitude of a point chage whose electic field 5 cm away has the magnitude of.n/c. E E 5.56 1 11 C.5 An atom of plutonium-39 has a nuclea adius of 6.64 fm and atomic numbe Z94. Assuming

### Seventh Edition DYNAMICS 15Ferdinand P. Beer E. Russell Johnston, Jr. Lecture Notes: J. Walt Oler Texas Tech University

Seenth 15Fedinand P. ee E. Russell Johnston, J. Kinematics of Lectue Notes: J. Walt Ole Texas Tech Uniesity Rigid odies CHPTER VECTOR MECHNICS FOR ENGINEERS: YNMICS 003 The McGaw-Hill Companies, Inc. ll

### Notes on Electric Fields of Continuous Charge Distributions

Notes on Electic Fields of Continuous Chage Distibutions Fo discete point-like electic chages, the net electic field is a vecto sum of the fields due to individual chages. Fo a continuous chage distibution

### 2. SCALARS, VECTORS, TENSORS, AND DYADS

2. SCALARS, VECTORS, TENSORS, AND DYADS This section is a eview of the popeties of scalas, vectos, and tensos. We also intoduce the concept of a dyad, which is useful in MHD. A scala is a quantity that

### Episode 401: Newton s law of universal gravitation

Episode 401: Newton s law of univesal gavitation This episode intoduces Newton s law of univesal gavitation fo point masses, and fo spheical masses, and gets students pactising calculations of the foce

### 12. Rolling, Torque, and Angular Momentum

12. olling, Toque, and Angula Momentum 1 olling Motion: A motion that is a combination of otational and tanslational motion, e.g. a wheel olling down the oad. Will only conside olling with out slipping.

### Chapter 13 Gravitation. Problems: 1, 4, 5, 7, 18, 19, 25, 29, 31, 33, 43

Chapte 13 Gavitation Poblems: 1, 4, 5, 7, 18, 19, 5, 9, 31, 33, 43 Evey object in the univese attacts evey othe object. This is called gavitation. We e use to dealing with falling bodies nea the Eath.

### Mechanics 1: Motion in a Central Force Field

Mechanics : Motion in a Cental Foce Field We now stud the popeties of a paticle of (constant) ass oving in a paticula tpe of foce field, a cental foce field. Cental foces ae ve ipotant in phsics and engineeing.

### Gauss Law. Physics 231 Lecture 2-1

Gauss Law Physics 31 Lectue -1 lectic Field Lines The numbe of field lines, also known as lines of foce, ae elated to stength of the electic field Moe appopiately it is the numbe of field lines cossing

### Nuclear models: Fermi-Gas Model Shell Model

Lectue Nuclea mode: Femi-Gas Model Shell Model WS/: Intoduction to Nuclea and Paticle Physics The basic concept of the Femi-gas model The theoetical concept of a Femi-gas may be applied fo systems of weakly

### Fluids Lecture 15 Notes

Fluids Lectue 15 Notes 1. Unifom flow, Souces, Sinks, Doublets Reading: Andeson 3.9 3.12 Unifom Flow Definition A unifom flow consists of a velocit field whee V = uî + vĵ is a constant. In 2-D, this velocit

### Coordinate Systems L. M. Kalnins, March 2009

Coodinate Sstems L. M. Kalnins, Mach 2009 Pupose of a Coodinate Sstem The pupose of a coodinate sstem is to uniquel detemine the position of an object o data point in space. B space we ma liteall mean

### Review Module: Dot Product

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Depatment of Physics 801 Fall 2009 Review Module: Dot Poduct We shall intoduce a vecto opeation, called the dot poduct o scala poduct that takes any two vectos and

### NURBS Drawing Week 5, Lecture 10

CS 43/585 Compute Gaphics I NURBS Dawing Week 5, Lectue 1 David Been, William Regli and Maim Pesakhov Geometic and Intelligent Computing Laboato Depatment of Compute Science Deel Univesit http://gicl.cs.deel.edu

### Basic Quantum Mechanics

Basic Quantum Mechanics Postulates of QM - The state of a system with n position variables q, q, qn is specified by a state (or wave) function Ψ(q, q, qn) - To every observable (physical magnitude) there

