Chapter 16 Angular Momentum


 Brianna Sparks
 11 months ago
 Views:
Transcription
1 Chapter 6 Angular Momentum Apart from the Hamiltonian, the angular momentum operator is of one of the most important Hermitian operators in quantum mechanics. In this chapter we consider its eigenvalues and eigenfunctions in more detail. We begin this chapter with the consideration of the orbital angular momentum. This gives rise to a general definition of angular momenta. We derive the eigenvalue spectrum of the orbital angular momentum with an algebraic method. After a brief presentation of the eigenfunctions of the orbital angular momentum in the position representation, we outline some concepts for the addition of angular momenta. 6. Orbital Angular Momentum Operator The orbital angular momentum is given by l = r p. (6.) As we have seen in Chap. 3, Vol., it is not necessary to symmetrize for the translation into quantum mechanics (spatial representation). It follows directly that l = r, (6.) i or, in components, l x = i ( y z z ) y (6.3) plus cyclic permutations (x y z x ). All the components of l are observables. J. Pade, Quantum Mechanics for Pedestrians : Applications and Extensions, 9 Undergraduate Lecture Notes in Physics, DOI:.7/ _6, Springer International Publishing Switzerland 4
2 3 6 Angular Momentum We know that one can measure two variables simultaneously if the corresponding operators commute. What about the components of the orbital angular momentum? A short calculation (see the exercises) yields lx, l y = i lz ; ly, l z = i lx ; lz, l x = i ly, (6.4) or, compactly, l x, l y = i lz and cyclic permutations. This term can be written still more compactly with the LeviCivita symbol (permutation symbol, epsilon tensor) ε ijk (see also Appendix F, Vol. ): ε ijk = if ijk is an even permutation of 3 ijk is an odd permutation of 3 otherwise (6.5) namely as li, l j = i l k ε ijk. (6.6) k Each component of the orbital angular momentum commutes with l = l x + l y + lz (see the exercises): l x, l = l y, l = l z, l =. (6.7) 6. Generalized Angular Momentum, Spectrum We now generalize these facts by the following definition: A vector operator J is a (generalized) angular momentum operator, if its components are observables and satisfy the commutation relation Jx, J y = i Jz. (6.8) and its cyclic permutations. It follows that J = Jx + J y + J z commutes with all the components: J x, J = ; J y, J = ; J z, J =. (6.9) Instead of J, one often finds j (whereby this is of course not be confused with the probability current density). In this section, we denote the operator by J and the eigenvalue by j. The factor is due to the choice of units, and would be replaced by a different constant if one were to choose different units. The essential factor is i.
3 6. Generalized Angular Momentum, Spectrum 3 The task that we address now is the calculation of the angular momentum spectrum. We will deduce that the angular momentum can assume only halfinteger and integer values. To this end, we need only Eqs. (6.8) and (6.9) and the fact that the J i are Hermitian operators, J i = J i (whereby it is actually quite amazing that one can extract so much information from such sparse initial data). We note first that the squares of the components of the angular momentum are positive operators. Thus, we have for an arbitrary state ϕ : ϕ J x ϕ = J x ϕ. (6.) Consequently, J as a sum of positive Hermitian operators is also positive, so it can have only nonnegative eigenvalues. For reasons which will become clear later, these eigenvalues are written in the special form j ( j + ) with j (and not just simply j or something similar). Equations (6.8) and (6.9) show that J and one of its components can be measured simultaneously. Traditionally, one chooses the zcomponent J z and denotes the eigenvalue associated with J z by m. 3 We are looking for eigenvectors j, m of J and J z with J j, m = j ( j + ) j, m ; J z j, m = m j, m. (6.) To continue, we must now use the commutation relations (6.8). It turns out that it is convenient to use two new operators instead of J x and J x, namely J ± = J x ± ij y. (6.) For reasons that will become apparent immediately, these two operators are called ladder operators; J + is the raising operator and J the lowering operator. The operators are adjoint to each other, since J x and J y are Hermitian: J ± = J. (6.3) With J + and J, the commutation relations (6.8) are written as Jz, J + = J+ ; Jz, J = J ; J+, J = Jz, (6.4) and for J,wehave J = (J + J + J J + ) + J z. (6.5) Together with J +, J = Jz, this equation leads to the expressions 3 j is also called the angular momentum quantum number and m the magnetic or directional quantum number.
4 3 6 Angular Momentum J + J = J J z (J z ) ; J J + = J J z (J z + ). (6.6) Furthermore, J commutes with J + and J (see the exercises). Our interest in the expressions J + J and J J + is essentially due to the fact that they are positive operators. This can be easily seen since, because of Eq. (6.3), the matrix element ϕ J + J ϕ is a norm and it follows that ϕ J + J ϕ = J ϕ. We apply J + J and J J + to the angular momentum states. With Eq. (6.6), it follows that J + J j, m = j ( j + ) m(m ) j, m and J J + j, m = j ( j + ) m(m + ) j, m. (6.7) Since the operators are positive, we obtain immediately the inequalities j ( j + ) m(m ) = ( j m)(j + m + ) and j ( j + ) m(m + ) = ( j + m)(j m + ) (6.8) which must be fulfilled simultaneously. This means that e.g. in the first inequality, the brackets ( j m) and ( j + m + ) must both be positive or both negative. If they were negative, we would have j m and m j. But this is a contradiction since j is positive. Hence we have j m and m j. If we consider in addition the second inequality, it follows that j m j. (6.9) In this way, the range of m is fixed. Now we have to determine the possible values of j. Let us consider the effect of J ± on the states j, m. Wehave J J ± j, m = j ( j + ) J ± j, m and J z J± j, m = (m ± ) J ± j, m. (6.) On the righthand sides, we have the numbers j ( j + ) and (m ± ). This means that J ± j, m is an eigenvector of J with the eigenvalue j ( j + ) as well as of J z with the eigenvalue (m ± ). It follows that J ± j, m = c ± j,m j, m ±. (6.) The proportionality constant c ± j,m can be fixed by using Eq. ( 6.7) (see the exercises). The simplest choice is J ± j, m = j ( j + ) m (m ± ) j, m ±. (6.)
