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Chapter: 5 -Problem: 18 >> Work out the quadratic Zeeman effect for the ground-state hydrogen atom [(x|0) = (1/??a03) e-r/a0] due to the usually neglected e2A2/2mec2- -term in the Hamiltonian taken to first order. Write the energy shift as and obtain an expression for diamagnetic susceptibility, X. The following definite integral may be useful: Tran
Answer Preview: The quadratic Zeeman effect refers to the splitting of atomic energy levels in the presence of a mag…

, Chapter: 3 -Problem: 21 >> The goal of this problem is to determine degenerate eigenstates of the threedimensional isotropic harmonic oscillator written as eigenstates of L2 and Lz, in terms of the Cartesian eigenstates |nxnynz) .(a) Show that the angular-momentum operators are given bywhere summation is implied over repeated indices, ?ijk is the totally antisymmetric symbol, and N = aj†aj counts the total number of quanta.
Answer Preview: To solve this problem, we need to find the eigenstates of the three-dimensional isotropic harmonic oscillator in terms of the Cartesian eigenstates |nxnynz). Let's go step by step: (a) Find the expres…

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, Chapter: 5 -Problem: 6 >> A slightly anisotropic three-dimensional harmonic oscillator has ?z ? ?x = ?y. A charged particle moves in the field of this oscillator and is at the same time exposed to a uniform magnetic field in the x-direction. Assuming that the Zeeman splitting is comparable to the splitting produced by the anisotropy, but small compared to h??, calculate to first order the energies of the components of the
Answer Preview: To solve this problem, we'll need to consider the effects of the anisotropic harmonic oscillator potential, the uniform magnetic field, and the Zeeman …

, Chapter: 1 -Problem: 15 >> Let A and B be observables. Suppose the simultaneous eigenkets of A and B {|a?, b??} form a complete orthonormal set of base kets. Can we always conclude that [A, B] = 0?If your answer is yes, prove the assertion. If your answer is no, give a counterexample.
Answer Preview: No, we cannot always conclude that [A, B] = 0 based on the fact that the simultaneous eigenkets of A …

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, Chapter: 1 -Problem: 13 >> A beam of spin 1/2 atoms goes through a series of Stem-Gerlach-type measurements as follows:(a) The first measurement accepts sz = h?/2 atoms and rejects sz = -h?2 atoms.(b) The second measurement accepts sn = h?/2 atoms and rejects sn = -h?/2 atoms, where sn is the eigenvalue of the operator S · n?, with ft making an angle ? in the xz-plane with respect to the z-axis.(c) The third measurement acc
Answer Preview: In this problem, we have a beam of spin 1/2 atoms that undergo a series of Stern-Gerlach-type measurements. Let's break down the problem step by step. …

, Chapter: 5 -Problem: 34 >> Consider the spontaneous emission of a photon by an excited atom. The process is known to be an E1 transition. Suppose the magnetic quantum number of the atom decreases by one unit. What is the angular distribution of the emitted photon? Also discuss the polarization of the photon, with attention to angular-momentum conservation for the whole (atom plus photon) system.
Answer Preview: When an excited atom undergoes a spontaneous emission of a photon through an E1 (electric dipole) transition, the magnetic quantum number of the atom …

, Chapter: 1 -Problem: 22 >> Estimate the rough order of magnitude of the length of time that an ice pick can be balanced on its point if the only limitation is that set by the Heisenberg uncertainty principle. Assume that the point is sharp and that the point and the surface on which it rests are hard. You may make approximations that do not alter the general order of magnitude of the result. Assume reasonable values for the
Answer Preview: The Heisenberg uncertainty principle states that there is a fundamental limit to how precisely we can simultaneously measure the position and momentum …

, Chapter: 6 -Problem: 10 >> Consider scattering by a repulsive ?-shell potential:(a) Set up an equation that determines the s-wave phase shift ?0 as a function of k(E = h?2k2 /2m).(b) Assume now that ? is very large,Show that if tan kR is not close to zero, the s-wave phase shift resembles the hard-sphere result discussed in the text. Show also that for tan kR close to (but not exactly equal to) zero, resonance behavior is p
Answer Preview: (a) The Schrdinger equation for a particle scattering by a repulsive -shell potential is [-h^2/2m] d…