### Forces & Magnetic Dipoles. r r τ = μ B r

Foces & Magnetic Dipoles x θ F θ F. = AI τ = U = Fist electic moto invented by Faaday, 1821 Wie with cuent flow (in cup of Hg) otates aound a a magnet Faaday s moto Wie with cuent otates aound a Pemanent

### Power and Sample Size Calculations for the 2-Sample Z-Statistic

Powe and Sample Size Calculations fo the -Sample Z-Statistic James H. Steige ovembe 4, 004 Topics fo this Module. Reviewing Results fo the -Sample Z (a) Powe and Sample Size in Tems of a oncentality Paamete.

### Mechanics 1: Work, Power and Kinetic Energy

Mechanics 1: Wok, Powe and Kinetic Eneg We fist intoduce the ideas of wok and powe. The notion of wok can be viewed as the bidge between Newton s second law, and eneg (which we have et to define and discuss).

### Samples of conceptual and analytical/numerical questions from chap 21, C&J, 7E

CHAPTER 1 Magnetism CONCEPTUAL QUESTIONS Cutnell & Johnson 7E 3. ssm A chaged paticle, passing though a cetain egion of space, has a velocity whose magnitude and diection emain constant, (a) If it is known

### Revision Guide for Chapter 11

Revision Guide fo Chapte 11 Contents Student s Checklist Revision Notes Momentum... 4 Newton's laws of motion... 4 Gavitational field... 5 Gavitational potential... 6 Motion in a cicle... 7 Summay Diagams

### UNIT CIRCLE TRIGONOMETRY

UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + = - - -

### Carter-Penrose diagrams and black holes

Cate-Penose diagams and black holes Ewa Felinska The basic intoduction to the method of building Penose diagams has been pesented, stating with obtaining a Penose diagam fom Minkowski space. An example

### Figure 2. So it is very likely that the Babylonians attributed 60 units to each side of the hexagon. Its resulting perimeter would then be 360!

1. What ae angles? Last time, we looked at how the Geeks intepeted measument of lengths. Howeve, as fascinated as they wee with geomety, thee was a shape that was much moe enticing than any othe : the

### 9. Mathematics Practice Paper for Class XII (CBSE) Available Online Tutoring for students of classes 4 to 12 in Physics, Chemistry, Mathematics

Available Online Tutoing fo students of classes 4 to 1 in Physics, 9. Mathematics Class 1 Pactice Pape 1 3 1. Wite the pincipal value of cos.. Wite the ange of the pincipal banch of sec 1 defined on the

### Skills Needed for Success in Calculus 1

Skills Needed fo Success in Calculus Thee is much appehension fom students taking Calculus. It seems that fo man people, "Calculus" is snonmous with "difficult." Howeve, an teache of Calculus will tell

### Lab M4: The Torsional Pendulum and Moment of Inertia

M4.1 Lab M4: The Tosional Pendulum and Moment of netia ntoduction A tosional pendulum, o tosional oscillato, consists of a disk-like mass suspended fom a thin od o wie. When the mass is twisted about the

### 2008 Quarter-Final Exam Solutions

2008 Quate-final Exam - Solutions 1 2008 Quate-Final Exam Solutions 1 A chaged paticle with chage q and mass m stats with an initial kinetic enegy K at the middle of a unifomly chaged spheical egion of

### Symmetric polynomials and partitions Eugene Mukhin

Symmetic polynomials and patitions Eugene Mukhin. Symmetic polynomials.. Definition. We will conside polynomials in n vaiables x,..., x n and use the shotcut p(x) instead of p(x,..., x n ). A pemutation

### The force between electric charges. Comparing gravity and the interaction between charges. Coulomb s Law. Forces between two charges

The foce between electic chages Coulomb s Law Two chaged objects, of chage q and Q, sepaated by a distance, exet a foce on one anothe. The magnitude of this foce is given by: kqq Coulomb s Law: F whee

### L19 Geomagnetic Field Part I

Intoduction to Geophysics L19-1 L19 Geomagnetic Field Pat I 1. Intoduction We now stat the last majo topic o this class which is magnetic ields and measuing the magnetic popeties o mateials. As a way o