5 6. Generalized Angular Momentum, Spectrum 33 Here, we see clearly the reason why J + and J are called ladder operators or raising and lowering operators; for if we apply J + or J several times to j, m, the magnetic quantum number is increased or decreased by each time. Hence, with the help of these operators, we can climb down or up step by step, as on a ladder. Especially for m = j or m = j,wehave J + j, j = or J j, m =. Since the norm of a vector vanishes iff it is the zero vector, it follows that J + j, j = ; J j, j =. (6.3) We now apply the ladder operators repeatedly and can conclude that J± N j, m is an eigenvector of J z with the eigenvalue (m ± N) (see the exercises), i.e. that it is proportional to j, m ± N with N N. In other words, if we start from any state j, m with j < m < j, then after a few steps 4 we obtain states whose norm is negative (or rather would obtain them), or whose magnetic quantum number m ± N violates the inequality (6.9). This can be avoided only if the following conditions are fulfilled, going up for J + and going down for J : m + N = j and m N = j; N, N N. (6.4) For as we know from Eq. (6.3), J + j, j =, and further applications of J + yield just zero again; and similarly for J. The addition of the two last equations (6.4) leads to j = N + N.Itfollows with j that the allowed values for j are given by j =,,, 3,,... (6.5) and for m by m = j, j +, j +,..., j, j, j (6.6) In this way, we have determined the possible eigenvalues of the general angular momentum operators J and J z. We remark that there are operators with only integer eigenvalues (e.g. the orbital angular momentum operator), or only halfinteger eigenvalues (e.g. the spin operator), but also those which have halfinteger and integer eigenvalues. One of the latter operators, for example, occurs in connection with the Lenz vector; see Appendix G, Vol.. As for elementary particles, nature has apparently set up two classes, which differ by their spins: those with halfinteger spin are called fermions, those with integer spin bosons. General quantum objects can, however, have halfinteger or integer spin, as we see in the example of helium, occurring as 3 He (fermion) and as 4 He (boson). We point out that a very general theorem (the spinstatistics theorem )shows the 4 For e.g. J +, j, m j, m + j, m +.
6 34 6 Angular Momentum connection between the spin and quantum statistics, proving that all fermions obey FermiDirac statistics and all bosons obey BoseEinstein statistics. We note that in the derivation of the angularmomentum eigenvalues, we have obtained little information about the eigenvectors but we do not need it in order to derive the spectrum. It is understandable if this leaves feelings of uncertainty at first sight; but on the other hand, it is simply superb that there is such an elegant technique! 6.3 Matrix Representation of Angular Momentum Operators With Eq. (6.), it follows that j, m J± j, m = j ( j + ) m (m ± )δ m,m±. (6.7) By means of this equation, we can represent the angular momentum operators as matrices. We assume that the eigenstates are given as column vectors. We consider the case of spin, where one usually writes s instead of J. We can represent the two possible states as 5, ( ) ( ) = and, =. (6.8) Only two matrix elements do not vanish, namely, s +, = and, s, = (6.9) or s + = ( ) and s = ( ). (6.3) With the definition of the ladder operators, s ± = s x ± is y or s x = (s + + s )/ and s y = (s + s )/i, it follows that s x = ( ) = σ x and s y = ( ) i = i σ y (6.3) i.e. two of the wellknown spin matrices (s) or Pauli matrices (σ). We obtain the third one by using the equation s z /, m = m /, m directly as 5 We omit here the distinction between = and =.
7 6.3 Matrix Representation of Angular Momentum Operators 35 s z = ( ) = σ z. (6.3) In the same way, we can obtain e.g. the matrix representation of the orbital angular momentum operator for l = ; see the exercises. We recall briefly some properties of the Pauli matrices (cf. Chap. 4, Vol. ). With a view to a more convenient notation, one often writes σ, σ, σ 3 instead of σ x, σ y, σ z. In this notation, 6 σi, σ j = i ε ijk σ k ; { } σ i, σ j = δij ; σi = ; σ i σ j = i k k ε ijk σ k (6.33) holds. Finally, we note that every matrix can be represented as a linear combination of the three Pauli matrices and the unit matrix. 6.4 Orbital Angular Momentum: Spatial Representation of the Eigenfunctions The eigenvalues of the orbital angular momentum are integers. For the position representation of the eigenfunctions, it is advantageous to use spherical coordinates. In these coordinates, we find that 7 : l = sin ϑ ϑ ( sin ϑ ) + ϑ sin ϑ ϕ ; l z = i ϕ. (6.34) The eigenvalue problem (6.) is written as ( sin ϑ ) + sin ϑ ϑ ϑ sin ϑ ϕ Yl m (ϑ, ϕ) = l (l + ) Yl m (ϑ, ϕ) ϕ Y l m (ϑ, ϕ) = imyl m (ϑ, ϕ), (6.35) or, more compactly, as l Yl m (θ, ϕ) = l(l + )Y m l z Y m l l (θ, ϕ) with l =,,, 3,... (θ, ϕ) = myl m (θ, ϕ) with l m l (6.36) 6 The anticommutator is defined as usual by {A, B} = AB + BA. 7 We recall the equality See also Appendix D, Vol.. = r + r r l r