, Chapter: 3 -Problem: 15 >> (a) Let J be angular momentum. (It may stand for orbital L, spin S, or Jtotal·) Using the fact that Jx, ly, lz(± ? Jx ± i Jy) satisfy the usual angular-momentum commutation relations, prove(b) Using (a) (or otherwise), derive the "famous" expression for the coefficient c_that appears in Transcribed Image Text:
Answer Preview: (a) To prove the given expression, let's first recall the commutation relations for the angular momentum operators Jx, Jy, and Jz: [Jx, Jy] = iJz [Jy, …

, Chapter: 3 -Problem: 5 >> Consider a spin 1 particle. Evaluate the matrix elements of Transcribed Image Text: S?(S?+h)(S?-?) and Sx(Sx+?)(Sx -?).
Answer Preview: To evaluate the matrix elements of the operators Sz(Sz+)(Sz-) and Sx(Sx +)(Sx ) for a spin 1 particle, we first need to express these operators in ter…

, Chapter: 5 -Problem: 16 >> Consider a particle bound to a fixed center by a spherically symmetrical potential V(r).(a) Provefor all s-states, ground and excited.(b) Check this relation for the ground state of a three-dimensional isotropic oscillator, the hydrogen atom, and so on. (This relation has actually been found to be useful in guessing the form of the potential between a quark and an antiquark.)
Answer Preview: To prove the given relation and then check it for specific cases, let's break down the problem step by step. (a) Proving the relation for all s-states (ground and excited): For a spherically symmetric …

, Chapter: 6 -Problem: 11 >> A spinless particle is scattered by a time-dependent potentialShow that if the potential is treated to first order in the transition amplitude, the energy of the scattered particle is increased or decreased by h??. Obtain d?/d?. Discuss qualitatively what happens if the higher-order terms are taken into account. Transcribe
Answer Preview: To analyze the scattering of a spinless particle by a time-dependent potential, we'll start by considering the first-order approximation in the transi…

, Chapter: 3 -Problem: 6 >> Let the Hamiltonian of a rigid body bewhere K is the angular momentum in the body frame. From this expression obtain the Heisenberg equation of motion for K, and then find Euler's equation of motion in the correspondence limit. Transcribed Image Text: H -1 (4+4+4). 2 13
Answer Preview: To obtain the Heisenberg equation of motion for K, we need to compute the time derivative of K using …

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, Chapter: 3 -Problem: 4 >> The spin-dependent Hamiltonian of an electron positron system in the presence of a uniform magnetic field in the z-direction can be written asSuppose the spin function of the system is given by(a) Is this an eigenfunction of H in the limit A ? 0, e B ? mc ? 0? If it is, what is the energy eigenvalue? If it is not, what is the expectation value of H?(b) Solve the same problem when e B ? mc ? 0, A ?
Answer Preview: The spin-dependent Hamiltonian of an electron-positron system in the presence of a uniform magnetic …

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, Chapter: 3 -Problem: 3 >> Consider the 2 x 2 matrix defined bywhere a0 is a real number and a is a three dimensional vector with real components.(a) Prove that U is unitary and unimodular.(b) In general, a 2 x 2 unitary unimodular matri x represents a rotation in three dimensions. Find the axis and angle of rotation appropriate for U in terms of a0 , a1, a2, and a3.
Answer Preview: (a) To prove that a 2x2 matrix U is unitary, we need to show that its conjugate transpose (U) is equal to its inverse (U). Let's calculate U and U: Th…

, Chapter: 2 -Problem: 23 >> A particle in one dimension is trapped between two rigid walls:At t = 0 it is known to b e exactly at x = L /2 with certainty. What are the relative probabilities for the particle to be found in various energy eigenstates? Write down the wave function for t ? 0. (You need not worry about absolute normalization, convergence, and other mathematical subtleties.)
Answer Preview: To find the relative probabilities for the particle to be found in various energy eigenstates, we need to calculate the probability amplitudes for eac…

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, Chapter: 5 -Problem: 15 >> Suppose the electron had a very small intrinsic electric dipole moment analogous to the spin-magnetic moment (that is, ?el proportional to ?). Treating the hypothetical -?el. E interaction as a small perturbation, discuss qualitatively how the energy levels of the Na atom (Z = 11) would be altered in the absence of any external electromagnetic field. Are the level shifts first order or second orde
Answer Preview: In this hypothetical scenario, we are considering the presence of a very small intrinsic electric dipole moment (el) for the electron in addition to i…