### 4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first non-zero digit to

. Simplify: 0 4 ( 8) 0 64 ( 8) 0 ( 8) = (Ode of opeations fom left to ight: Paenthesis, Exponents, Multiplication, Division, Addition Subtaction). Simplify: (a 4) + (a ) (a+) = a 4 + a 0 a = a 7. Evaluate

### PY1052 Problem Set 8 Autumn 2004 Solutions

PY052 Poblem Set 8 Autumn 2004 Solutions H h () A solid ball stats fom est at the uppe end of the tack shown and olls without slipping until it olls off the ight-hand end. If H 6.0 m and h 2.0 m, what

### Today in Physics 217: multipole expansion

Today in Physics 17: multipole expansion Multipole expansions Electic multipoles and thei moments Monopole and dipole, in detail Quadupole, octupole, Example use of multipole expansion as appoximate solution

### mv2. Equating the two gives 4! 2. The angular velocity is the angle swept per GM (2! )2 4! 2 " 2 = GM . Combining the results we get !

Chapte. he net foce on the satellite is F = G Mm and this plays the ole of the centipetal foce on the satellite i.e. mv mv. Equating the two gives = G Mm i.e. v = G M. Fo cicula motion we have that v =!

### Chapter F. Magnetism. Blinn College - Physics Terry Honan

Chapte F Magnetism Blinn College - Physics 46 - Tey Honan F. - Magnetic Dipoles and Magnetic Fields Electomagnetic Duality Thee ae two types of "magnetic chage" o poles, Noth poles N and South poles S.

### 2. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES

. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES In ode to etend the definitions of the si tigonometic functions to geneal angles, we shall make use of the following ideas: In a Catesian coodinate sstem, an

### Week 3-4: Permutations and Combinations

Week 3-4: Pemutations and Combinations Febuay 24, 2016 1 Two Counting Pinciples Addition Pinciple Let S 1, S 2,, S m be disjoint subsets of a finite set S If S S 1 S 2 S m, then S S 1 + S 2 + + S m Multiplication

### 12.1. FÖRSTER RESONANCE ENERGY TRANSFER

ndei Tokmakoff, MIT epatment of Chemisty, 3/5/8 1-1 1.1. FÖRSTER RESONNCE ENERGY TRNSFER Föste esonance enegy tansfe (FR) efes to the nonadiative tansfe of an electonic excitation fom a dono molecule to

### General Physics (PHY 2130)

Geneal Physics (PHY 130) Lectue 11 Rotational kinematics and unifom cicula motion Angula displacement Angula speed and acceleation http://www.physics.wayne.edu/~apetov/phy130/ Lightning Review Last lectue:

### Exam 3: Equation Summary

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Depatment of Physics Physics 8.1 TEAL Fall Tem 4 Momentum: p = mv, F t = p, Fext ave t= t f t= Exam 3: Equation Summay total = Impulse: I F( t ) = p Toque: τ = S S,P

### Trigonometry in the Cartesian Plane

Tigonomet in the Catesian Plane CHAT Algeba sec. 0. to 0.5 *Tigonomet comes fom the Geek wod meaning measuement of tiangles. It pimail dealt with angles and tiangles as it petained to navigation astonom

### Classical Mechanics (CM):

Classical Mechanics (CM): We ought to have some backgound to aeciate that QM eally does just use CM and makes one slight modification that then changes the natue of the oblem we need to solve but much

### Gravitation. AP Physics C

Gavitation AP Physics C Newton s Law of Gavitation What causes YOU to be pulled down? THE EARTH.o moe specifically the EARTH S MASS. Anything that has MASS has a gavitational pull towads it. F α Mm g What

### PHYSICS 111 HOMEWORK SOLUTION #13. May 1, 2013

PHYSICS 111 HOMEWORK SOLUTION #13 May 1, 2013 0.1 In intoductoy physics laboatoies, a typical Cavendish balance fo measuing the gavitational constant G uses lead sphees with masses of 2.10 kg and 21.0

### PHYSICS 111 HOMEWORK SOLUTION #5. March 3, 2013

PHYSICS 111 HOMEWORK SOLUTION #5 Mach 3, 2013 0.1 You 3.80-kg physics book is placed next to you on the hoizontal seat of you ca. The coefficient of static fiction between the book and the seat is 0.650,

### Exam I. Spring 2004 Serway & Jewett, Chapters 1-5. Fill in the bubble for the correct answer on the answer sheet. next to the number.