8 36 6 Angular Momentum where the functions Yl m (ϑ, ϕ) are the eigenfunctions of the orbital angular momentum in the position representation 8 ; they are called spherical functions (or spherical harmonics). The number l is the orbital angular momentum (quantum) number, m the magnetic (or directional) (quantum) number. With the separation ansatz Yl m (ϑ, ϕ) = (ϑ) (ϕ), we obtain ( sin ϑ ) + sin ϑ ϑ ϑ sin + l (l + ) (ϑ) = ϑ ϕ (ϕ) = im (ϕ). (6.37) ϕ The solutions of the first equation are wellknown special functions (associated Legendre functions). The solutions of the second equation can immediately be written down as (ϕ) = e imϕ. We will not deal with the general form of the spherical functions (for details see Appendix B, Vol. ), but just note here some important features as well as the simplest cases. The spherical harmonics form a CONS. They are orthonormal π dϑ sin ϑ π dϕ Y m l (ϑ, ϕ) Y m l (ϑ, ϕ) = δ ll δ mm (6.38) and complete l l= m= l Y m l (ϑ, ϕ) Y m l ( ϑ, ϕ ) δ ( ϑ ϑ ) δ ( ϕ ϕ ) =. (6.39) sin ϑ With the notation = (ϑ, ϕ) for the solid angle and d = sin ϑdϑdϕ (also written as d ˆr or d ˆr, see Appendix D, Vol. ) for its differential element, we can write the orthogonality relation as Y m l (ϑ, ϕ) Y m l (ϑ, ϕ) d = δ ll δ mm. (6.4) Because of the completeness of the spherical harmonics, we can expand any (sufficiently wellbehaved) function f (ϑ, ϕ) in terms of them: f (ϑ, ϕ) = l,m c lm Y m l (ϑ, ϕ) (6.4) with 8 Other notations: Yl m (ϑ, ϕ) = Yl m ) (ˆr = ˆr l, m. ˆr is the unit vector and is an abbreviation for the pair (ϑ, ϕ). Moreover, the notation Y lm (ϑ, ϕ) is also common.
9 6.4 Orbital Angular Momentum: Spatial Representation of the Eigenfunctions 37 c lm = Y m l (ϑ, ϕ) f (ϑ, ϕ) d. (6.4) Such an expansion is called a multipole expansion; the contribution for l = is called the monopole (term), for l = the dipole, for l = the quadrupole, and generally, that for l = n the n pole. Finally, we write down the first spherical harmonics explicitly (more spherical functions may be found in Appendix B, Vol., where there are also some graphs): Y = ; Y 3 = 4π 4π cos ϑ; Y 3 = 8π ) Y 5 ( = 3cos ϑ 6π For negative m, the functions are given by Y m l sin ϑeiϕ ; Y = 5 8π sin ϑ cos ϑeiϕ ; Y = 5 3π sin ϑe iϕ. (6.43) (ϑ, ϕ) = ( ) m Y m l (ϑ, ϕ). (6.44) 6.5 Addition of Angular Momenta This section is intended to give a brief overview of the topic. Therefore, only some results are given, without derivation. In this section, we denote the angularmomentum operators by lowercase letters j. The addition theorem for angular momentum states that: If one adds two angular momenta j and j, then j, the quantum number of the total angular momentum j = j + j can assume only one of the values 9 : j + j, j + j, j + j,..., j j (6.45) while the projections onto the zaxis are added directly: m = m + m. (6.46) Total angular momentum states jm can be obtained from the individual states j, m and j, m through the relation: jm; j j = j j m m jm j, m j, m (6.47) m +m =m 9 Intuitivelyclear reason: If the two angular momentum vectors are parallel to each other, then their angularmomentum quantum numbers are added to give j = j + j. For other arrangements, j is smaller. The smallest value is j j, because of j.
10 38 6 Angular Momentum where of course j j j j + j has to be satisfied. The numbers j j m m jm are called ClebschGordan coefficients ; they are real and are tabulated in relevant works on the angular momentum in quantum mechanics. The inverse of the last equation is j, m j, m = j + j j= j j j j m m jm jm; j j. (6.48) As an application, we consider the spinorbit coupling, i.e. the coupling of the orbital angular momentum l of electrons with their spins s. The total angular momentum of the electron is j = l + s (6.49) and its possible values are j = l + and j = l (except for l = in which case only j = occurs). Intuitively, this means that orbital angular momentum and spin are either parallel ( j = l + ) or antiparallel ( j = l ). The spin is a relativistic phenomenon and can, if one argues from the nonrelativistic SEq, at best be introduced heuristically. A clearcut approach starts e.g. from the relativistically correct Dirac equation i t ψ = ( cαp + βmc + eφ ) ψ = H Dirac ψ. Here, p is the threedimensional momentum operator p = i, while β and the three components of the vector α are certain 4 4 matrices, the Dirac matrices. j j There are several different notations for these coefficients, e.g. C 3jsymbols are related coefficients: ( ) j j j = ( ) m m m j j m j j m m j m. j + The 3jsymbols are invariant against cyclic permutation: ( ) ( ) j j j j j j =, etc. m m m mm m m m ; jm. The socalled The two conditions j j j j + j and m = m + m must be met in order that a ClebschGordan coefficient is nonzero. The CGC satisfy the orthogonality relations j j m m jm j j m m j m = δ jj δ mm m,m j j m m jm j j m m jm = δ m m δ m m. j,m A particular CGC can in principle be calculated using the ladder operators j ± + j ±, whereby one starts e.g. from j j j j j + j j + j =. In this way, one obtains for example ( ) ABA+ B = ( ) a b c A B c (A)! (B)! (A + B + c)! (A + B c)!. (A + B + )! (A + a)! (A a)! (B + b)! (B b)!