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, Chapter: 3 -Problem: 14 >> Show that the 3 x 3 matrices Gi(i = 1, 2, 3) whose elements are given bywhere j and k are the row and column indices, satisfy the angular-momentum commutation relations. What is the physical (or geometric) significance of the transformation matrix that connects Gi to the more usual 3 x 3 representations of the angular-momentum operator Ji with j3 taken to be diagonal? Relate your result tounder in
Answer Preview: The angular momentum commutation relations for the components of angular momentum operators are given by: [Ji, Jj] = iijkJk Where i, j, k are indices …

, Chapter: 3 -Problem: 32 >> (a) Write xy, xz, and (x2 - y2) as components of a spherical (irreducible) tensor of rank 2.(b) The expectation valueis known as the quadrupole moment. Evaluatewhere m? = j, j - 1, j - 2, . . . , in terms of Q and appropriate Clebsch-Gordan coefficients. Transcribed Image Text: Q = ela, j,m=j|(3z² —
Answer Preview: (a) To express xy, xz, and (x^2 - y^2) as components of a spherical (irreducible) tensor of rank 2, …

, Chapter: 2 -Problem: 25 >> A particle of mass m in one dimension is bound to a fixed center by an attractive ?-function potential:At t = 0, the potential is suddenly switched off (that is, V = 0 for t > 0). Find the wave function for t > 0. (Be quantitative! But you need not attempt to evaluate an integral that may appear.) Transcribed Image Text:
Answer Preview: To find the wave function for the particle with mass 'm' after the potential is suddenly switched off at t = 0, we can use the time-dependent Schrding…

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, Chapter: 5 -Problem: 9 >> Ap-orbital electron characterized by |n, l = l, m = ±1,0) (ignore spin) is subjected to a potential(a) Obtain the "correct" zeroth-order energy eigenstates that diagonalize the perturbation. You need not evaluate the energy shifts in detail, but show that the original threefold degeneracy is now completely removed.(b) Because V is invariant under time reversal and because there is no longer any de
Answer Preview: To find the "correct" zeroth-order energy eigenstates that diagonalize the perturbation, we first need to consider the unperturbed state and the perturbation potential. The unperturbed state for an el…

, Chapter: 5 -Problem: 26 >> A one-dimensional simple harmonic oscillator of angular frequency ? is acted upon by a spatially uniform but time-dependent force (not potential)At t = -?, the oscillator is known to be in the ground state. Using the timedependent perturbation theory to first order, calculate the probability that the oscillator is found in the first excited state at t = +?.Challenge for experts: F(t) is so normali
Answer Preview: To calculate the probability of the one-dimensional simple harmonic oscillator being in the first ex…

, Chapter: 5 -Problem: 22 >> Consider a one-dimensional simple harmonic oscillator whose classical angular frequency is wo. For t 0 there is also a time-dependent potentialwhere F0 is constant in both space and time. Obtain an expression for the expectation value ?x? as a function of time using time-dependent perturbation theory to lowest nonvanishing order. Is this procedure valid for ? ? ?0? [You may use
Answer Preview: To calculate the expectation value x as a function of time for the one-dimensional simple harmonic oscillator with the time-dependent potential V(t) = …

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, Chapter: 1 -Problem: 4 >> Using the rules of bra-ket algebra, prove or evaluate the following:(a) tr(XY) = tr(Y X), where X and Y are operators.(b) (XY)† = Y† X†, where X and Y are operators.(c) exp[if(A)] =? in ket-bra form, where A is a Hermitian operator whose eigenvalues are known.(d) Transcribed Image Text: ?a (x) (x"),
Answer Preview: Sure, let's go through each part one by one: (a) Proof of tr(XY) = tr(YX): In order to prove this, we need to use the properties of the trace operation and the commutation of operators. The trace of a…

, Chapter: 5 -Problem: 30 >> Consider a two-level system with E1 2. There is a time-dependent potential that connects the two levels as follows:At t = 0, it is known that only the lower level is populated-that is, c1(0) = 1, c2(0) = 0.(a) Findby exactly solving the coupled differential equation(b) Do the same problem using time-dependent perturbation theory to lowest nonvanishing order. Compare the two approaches for small va
Answer Preview: To solve this problem, we will first solve the coupled differential equations using the time-evolution operator method and then use time-dependent per…

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, Chapter: 2 -Problem: 3 >> An electron is subject to a uniform, time-independent magnetic field of strength B in the positive z-direction. At t = 0 the electron is known to be in an eigenstate of S • n? with eigenvalue h?/2, where n? is a unit vector, lying in the xz-plane, that makes an angle ? with the z-axis.(a) Obtain the probability for finding the electron in the sx = h? /2 state as a function of time.(b) Find the exp
Answer Preview: To solve this problem, we need to work with the time-evolution of the state of the electron in the p…