Agin/Meye PART I: QUALITATIVE Exam I Sping 2004 Seway & Jewett, Chaptes 1-5 Assigned Seat Numbe Fill in the bubble fo the coect answe on the answe sheet. next to the numbe. NO PARTIAL CREDIT: SUBMIT ONE

### TALLINN UNIVERSITY OF TECHNOLOGY, INSTITUTE OF PHYSICS 14. NEWTON'S RINGS

4. NEWTON'S RINGS. Obective Detemining adius of cuvatue of a long focal length plano-convex lens (lage adius of cuvatue).. Equipment needed Measuing micoscope, plano-convex long focal length lens, monochomatic

### Physics 111 Fall 2007 Electrostatic Forces and the Electric Field - Solutions

Physics 111 Fall 007 Electostatic Foces an the Electic Fiel - Solutions 1. Two point chages, 5 µc an -8 µc ae 1. m apat. Whee shoul a thi chage, equal to 5 µc, be place to make the electic fiel at the

### Chapter 17 The Kepler Problem: Planetary Mechanics and the Bohr Atom

Chapte 7 The Keple Poblem: Planetay Mechanics and the Boh Atom Keple s Laws: Each planet moves in an ellipse with the sun at one focus. The adius vecto fom the sun to a planet sweeps out equal aeas in

### In the lecture on double integrals over non-rectangular domains we used to demonstrate the basic idea

Double Integals in Pola Coodinates In the lectue on double integals ove non-ectangula domains we used to demonstate the basic idea with gaphics and animations the following: Howeve this paticula example

### The LCOE is defined as the energy price (\$ per unit of energy output) for which the Net Present Value of the investment is zero.

Poject Decision Metics: Levelized Cost of Enegy (LCOE) Let s etun to ou wind powe and natual gas powe plant example fom ealie in this lesson. Suppose that both powe plants wee selling electicity into the

### TORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION

MISN-0-34 TORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION shaft TORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION by Kiby Mogan, Chalotte, Michigan 1. Intoduction..............................................

### Voltage ( = Electric Potential )

V-1 Voltage ( = Electic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage is

### Chapter 5 Review - Part I

Math 17 Chate Review Pat I Page 1 Chate Review - Pat I I. Tyes of Polynomials A. Basic Definitions 1. In the tem b m, b is called the coefficient, is called the vaiable, and m is called the eonent on the

### Problems on Force Exerted by a Magnetic Fields from Ch 26 T&M

Poblems on oce Exeted by a Magnetic ields fom Ch 6 TM Poblem 6.7 A cuent-caying wie is bent into a semicicula loop of adius that lies in the xy plane. Thee is a unifom magnetic field B Bk pependicula to

### Unit Vectors. the unit vector rˆ. Thus, in the case at hand, 5.00 rˆ, means 5.00 m/s at 36.0.

Unit Vectos What is pobabl the most common mistake involving unit vectos is simpl leaving thei hats off. While leaving the hat off a unit vecto is a nast communication eo in its own ight, it also leads

### Psychology 282 Lecture #2 Outline. Review of Pearson correlation coefficient:

Psychology 282 Lectue #2 Outline Review of Peason coelation coefficient: z z ( n 1) Measue of linea elationship. Magnitude Stength Sign Diection Bounded by +1.0 and -1.0. Independent of scales of measuement.