11 6.5 Addition of Angular Momenta 39 Essentially, one performs an expansion of this equation in powers of ( v c ) and retains only the lowest term for the nonrelativistic description. Then it turns out that in the SEq, a spinorbit term F(r) l s appears (see also Chap. 9), with F(r) a radiallysymmetric function. Because of l s has the values (l + s) = j = l + ls + s, (6.5) l s = j ( j + ) l (l + ) s (s + ). (6.5) It follows for the spinorbit term in the SEq: F(r) l s = { lf(r) for (l + ) F(r) j = l + j = l. (6.5) Next, we briefly discuss the corresponding eigenfunctions and write down their explicit form. We start from the states for the spin sm s and the orbital angular momentum Q; lm l, where Q stands for possible additional quantum numbers (e.g. the principal quantum number of the hydrogen atom). The total angular momentum state is then described by Q; j, m j; l ; j = l ± for l ; j = for l = (6.53) and is composed of spin and orbital angular momentum states: Q; j, m j; l = m l +m s =m j lsm l m s Calculating the ClebschGordan coefficients yields Q; j = l + /, m j; l l+m = j +/ l+ Q; j = l /, m j; l = l m j +/ l+ jmj Q; lml sm s (6.54) Q; l, m j / l m s, / + j +/ l+ Q; l, m j + / s, / Q; l, m j / l+m s, / + j +/ l+ Q; l, m j + / s, / (6.55) with m j = l + /,..., (l + /) in the upper line and m j = l /,..., (l /) in the lower line. With the notation as column vectors, we obtain using ( ) s, / /, / = ( ) and s, / /, / = (6.56)
12 4 6 Angular Momentum the explicit formulations Q; j = l + /, m j; l = l+m j +/ l+ l m j +/ l+ Q; l, m j / Q; l, m j + / with m j = l + /,..., (l + /) (6.57) and Q; j = l /, m j; l = l m j +/ l+ l+m j +/ l+ Q; l, m j / Q; l, m j + / with m j = l /,..., (l /). (6.58) 6.6 Exercises. For which K, N, M are the spherical harmonics (in spherical coordinates) f (ϑ, ϕ) = cos K ϑ sin M ϑ e inϕ (6.59) eigenfunctions of l?. Write out the spherical harmonics for l = using Cartesian coordinates, x, y, z. 3. Show that: l ˆr = ˆr l = (6.6) 4. Show that the components of l are Hermitian. 5. Show that for the orbital angular momentum, it holds that lx, l y = i lz ; l y, l z = i lx ; l z, l x = i ly. (6.6) 6. Show that A, BC = B A, C + A, B C holds. Using this identity and the commutators l x, l y = i lz plus cyclic permutations, prove that l x, l =. 7. Show that: J, J ± =. (6.6) 8. We have seen in the text that J ± j, m = c ± j,m j, m ±. (6.63)
13 6.6 Exercises 4 Using J + J j, m = j ( j + ) m(m ) j, m J J + j, m = j ( j + ) m(m + ) j, m, (6.64) show that for the coefficients c ± j,m, holds. 9. Given the Pauli matrices σ k, (a) Show (once more) that σi, σ j = iεijk σ k ; (b) Prove that c ± j,m = j ( j + ) m (m ± ) (6.65) { σi, σ j } = δij ; σ i = ; σ i σ j = iε ijk σ k ; (6.66) (σ A)(σ B) = AB + iσ (A B) (6.67) where σ is the vector σ = (σ, σ, σ 3 ) and A, B are threedimensional vectors; (c) Show that every x matrix can be expressed as a linear combination of the three Pauli matrices and the unit matrix.. Given the orbital angular momentum operator l and the spin operator s, show that l z, s l = ; s z, s l = ; l z + s z, s l =.. The ladder operators for a generalized angular momentum are given as J ± = J x ± ij y. (a) Show that J z, J + = J+, J z, J = J, J +, J = Jz, as well as J = (J + J + J J+) + J z. (b) Show that it follows from the last equation that: and hence J + J = J J z (J z ) ; J J + = J J z (J z + ) (6.68) J + J j, m = j ( j + ) m(m ) j, m J J + j, m = j ( j + ) m(m + ) j, m. (6.69)
14 4 6 Angular Momentum Fig. 6. Rotation about an axis â z â x ϕ ϑ y (c) Show that from the last two equations, it follows that: j ( j + ) m(m ) = ( j m)(j + m + ) j ( j + ) m(m + ) = ( j + m)(j m + ) (6.7) and hence j m j. (6.7). What is the matrix representation of the orbital angular momentum for l =? 3. Consider the orbital angular momentum l =. Express the operator e iαl z/ as sum over dyadic products (representationfree). Specify this for the bases and, = ;, = ;, = (6.7), = ± i ;, = ;, = i. (6.73) 4. Calculate the term γâl i e = e iγâl (6.74) for the orbital angular momentum l = and the basis (6.73). â is the rotation axis (a unit vector), γ the rotation angle. For reasons of economy, use the simplified angular momentum l = L/, i.e. the theoretical units system. (a) Express the rotations around the x, y and zaxis as matrices. (b) Express the rotations about an axis â with rotation angle γ as matrices (the angles in spherical coordinates are ϑ and ϕ; see Fig. 6.). Of course, all the calculations may also be performed representationfree.