, Chapter: 2 -Problem: 24 >> Consider a particle in one dimension bound to a fixed center by a ?-function potential of the formFind the wave function and the binding energy of the ground state. Are there excited bound states? Transcribed Image Text: V(x)=-vod(x), (vo real and positive).
Answer Preview: To find the wave function and binding energy of the ground state for a particle bound to a fixed center by a -function potential, we need to solve the …

, Chapter: 5 -Problem: 36 >> Show that An (R) defined in (5.6.23) is a purely real quantity. Transcribed Image Text: An (R) i (n; t| [VR;t)], (5.6.23)
Answer Preview: To show that the quantity A_n(R) defined in equation (5 6 23) is purely real, we need to demonstrate …

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, Chapter: 5 -Problem: 39 >> A particle of mass m constrained to move in one dimension is confined within 0 < x < L by an infinite-wall potentialObtain an expression for the density of states (that is, the number of states per unit energy interval) for high energies as a function of E. Transcribed Image Text: V = ? V = 0 for x
Answer Preview: To obtain an expression for the density of states (DOS) for high energies as a function of e…

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, Chapter: 5 -Problem: 20 >> Estimate the ground-state energy of a one-dimensional simple harmonic oscillator usingas a trial function with ? to be varied. You may use Transcribed Image Text: (x|0) = e-Blx|
Answer Preview: To estimate the ground-state energy of a one-dimensional simple harmonic oscillator using the given trial function, we can use the variational principle. The variational principle states that for any …

, Chapter: 3 -Problem: 31 >> Consider a spinless particle bound to a fixed center by a central force potential.(a) Relate, as much as possible, the matrix elementsusing only the Wigner-Eckart theorem. Make sure to state under what conditions the matrix elements are nonvanishing.(b) Do the same problem using wave functions Transcribed Image Text:
Answer Preview: (a) Using the Wigner-Eckart theorem, we can relate the matrix elements (n', l', m' | F1(x + iy) | n, l, m) and (n', l', m' | z | n, l, m) for a spinle…

, Chapter: 6 -Problem: 6 >> Check explicitly the x - px uncertainty relation for the ground state of a particle confined inside a hard sphere: V = ? for r > a, V = 0 for r < a.
Answer Preview: To find the uncertainty relation for a particle confined inside a hard sphere potential, we need to determine the ground state wavefunction and then calculate the position and momentum uncertainties. …

, Chapter: 1 -Problem: 8 >> Using the orthonormality of |+? and | - ?, provewhere Transcribed Image Text: [Si, Sj]=i&ijkh Sk. (Si, Sj): = 7/2 ?² Sij,
Answer Preview: |+ and | - are likely representing basis states. [Si, Sj] represents the commutator of two operators Si and Sj. iijk is the Levi-Civita symbol. Sk, Si…

, Chapter: 5 -Problem: 13 >> Compute the Stark effect for the 2S1/2 and 2P1/2 levels of hydrogen for a field ? sufficiently weak that e?a0 is small compared to the fine structure, but take the Lamb shift ? (? =  1,057 MHz) into account (that is, ignore lP3/2 in this calculation).Show that for the energy shifts are quadratic in ?, whereas for they are linear in ?. (The radial integral you need isBriefly discuss the consequence
Answer Preview: you're discussing a quantum mechanical problem involving energy shifts and integrals related to perturbation theory. The problem involves analyzing th…

, Chapter: 3 -Problem: 16 >> Show that the orbital angular-momentum operator L commutes with both the operators p2 and x2; that is, prove (3.7.2). Transcribed Image Text: [L,p²] = [L,x²] = 0 (3.7.2)
Answer Preview: To show that the orbital angular momentum operator L commutes with both the operators p^2 and x^2, w…

, Chapter: 5 -Problem: 19 >> For the He wave function, usewith Zeff = 2 - 5/16, as obtained by the variational method. The measured value of the diamagnetic susceptibility is 1.88 x 10-6 cm3/mole.Using the Hamiltonian for an atomic electron in a magnetic field, determine, for a state of zero angular momentum, the energy, change to order B2 if the system is in a uniform magnetic field represented by the vector potential A = 1/
Answer Preview: The trial wave function given in problem 5 19 for the helium atom is: (x1, x2) = (Zeff / na) exp(-Zeff(|r1| + |r2|) / 16) where Zeff is the effective …