### CHAPTER 5 GRAVITATIONAL FIELD AND POTENTIAL

CHATER 5 GRAVITATIONAL FIELD AND OTENTIAL 5. Intoduction. This chapte deals with the calculation of gavitational fields and potentials in the vicinity of vaious shapes and sizes of massive bodies. The

### Model Question Paper Mathematics Class XII

Model Question Pape Mathematics Class XII Time Allowed : 3 hous Maks: 100 Ma: Geneal Instuctions (i) The question pape consists of thee pats A, B and C. Each question of each pat is compulsoy. (ii) Pat

### Universal Cycles. Yu She. Wirral Grammar School for Girls. Department of Mathematical Sciences. University of Liverpool

Univesal Cycles 2011 Yu She Wial Gamma School fo Gils Depatment of Mathematical Sciences Univesity of Livepool Supeviso: Pofesso P. J. Giblin Contents 1 Intoduction 2 2 De Buijn sequences and Euleian Gaphs

### Relativistic Quantum Mechanics

Chapte Relativistic Quantum Mechanics In this Chapte we will addess the issue that the laws of physics must be fomulated in a fom which is Loentz invaiant, i.e., the desciption should not allow one to

### Magnetic Field and Magnetic Forces. Young and Freedman Chapter 27

Magnetic Field and Magnetic Foces Young and Feedman Chapte 27 Intoduction Reiew - electic fields 1) A chage (o collection of chages) poduces an electic field in the space aound it. 2) The electic field

### Algebra and Trig. I. A point is a location or position that has no size or dimension.

Algeba and Tig. I 4.1 Angles and Radian Measues A Point A A B Line AB AB A point is a location o position that has no size o dimension. A line extends indefinitely in both diections and contains an infinite

### Infinite-dimensional Bäcklund transformations between isotropic and anisotropic plasma equilibria.

Infinite-dimensional äcklund tansfomations between isotopic and anisotopic plasma equilibia. Infinite symmeties of anisotopic plasma equilibia. Alexei F. Cheviakov Queen s Univesity at Kingston, 00. Reseach

### Solution Derivations for Capa #8

Solution Deivations fo Capa #8 1) A ass spectoete applies a voltage of 2.00 kv to acceleate a singly chaged ion (+e). A 0.400 T field then bends the ion into a cicula path of adius 0.305. What is the ass

### Review of Vectors. Appendix A A.1 DESCRIBING THE 3D WORLD: VECTORS. 3D Coordinates. Basic Properties of Vectors: Magnitude and Direction.

Appendi A Review of Vectos This appendi is a summa of the mathematical aspects of vectos used in electicit and magnetism. Fo a moe detailed intoduction to vectos, see Chapte 1. A.1 DESCRIBING THE 3D WORLD:

### Nontrivial lower bounds for the least common multiple of some finite sequences of integers

J. Numbe Theoy, 15 (007), p. 393-411. Nontivial lowe bounds fo the least common multiple of some finite sequences of integes Bai FARHI bai.fahi@gmail.com Abstact We pesent hee a method which allows to

### Lecture 16: Color and Intensity. and he made him a coat of many colours. Genesis 37:3

Lectue 16: Colo and Intensity and he made him a coat of many colous. Genesis 37:3 1. Intoduction To display a pictue using Compute Gaphics, we need to compute the colo and intensity of the light at each

### Single Particle State A A

LECTURE 3 Maxwell Boltzmann, Femi, and Bose Statistics Suppose we have a gas of N identical point paticles in a box of volume V. When we say gas, we mean that the paticles ae not inteacting with one anothe.

### Lesson 7 Gauss s Law and Electric Fields

Lesson 7 Gauss s Law and Electic Fields Lawence B. Rees 7. You may make a single copy of this document fo pesonal use without witten pemission. 7. Intoduction While it is impotant to gain a solid conceptual

### Questions & Answers Chapter 10 Software Reliability Prediction, Allocation and Demonstration Testing

M13914 Questions & Answes Chapte 10 Softwae Reliability Pediction, Allocation and Demonstation Testing 1. Homewok: How to deive the fomula of failue ate estimate. λ = χ α,+ t When the failue times follow

### In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature.

Radians At school we usually lean to measue an angle in degees. Howeve, thee ae othe ways of measuing an angle. One that we ae going to have a look at hee is measuing angles in units called adians. In

### (References to) "Introduction to Electrodynamics" by David J. Griffiths 3rd ed., Prentice-Hall, 1999. ISBN 0-13-805326-X

lectic Fields in Matte page. Capstones in Physics: lectomagnetism. LCTRIC FILDS IN MATTR.. Multipole xpansion A. Multipole expansion of potential B. Dipole moment C. lectic field of dipole D. Dipole in