15
For the case of an Ndimensional spinor the vector α is associated to the onedimensional . N
1 CHAPTER 1 Review of basic Quantum Mechanics concepts Introduction. Hermitian operators. Physical meaning of the eigenvectors and eigenvalues of Hermitian operators. Representations and their use. onhermitian
More information1D 3D 1D 3D. is called eigenstate or state function. When an operator act on a state, it can be written as
Chapter 3 (Lecture 45) Postulates of Quantum Mechanics Now we turn to an application of the preceding material, and move into the foundations of quantum mechanics. Quantum mechanics is based on a series
More informationHermitian Operators An important property of operators is suggested by considering the Hamiltonian for the particle in a box: d 2 dx 2 (1)
CHAPTER 4 PRINCIPLES OF QUANTUM MECHANICS In this Chapter we will continue to develop the mathematical formalism of quantum mechanics, using heuristic arguments as necessary. This will lead to a system
More informationNotes: Tensor Operators
Notes: Tensor Operators Ben Baragiola I. VECTORS We are already familiar with the concept of a vector, but let s review vector properties to refresh ourselves. A component Cartesian vector is v = v x
More information13 Addition of angular momenta
TFY450/FY045 Tillegg 13  Addition of angular momenta 1 TILLEGG 13 13 Addition of angular momenta (8.4 in Hemmer, 6.10 in B&J, 4.4 in Griffiths) Addition of angular momenta enters the picture when we consider
More information1. The quantum mechanical state of a hydrogen atom is described by the following superposition: ψ = (2ψ 1,0,0 3ψ 2,0,0 ψ 3,2,2 )
CHEM 352: Examples for chapter 2. 1. The quantum mechanical state of a hydrogen atom is described by the following superposition: ψ = 1 14 2ψ 1,, 3ψ 2,, ψ 3,2,2 ) where ψ n,l,m are eigenfunctions of the
More informationA. The wavefunction itself Ψ is represented as a socalled 'ket' Ψ>.
Quantum Mechanical Operators and Commutation C I. BraKet Notation It is conventional to represent integrals that occur in quantum mechanics in a notation that is independent of the number of coordinates
More information3. A LITTLE ABOUT GROUP THEORY
3. A LITTLE ABOUT GROUP THEORY 3.1 Preliminaries It is an apparent fact that nature exhibits many symmetries, both exact and approximate. A symmetry is an invariance property of a system under a set of
More informationCHAPTER 12 MOLECULAR SYMMETRY
CHAPTER 12 MOLECULAR SYMMETRY In many cases, the symmetry of a molecule provides a great deal of information about its quantum states, even without a detailed solution of the Schrödinger equation. A geometrical
More informationState of Stress at Point
State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,
More informationWe consider a hydrogen atom in the ground state in the uniform electric field
Lecture 13 Page 1 Lectures 1314 Hydrogen atom in electric field. Quadratic Stark effect. Atomic polarizability. Emission and Absorption of Electromagnetic Radiation by Atoms Transition probabilities and
More informationHow is a vector rotated?
How is a vector rotated? V. Balakrishnan Department of Physics, Indian Institute of Technology, Madras 600 036 Appeared in Resonance, Vol. 4, No. 10, pp. 6168 (1999) Introduction In an earlier series
More informationH = = + H (2) And thus these elements are zero. Now we can try to do the same for time reversal. Remember the
1 INTRODUCTION 1 1. Introduction In the discussion of random matrix theory and information theory, we basically explained the statistical aspect of ensembles of random matrices, The minimal information
More informationUNIVERSITETET I OSLO Det matematisknaturvitenskapelige fakultet
UNIVERSITETET I OSLO Det matematisknaturvitenskapelige fakultet Solution for take home exam: FYS311, Oct. 7, 11. 1.1 The Hamiltonian of a charged particle in a weak magnetic field is Ĥ = P /m q mc P A
More informationQuick Reference Guide to Linear Algebra in Quantum Mechanics
Quick Reference Guide to Linear Algebra in Quantum Mechanics Scott N. Walck September 2, 2014 Contents 1 Complex Numbers 2 1.1 Introduction............................ 2 1.2 Real Numbers...........................
More information1 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
More information1 Spherical Kinematics
ME 115(a): Notes on Rotations 1 Spherical Kinematics Motions of a 3dimensional rigid body where one point of the body remains fixed are termed spherical motions. A spherical displacement is a rigid body
More informationWe can represent the eigenstates for angular momentum of a spin1/2 particle along each of the three spatial axes with column vectors: 1 +y =
Chapter 0 Pauli Spin Matrices We can represent the eigenstates for angular momentum of a spin/ particle along each of the three spatial axes with column vectors: +z z [ ] 0 [ ] 0 +y y [ ] / i/ [ ] i/
More informationBasic 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
More informationChapter 4. Rotations
Chapter 4 Rotations 1 CHAPTER 4. ROTATIONS 2 4.1 Geometrical rotations Before discussing rotation operators acting the state space E, we want to review some basic properties of geometrical rotations. 4.1.1
More informationMatrix Algebra LECTURE 1. Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 = a 11 x 1 + a 12 x 2 + +a 1n x n,
LECTURE 1 Matrix Algebra Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 a 11 x 1 + a 12 x 2 + +a 1n x n, (1) y 2 a 21 x 1 + a 22 x 2 + +a 2n x n, y m a m1 x 1 +a m2 x
More informationChapter 9 Unitary Groups and SU(N)
Chapter 9 Unitary Groups and SU(N) The irreducible representations of SO(3) are appropriate for describing the degeneracies of states of quantum mechanical systems which have rotational symmetry in three
More informationHomework One Solutions. Keith Fratus
Homework One Solutions Keith Fratus June 8, 011 1 Problem One 1.1 Part a In this problem, we ll assume the fact that the sum of two complex numbers is another complex number, and also that the product
More informationThe Matrix Elements of a 3 3 Orthogonal Matrix Revisited
Physics 116A Winter 2011 The Matrix Elements of a 3 3 Orthogonal Matrix Revisited 1. Introduction In a class handout entitled, ThreeDimensional Proper and Improper Rotation Matrices, I provided a derivation
More informationMathematical Formulation of the Superposition Principle
Mathematical Formulation of the Superposition Principle Superposition add states together, get new states. Math quantity associated with states must also have this property. Vectors have this property.