, Chapter: 5 -Problem: 35 >> Consider an atom made up of an electron and a singly charged (Z = 1) triton (3H). Initially the system is in its ground state (n = 1, l = 0). Suppose the system undergoes beta decay, in which the nuclear charge suddenly increases by one unit (realistically by emitting an electron and an antineutrino). This means that the tritium nucleus (called a triton) turns into a helium (Z = 2) nucleus of mass
Answer Preview: To find the probability for the system to be found in the ground state of the resulting helium ion after beta decay, we need to calculate the transition probability from the initial state (tritium wit…

, Chapter: 3 -Problem: 10 >> (a) Consider a pure ensemble of identically prepared spin 1/2 systems. Suppose the expectation values ?Sx? and ?Sz? and the sign of ?Sy? are known. Show how we may determine the state vector. Why is it unnecessary to know the magnitude of ?Sy??(b) Consider a mixed ensemble of spin 1/2 systems. Suppose the ensemble averages [Sx] , [Sy], and [Sz] are all known. Show how we may construct the 2 x 2 en
Answer Preview: (a) To determine the state vector of a pure ensemble of identically prepared spin 1/2 systems, we can start by considering the general form of the spi…

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, Chapter: 5 -Problem: 21 >> Estimate the lowest eigenvalue (?) of the differential equationusing the variational method withas a trial function. Numerical data that may be useful for this problem areThe exact value of the lowest eigenvalue can be shown to be 1 .019. Transcribed Image Text: d² y dx² +(2-x) = 0, 0 for|x|? ?
Answer Preview: In this problem, you're asked to estimate the lowest eigenvalue of a given differential equation using the variational method with a specific trial fu…

, Chapter: 1 -Problem: 1 >> Prove[AB,C D] =- AC{D, B} + A{C, B}D - C{D, A}B + {C, A}DB.
Answer Preview: To prove the given expression [AB,CD] = -AC{D,B} + A{C,B}D - C{D,A}B + {C,A}DB, we can use the defin…

, Chapter: 5 -Problem: 32 >> (a) In the presence of a uniform and static magnetic field B along the z-axis, the Hamiltonian is given bySolve this problem to obtain the energy levels of all four states using degenerate time-independent perturbation theory (instead of diagonalizing the Hamiltonian matrix). Regard the first and second terms in the expression for H as H0 and V, respectively. Compare your results with the exact ex
Answer Preview: (a) The Hamiltonian for the positronium in a uniform and static magnetic field B along the z-axis is given by H = AS1 S2 + ((eB / mec))(S1z - S2z) whe…

, Chapter: 5 -Problem: 24 >> Consider a particle bound in a simple harmonic-oscillator potential. Initially (t < 0), it is in the ground state. At t = 0 a perturbation of the formis switched on. Using time-dependent perturbation theory, calculate the probability that after a sufficiently long time (t ? ?), the system will have made a transition to a given excited state. Consider all final states.
Answer Preview: To calculate the probability of making a transition to a given excited state after a sufficiently long time using time-dependent perturbation theory, …

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, Chapter: 5 -Problem: 23 >> A one-dimensional harmonic oscillator is in its ground state for t (a) Using time-dependent perturbation theory to first order, obtain the probability of finding the oscillator in its first excited state for t > 0. Show that the t ? ? (? finite) limit of your expression is independent of time. Is this reasonable or surprising?(b) Can we find higher excited states? You may use
Answer Preview: (a) To find the probability of finding the one-dimensional harmonic oscillator in its first excited …

, Chapter: 3 -Problem: 27 >> Express the matrix elementin terms of a series in Transcribed Image Text: (?2?2?2J3|??????)
Answer Preview: To express the matrix element (2B2Y2J3 | 1B171) in terms of a series involving the Clebsch-Gordan co…

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, Chapter: 5 -Problem: 8 >> Evaluate the matrix elements (or expectation values) given below. If any vanishes, explain why it vanishes using simple symmetry (or other) arguments.[In (a) and (b), |nlm? stands for the energy eigenket of a nonrelativistic hydrogen atom with spin ignored.] Transcribed Image Text: (a) (n=2,1 1,m=0|
Answer Preview: (a) To evaluate the matrix element langle n = 2, l = 1, m = 0 | x | n = 2, l = 0, m = 0 angle, we need to consider the selection rules for the matrix elements of the position operator between differe…

, Chapter: 5 -Problem: 14 >> Work out the Stark effect to lowest nonvanishing order for the n = 3 level of the hydrogen atom. Ignoring the spin-orbit force and relativistic correction (Lamb shift), obtain not only the energy shifts to lowest nonvanishing order but also the corresponding zeroth-order eigenket.
Answer Preview: The Stark effect describes the energy shift of atomic energy levels in the presence of an external electric field. To calculate the Stark effect for t…