### FXA 2008. Candidates should be able to : Describe how a mass creates a gravitational field in the space around it.

Candidates should be able to : Descibe how a mass ceates a gavitational field in the space aound it. Define gavitational field stength as foce pe unit mass. Define and use the peiod of an object descibing

### A r. (Can you see that this just gives the formula we had above?)

24-1 (SJP, Phys 1120) lectic flux, and Gauss' law Finding the lectic field due to a bunch of chages is KY! Once you know, you know the foce on any chage you put down - you can pedict (o contol) motion

### Some text, some maths and going loopy. This is a fun chapter as we get to start real programming!

Chapte Two Some text, some maths and going loopy In this Chapte you ae going to: Lean how to do some moe with text. Get Python to do some maths fo you. Lean about how loops wok. Lean lots of useful opeatos.

### Physics 202, Lecture 4. Gauss s Law: Review

Physics 202, Lectue 4 Today s Topics Review: Gauss s Law Electic Potential (Ch. 25-Pat I) Electic Potential Enegy and Electic Potential Electic Potential and Electic Field Next Tuesday: Electic Potential

### The Binomial Distribution

The Binomial Distibution A. It would be vey tedious if, evey time we had a slightly diffeent poblem, we had to detemine the pobability distibutions fom scatch. Luckily, thee ae enough similaities between

### 1.4 Phase Line and Bifurcation Diag

Dynamical Systems: Pat 2 2 Bifucation Theoy In pactical applications that involve diffeential equations it vey often happens that the diffeential equation contains paametes and the value of these paametes

### MULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION

MULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION K.C. CHANG AND TAN ZHANG In memoy of Pofesso S.S. Chen Abstact. We combine heat flow method with Mose theoy, supe- and subsolution method with

### Chapter 13. Vector-Valued Functions and Motion in Space 13.6. Velocity and Acceleration in Polar Coordinates

13.6 Velocity and Acceleation in Pola Coodinates 1 Chapte 13. Vecto-Valued Functions and Motion in Space 13.6. Velocity and Acceleation in Pola Coodinates Definition. When a paticle P(, θ) moves along

### CHAPTER 3 DIPOLE AND QUADRUPOLE MOMENTS

CHAPTER DIPOLE AND QUADRUPOLE MOMENTS Intoduction + τ p + + E FIGURE III Conside a body which is on the whole electically neutal, but in which thee is a sepaation of chage such that thee is moe positive

### International Monetary Economics Note 1

36-632 Intenational Monetay Economics Note Let me biefly ecap on the dynamics of cuent accounts in small open economies. Conside the poblem of a epesentative consume in a county that is pefectly integated

### Physics 107 HOMEWORK ASSIGNMENT #14

Physics 107 HOMEWORK ASSIGNMENT #14 Cutnell & Johnson, 7 th edition Chapte 17: Poblem 44, 60 Chapte 18: Poblems 14, 18, 8 **44 A tube, open at only one end, is cut into two shote (nonequal) lengths. The

### Gauss Law in dielectrics

Gauss Law in dielectics We fist deive the diffeential fom of Gauss s law in the pesence of a dielectic. Recall, the diffeential fom of Gauss Law is This law is always tue. E In the pesence of dielectics,

### Moment and couple. In 3-D, because the determination of the distance can be tedious, a vector approach becomes advantageous. r r

Moment and couple In 3-D, because the detemination of the distance can be tedious, a vecto appoach becomes advantageous. o k j i M k j i M o ) ( ) ( ) ( + + M o M + + + + M M + O A Moment about an abita

### Continuous Compounding and Annualization

Continuous Compounding and Annualization Philip A. Viton Januay 11, 2006 Contents 1 Intoduction 1 2 Continuous Compounding 2 3 Pesent Value with Continuous Compounding 4 4 Annualization 5 5 A Special Poblem

### 1 Lecture 3: Operators in Quantum Mechanics

1 Lecture 3: Operators in Quantum Mechanics 1.1 Basic notions of operator algebra. In the previous lectures we have met operators: ˆx and ˆp = i h they are called fundamental operators. Many operators