More informationLinear Algebra In Dirac Notation
Chapter 3 Linear Algebra In Dirac Notation 3.1 Hilbert Space and Inner Product In Ch. 2 it was noted that quantum wave functions form a linear space in the sense that multiplying a function by a complex
More informationContinuous Groups, Lie Groups, and Lie Algebras
Chapter 7 Continuous Groups, Lie Groups, and Lie Algebras Zeno was concerned with three problems... These are the problem of the infinitesimal, the infinite, and continuity... Bertrand Russell The groups
More information1 Scalars, Vectors and Tensors
DEPARTMENT OF PHYSICS INDIAN INSTITUTE OF TECHNOLOGY, MADRAS PH350 Classical Physics Handout 1 8.8.2009 1 Scalars, Vectors and Tensors In physics, we are interested in obtaining laws (in the form of mathematical
More informationPrimer on Index Notation
Massachusetts Institute of Technology Department of Physics Physics 8.07 Fall 2004 Primer on Index Notation c 20032004 Edmund Bertschinger. All rights reserved. 1 Introduction Equations involving vector
More informationSymbols, conversions, and atomic units
Appendix C Symbols, conversions, and atomic units This appendix provides a tabulation of all symbols used throughout this thesis, as well as conversion units that may useful for reference while comparing
More informationQuantum Mechanics: Fundamental Principles and Applications
Quantum Mechanics: Fundamental Principles and Applications John F. Dawson Department of Physics, University of New Hampshire, Durham, NH 0384 October 14, 009, 9:08am EST c 007 John F. Dawson, all rights
More informationLecture 34: Principal Axes of Inertia
Lecture 34: Principal Axes of Inertia We ve spent the last few lectures deriving the general expressions for L and T rot in terms of the inertia tensor Both expressions would be a great deal simpler if
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationELEMENTS OF VECTOR ALGEBRA
ELEMENTS OF VECTOR ALGEBRA A.1. VECTORS AND SCALAR QUANTITIES We have now proposed sets of basic dimensions and secondary dimensions to describe certain aspects of nature, but more than just dimensions
More informationModern Geometry Homework.
Modern Geometry Homework. 1. Rigid motions of the line. Let R be the real numbers. We define the distance between x, y R by where is the usual absolute value. distance between x and y = x y z = { z, z
More informationContents Quantum states and observables
Contents 2 Quantum states and observables 3 2.1 Quantum states of spin1/2 particles................ 3 2.1.1 State vectors of spin degrees of freedom.......... 3 2.1.2 Pauli operators........................
More informationAppendix E  Elements of Quantum Mechanics
1 Appendix E  Elements of Quantum Mechanics Quantum mechanics provides a correct description of phenomena on the atomic or sub atomic scale, where the ideas of classical mechanics are not generally applicable.
More informationAverage Angular Velocity
Average Angular Velocity Hanno Essén Department of Mechanics Royal Institute of Technology S100 44 Stockholm, Sweden 199, December Abstract This paper addresses the problem of the separation of rotational
More information2. Introduction to quantum mechanics
2. Introduction to quantum mechanics 2.1 Linear algebra Dirac notation Complex conjugate Vector/ket Dual vector/bra Inner product/bracket Tensor product Complex conj. matrix Transpose of matrix Hermitian
More informationHarmonic Oscillator Physics
Physics 34 Lecture 9 Harmonic Oscillator Physics Lecture 9 Physics 34 Quantum Mechanics I Friday, February th, 00 For the harmonic oscillator potential in the timeindependent Schrödinger equation: d ψx
More informationPHY4604 Introduction to Quantum Mechanics Fall 2004 Practice Test 3 November 22, 2004
PHY464 Introduction to Quantum Mechanics Fall 4 Practice Test 3 November, 4 These problems are similar but not identical to the actual test. One or two parts will actually show up.. Short answer. (a) Recall
More informationSummary of week 8 (Lectures 22, 23 and 24)
WEEK 8 Summary of week 8 (Lectures 22, 23 and 24) This week we completed our discussion of Chapter 5 of [VST] Recall that if V and W are inner product spaces then a linear map T : V W is called an isometry
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationSimilarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
More informationUNIT 2 MATRICES  I 2.0 INTRODUCTION. Structure
UNIT 2 MATRICES  I Matrices  I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress
More informationSubatomic Physics: Particle Physics Lecture 4. Quantum ElectroDynamics & Feynman Diagrams. Antimatter. Virtual Particles. Yukawa Potential for QED
Ψ e (r, t) exp i/ (E)(t) (p) (r) Subatomic Physics: Particle Physics Lecture Quantum ElectroDynamics & Feynman Diagrams Antimatter Feynman Diagram and Feynman Rules Quantum description of electromagnetism
More informationMixed states and pure states
Mixed states and pure states (Dated: April 9, 2009) These are brief notes on the abstract formalism of quantum mechanics. They will introduce the concepts of pure and mixed quantum states. Some statements
More informationANALYTICAL MATHEMATICS FOR APPLICATIONS 2016 LECTURE NOTES Series
ANALYTICAL MATHEMATICS FOR APPLICATIONS 206 LECTURE NOTES 8 ISSUED 24 APRIL 206 A series is a formal sum. Series a + a 2 + a 3 + + + where { } is a sequence of real numbers. Here formal means that we don
More informationThe threedimensional rotations are defined as the linear transformations of the vector x = (x 1, x 2, x 3 ) x i = R ij x j, (2.1) x 2 = x 2. (2.
2 The rotation group In this Chapter we give a short account of the main properties of the threedimensional rotation group SO(3) and of its universal covering group SU(2). The group SO(3) is an important
More informationMATH 304 Linear Algebra Lecture 24: Scalar product.
MATH 304 Linear Algebra Lecture 24: Scalar product. Vectors: geometric approach B A B A A vector is represented by a directed segment. Directed segment is drawn as an arrow. Different arrows represent
More informationThe Mathematics of Origami
The Mathematics of Origami Sheri Yin June 3, 2009 1 Contents 1 Introduction 3 2 Some Basics in Abstract Algebra 4 2.1 Groups................................. 4 2.2 Ring..................................