, Chapter: 1 -Problem: 21 >> Evaluate the x-p uncertainty product ?(?x)2 ? ?(?p)2? for a one-dimensional particle confined between two rigid walls,Do this for both the ground and excited states. Transcribed Image Text: 0 v = { % V for 0 < x
Answer Preview: To evaluate the uncertainty product (x) (p) for a one-dimensional particle confined between two rigi…

, Chapter: 5 -Problem: 27 >> Consider a particle in one dimension moving under the influence of some timeindependent potential. The energy levels and the corresponding eigenfunctions for this problem are assumed to be known . We now subject the particle to a traveling pulse represented by a time-dependent potential,(a) Suppose that at t = -? the particle is known to be in the ground state whose energy eigenfunction isObtain t
Answer Preview: In this problem, you're dealing with a particle in one dimension initially in a known ground state and then subjecting it to a traveling pulse represe…

, Chapter: 3 -Problem: 30 >> (a) Construct a spherical tensor of rank 1 out of two different vectors U = (Ux, Uy, Uz) and V = (Vx, Vy, Vz). Explicitly write T±(1)1,0 in terms of Ux,y,z and Vx,y,z.(b) Construct a spherical tensor of rank 2 out of two different vectors U and V. Write down explicitly T±2(2)±1,0 in terms of Ux,y,z and Vx,y,z.
Answer Preview: (a) To construct a spherical tensor of rank 1 out of two different vectors U = (Ux, Uy, Uz) and V = …

, Chapter: 3 -Problem: 19 >> Suppose a half-integer l-value, say 1/2, were allowed for orbital angular momentum. Fromwe may deduce, as usual,Now try to construct Y1/2,-1/2(?,?) by (a) applying L_ to Y1/2,-1/2(?,?); and (b) using L_ to Y1/2,-1/2(?,?) = 0. Show that the two procedures lead to contradictory results . (This gives an argument against half-integer !-values for orbital angular momentum.)
Answer Preview: Let's work through the two procedures to construct Y1/2, -1/2(, ) and show that they lead to contradictory results. Applying L+ to Y1/2, -1/2(, ): We …

, Chapter: 3 -Problem: 24 >> We are to add angular momenta j1 = 1 and j2 = 1 to form j = 2, 1 , and 0 states. Using either the ladder operator method or the recursion relation, express all (nine) {j,m} eigenkets in terms of |j1j2;m1m2?. Write your answer aswhere + and 0 stand for m1,2 = 1 , 0, respectively. Transcribed Image Text:
Answer Preview: To express the {j, m} eigenkets in terms of the given basis |j1 j2; m1 m2, we can use the ladder ope…

, Chapter: 5 -Problem: 17 >> (a) Suppose the Hamiltonian of a rigid rotator in a magnetic field perpendicular to the axis is of the formif terms quadratic in the field are neglected . Assuming B ? C, use perturbation theory to lowest nonvanishing order to get approximate energy eigenvalues.(b) Consider the matrix elementsof a one-electron (for example, alkali) atom. Write the selection rules for ?l, ?ml, and ?ms . Justify you
Answer Preview: (a) To find the approximate energy eigenvalues using perturbation theory, we consider the unperturbed Hamiltonian of the rigid rotator without the mag…

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, Chapter: 3 -Problem: 7 >> Let U = eiG3?eiG2? eiG3?, where (a ,?, ?) are the Eulerian angles. In order that U represent a rotation (a ,?, ?), what are the commutation rules that must be satisfied by the Gk? Relate G to the angular-momentum operators.
Answer Preview: To understand the commutation rules that the Gk operators must satisfy in order for U to represent a rotation, we need to examine the properties of th…

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, Chapter: 3 -Problem: 17 >> The wave function of a particle subjected to a spherically symmetrical potential V (r) is given by(a) Is ? an eigenfunction of L2? If so, what is the ! value? If not, what are the possible values of l that we may obtain when L2 is measured?(b) What are the probabilities for the particle to be found in vari ous ml states?(c) Suppose it is known somehow that ?(x) is an energy eigenfunction with eige
Answer Preview: To analyze the given wave function and its properties, let's start by expressing the wave function in spherical coordinates. The spherical coordinates are related to Cartesian coordinates as follows: …

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14 Reviews for Modern Quantum Mechanics Textbook Questions And Answers