More informationTangent and normal lines to conics
4.B. Tangent and normal lines to conics Apollonius work on conics includes a study of tangent and normal lines to these curves. The purpose of this document is to relate his approaches to the modern viewpoints
More informationPHY4604 Introduction to Quantum Mechanics Fall 2004 Practice Exam Solutions Dec. 13, 2004
PHY4604 Introduction to Quantum Mechanics Fall 2004 Practice Exam Solutions Dec. 1, 2004 No materials allowed. If you can t remember a formula, ask and I might help. If you can t do one part of a problem,
More informationQuaternions and isometries
6 Quaternions and isometries 6.1 Isometries of Euclidean space Our first objective is to understand isometries of R 3. Definition 6.1.1 A map f : R 3 R 3 is an isometry if it preserves distances; that
More information(a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular.
Theorem.7.: (Properties of Triangular Matrices) (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. (b) The product
More informationIndex notation in 3D. 1 Why index notation?
Index notation in 3D 1 Why index notation? Vectors are objects that have properties that are independent of the coordinate system that they are written in. Vector notation is advantageous since it is elegant
More informationIntroduces the bra and ket notation and gives some examples of its use.
Chapter 7 ket and bra notation Introduces the bra and ket notation and gives some examples of its use. When you change the description of the world from the inutitive and everyday classical mechanics to
More informationDMATH Algebra I HS 2013 Prof. Brent Doran. Solution 5
DMATH Algebra I HS 2013 Prof. Brent Doran Solution 5 Dihedral groups, permutation groups, discrete subgroups of M 2, group actions 1. Write an explicit embedding of the dihedral group D n into the symmetric
More information221A Lecture Notes on Spin
22A Lecture Notes on Spin This lecture note is just for the curious. It is not a part of the standard quantum mechanics course. True Origin of Spin When we introduce spin in nonrelativistic quantum mechanics,
More informationDerivatives of Matrix Functions and their Norms.
Derivatives of Matrix Functions and their Norms. Priyanka Grover Indian Statistical Institute, Delhi India February 24, 2011 1 / 41 Notations H : ndimensional complex Hilbert space. L (H) : the space
More informationChapter 1. Group and Symmetry. 1.1 Introduction
Chapter 1 Group and Symmetry 1.1 Introduction 1. A group (G) is a collection of elements that can multiply and divide. The multiplication is a binary operation that is associative but not necessarily commutative.
More informationA matrix over a field F is a rectangular array of elements from F. The symbol
Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F) denotes the collection of all m n matrices over F Matrices will usually be denoted
More information6. ISOMETRIES isometry central isometry translation Theorem 1: Proof:
6. ISOMETRIES 6.1. Isometries Fundamental to the theory of symmetry are the concepts of distance and angle. So we work within R n, considered as an innerproduct space. This is the usual n dimensional
More informationMATH36001 Background Material 2015
MATH3600 Background Material 205 Matrix Algebra Matrices and Vectors An ordered array of mn elements a ij (i =,, m; j =,, n) written in the form a a 2 a n A = a 2 a 22 a 2n a m a m2 a mn is said to be
More informationChapter 6. Orthogonality
6.3 Orthogonal Matrices 1 Chapter 6. Orthogonality 6.3 Orthogonal Matrices Definition 6.4. An n n matrix A is orthogonal if A T A = I. Note. We will see that the columns of an orthogonal matrix must be
More information13 Solutions for Section 6
13 Solutions for Section 6 Exercise 6.2 Draw up the group table for S 3. List, giving each as a product of disjoint cycles, all the permutations in S 4. Determine the order of each element of S 4. Solution
More informationFrom Einstein to KleinGordon Quantum Mechanics and Relativity
From Einstein to KleinGordon Quantum Mechanics and Relativity Aline Ribeiro Department of Mathematics University of Toronto March 24, 2002 Abstract We study the development from Einstein s relativistic
More informationLecture Notes for Ph219/CS219: Quantum Information and Computation Chapter 2. John Preskill California Institute of Technology
Lecture Notes for Ph19/CS19: Quantum Information and Computation Chapter John Preskill California Institute of Technology Updated July 015 Contents Foundations I: States and Ensembles 3.1 Axioms of quantum
More informationRutgers  Physics Graduate Qualifying Exam Quantum Mechanics: September 1, 2006
Rutgers  Physics Graduate Qualifying Exam Quantum Mechanics: September 1, 2006 QA J is an angular momentum vector with components J x, J y, J z. A quantum mechanical state is an eigenfunction of J 2 J
More information5. Möbius Transformations
5. Möbius Transformations 5.1. The linear transformation and the inversion. In this section we investigate the Möbius transformation which provides very convenient methods of finding a onetoone mapping
More informationLecture 2. Observables
Lecture 2 Observables 13 14 LECTURE 2. OBSERVABLES 2.1 Observing observables We have seen at the end of the previous lecture that each dynamical variable is associated to a linear operator Ô, and its expectation
More informationThe Characteristic Polynomial
Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem
More informationLecture L3  Vectors, Matrices and Coordinate Transformations
S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3  Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between
More information2. Spin Chemistry and the Vector Model
2. Spin Chemistry and the Vector Model The story of magnetic resonance spectroscopy and intersystem crossing is essentially a choreography of the twisting motion which causes reorientation or rephasing
More informationSolid State Theory. (Theorie der kondensierten Materie) Wintersemester 2015/16
Martin Eckstein Solid State Theory (Theorie der kondensierten Materie) Wintersemester 2015/16 MaxPlanck Research Departement for Structural Dynamics, CFEL Desy, Bldg. 99, Rm. 02.001 Luruper Chaussee 149
More informationThe Central Equation
The Central Equation Mervyn Roy May 6, 015 1 Derivation of the central equation The single particle Schrödinger equation is, ( H E nk ψ nk = 0, ) ( + v s(r) E nk ψ nk = 0. (1) We can solve Eq. (1) at given
More informationOn the general equation of the second degree
On the general equation of the second degree S Kesavan The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai  600 113 email:kesh@imscresin Abstract We give a unified treatment of the
More informationThe Solution of Linear Simultaneous Equations
Appendix A The Solution of Linear Simultaneous Equations Circuit analysis frequently involves the solution of linear simultaneous equations. Our purpose here is to review the use of determinants to solve
More informationFourier Series and SturmLiouville Eigenvalue Problems
Fourier Series and SturmLiouville Eigenvalue Problems 2009 Outline Functions Fourier Series Representation Halfrange Expansion Convergence of Fourier Series Parseval s Theorem and Mean Square Error Complex
More informationLinear Algebra: Vectors
A Linear Algebra: Vectors A Appendix A: LINEAR ALGEBRA: VECTORS TABLE OF CONTENTS Page A Motivation A 3 A2 Vectors A 3 A2 Notational Conventions A 4 A22 Visualization A 5 A23 Special Vectors A 5 A3 Vector
More informationWHICH LINEARFRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE?
WHICH LINEARFRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE? JOEL H. SHAPIRO Abstract. These notes supplement the discussion of linear fractional mappings presented in a beginning graduate course
More informationTopic 1: Matrices and Systems of Linear Equations.
Topic 1: Matrices and Systems of Linear Equations Let us start with a review of some linear algebra concepts we have already learned, such as matrices, determinants, etc Also, we shall review the method
More informationNear horizon black holes in diverse dimensions and integrable models
Near horizon black holes in diverse dimensions and integrable models Anton Galajinsky Tomsk Polytechnic University LPI, 2012 A. Galajinsky (TPU) Near horizon black holes and integrable models LPI, 2012
More informationLecture 22 Relevant sections in text: 3.1, 3.2. Rotations in quantum mechanics
Lecture Relevant sections in text: 3.1, 3. Rotations in quantum mechanics Now we will discuss what the preceding considerations have to do with quantum mechanics. In quantum mechanics transformations in
More informationTheory of Angular Momentum and Spin
Chapter 5 Theory of Angular Momentum and Spin Rotational symmetry transformations, the group SO3 of the associated rotation matrices and the corresponding transformation matrices of spin states forming
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationElectronic calculators may not be used. This text is not the text of solutions of exam papers! Here we will discuss the solutions of the exampapers.
EXAM FEEDBACK INTRODUCTION TO GEOMETRY (Math 20222). Spring 2014 ANSWER THREE OF THE FOUR QUESTIONS If four questions are answered credit will be given for the best three questions. Each question is worth
More informationSmall oscillations with many degrees of freedom
Analytical Dynamics  Graduate Center CUNY  Fall 007 Professor Dmitry Garanin Small oscillations with many degrees of freedom General formalism Consider a dynamical system with N degrees of freedom near
More informationd(f(g(p )), f(g(q))) = d(g(p ), g(q)) = d(p, Q).
10.2. (continued) As we did in Example 5, we may compose any two isometries f and g to obtain new mappings f g: E E, (f g)(p ) = f(g(p )), g f: E E, (g f)(p ) = g(f(p )). These are both isometries, the
More informationThe Sixfold Path to understanding bra ket Notation
The Sixfold Path to understanding bra ket Notation Start by seeing the three things that bra ket notation can represent. Then build understanding by looking in detail at each of those three in a progressive
More informationDETERMINANTS. b 2. x 2
DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in
More information5.61 Physical Chemistry 25 Helium Atom page 1 HELIUM ATOM
5.6 Physical Chemistry 5 Helium Atom page HELIUM ATOM Now that we have treated the Hydrogen like atoms in some detail, we now proceed to discuss the next simplest system: the Helium atom. In this situation,
More informationEXPERIMENT 13. Balmer Series of Hydrogen
EXPERIMENT 13 Balmer Series of Hydrogen Any atomic gas or element (heated to vapor form) can be made to radiate light when suitably "excited" by an electric discharge, spark, or flame. If this light is
More information1 The Fourier Transform
Physics 326: Quantum Mechanics I Prof. Michael S. Vogeley The Fourier Transform and Free Particle Wave Functions 1 The Fourier Transform 1.1 Fourier transform of a periodic function A function f(x) that
More informationDefinition 12 An alternating bilinear form on a vector space V is a map B : V V F such that
4 Exterior algebra 4.1 Lines and 2vectors The time has come now to develop some new linear algebra in order to handle the space of lines in a projective space P (V ). In the projective plane we have seen
More informationIntroduction to Matrix Algebra
Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra  1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary
More informationDeriving character tables: Where do all the numbers come from?
3.4. Characters and Character Tables 3.4.1. Deriving character tables: Where do all the numbers come from? A general and rigorous method for deriving character tables is based on five theorems which in
More informationMITES 2010: Physics III Survey of Modern Physics Final Exam Solutions
MITES 2010: Physics III Survey of Modern Physics Final Exam Solutions Exercises 1. Problem 1. Consider a particle with mass m that moves in onedimension. Its position at time t is x(t. As a function of
More informationWe have seen in the previous Chapter that there is a sense in which the state of a quantum
Chapter 8 Vector Spaces in Quantum Mechanics We have seen in the previous Chapter that there is a sense in which the state of a quantum system can be thought of as being made up of other possible states.
More information= [a ij ] 2 3. Square matrix A square matrix is one that has equal number of rows and columns, that is n = m. Some examples of square matrices are
This document deals with the fundamentals of matrix algebra and is adapted from B.C. Kuo, Linear Networks and Systems, McGraw Hill, 1967. It is presented here for educational purposes. 1 Introduction In
More information