Quantum Mechanics: A Paradigms Approach David H. McIntyre. 27 July Corrections to the 1 st printing

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1 Quantum Mechanics: A Paradigms Approach David H. McIntyre 7 July 6 Corrections to the st printing page xxi, last paragraph before "The End", nd line: Change "become" to "became". page, first line after Eq. (.4): Change "gyroscopic" to "gyromagnetic". page, Problem.9, first line: Change "A pair of dice" to "Two dice". page 5, paragraph after Eq. (.), penultimate line: Change " " to " ". page 44, Eq. (.5): Change the "+" superscript in the nd and rd lines to the proper adjoint symbol " ": φ β β φ * { } * ψ A β β A ψ (.5) ψ A β β A ψ * page 7, Eq. (.8): Change the " + " ket in the nd line to " ": H + ω S z + ω + E + + H ω S z ω E (.8) page 8, Eq. (.65): In the st line, change the vertical bar on the st bra to a left angle bracket to make a proper bra symbol: ψ + y y + + y y + y y + + y y y + + y + y y (.65) i + y + i y page 84, last sentence of paragraph after Eq. (.68): Change sentence to read: "Neutrinos are so elusive because they interact via the weak force or weak interaction, which is much weaker than the strong nuclear force and electromagnetism, but not as weak as gravity." page 88, line before Eq. (.84): Delete duplicate "the".

2 page 96, Problem.6: (i) st line: After the term "Rabi oscillations", insert the parenthetic comment "(i.e., Rabi flopping)"; (ii) Part (b), Delete "complete" and after "flip" insert the parenthetic comment "(i.e., P + ( t) P and P ( t) P + )". page 4, lines before Eq. (4.5): Change "π-pulse" to "π/-pulse". page 6, rd reference (Mermin's moon article): Change issue number from "(5)" to "(4)". page 8, Eq. (5.): Change to π : f ij ω ij π E E i j h λ ij c hc f ij E i E j (5.) page 7, Eq. (5.7): Change the "ϕ " symbol to the "φ " symbol all places: P ψ >φ φ ψ φ * ( x)ψ ( x)dx (5.7) page 4, nd line: Change x < L to x > L. page 6, Eq. (5.97): (i) Change ϕ symbols in bra-ket to E, (ii) change last subscript from "m" to "n": page 4, Example 5.4: renumber to 5.5. E m qx E n ϕ m ( x) qx ϕ n ( x)d r (5.97) page 4, Eq. (5.): Insert in denominator of exponent term e ie nt in line : c n ϕ n x ψ x,t e ie nt c n L n 6 L n nπ n n sin nπx nπx sin L e in π t ml L e in π t ml page 44, nd line after Eq. (5.): Change "Example 5." to "Example 5.4". page 47, st line after Fig. 5.4: Change ".67 ev" to ".4 ev.4 ev". page 56, Problem 5.: Change ϕ symbols inside kets to E: and E n ψ ( t ) A E E + i E page 57, Problem 5.7: Change ϕ symbols inside kets to E: 5 and E n ψ ( t ) E i E page 58, Problem 5., end of second line: Change " x > L " to " x > a ". (5.) 7/7/6

3 page 7, Eq. (6.7): Add curly brackets to match the text in the subsequent paragraph: ψ (x,t) ( x pt m) { } + cos δ p π eip x p m t (6.7) page 74, Eq. (6.5): Add curly and square brackets to match the text in the subsequent paragraph: ψ (x,t) πα 4 γ e ip ( x p t m) { } e x p t m 4α γ (6.5) page 8, Eq. (6.75): Move zero subscript on momentum space representation of position eigenstate from p to x: Position space Position eigenstate x δ ( x x ) x Momentum eigenstate p π ei p x Momentum space π ei px p δ ( p p ) (6.75) page 99, Problem 6., centered equation: move "x" before division symbol: ψ (x) Ae i p x e x 4α page 7, Eqs. (7.77) and (7.78): Change the sign of the second term on the left-hand side of both equations: R( r) d dr r dr r + µ (E V (r))r dr Y ( θ,φ) L Y ( θ,φ) d dr r dr ( r ) dr R r function of r only + µ (E V (r))r Y θ,φ function of θ,φ only (7.77) L Y ( θ,φ) A (7.78) page 5, paragraph after Eq. (7.5), penultimate line: After the word "determine", insert the parenthetic comment "(with some ambiguity)". page, Eq. (7.48): Remove "*" indicating complex conjugate: P k (z)p (z)dz + δ (7.48) k page 5, Eq. (7.56): Place the normalization factor inside a square root: Θ m m ( +) ( m)! ( θ ) ( ) ( + m)! P m (cosθ), m (7.56) 7/7/6

4 page 9, Eq. (7.7): Change lower limit of sum to absolute value of m: P Lz m m ψ (7.7) m page 4, Spherical harmonic terms throughout this example are mislabeled, so (i) change to in Eqs. (7.7), (7.74), (7.75), and (7.76); and (ii) change energy and angular momentum values in sentence after Eq. (7.76): ψ ( θ,φ) Y ( θ,φ) + Y ( θ,φ) (7.7) ψ +, (7.74) c m m ψ δ δ m + δ δ m, (7.75) P E m ψ m δ δ + δ (7.76) New Sentence after Eq. (7.76): The probability of measuring the energy to be E I is %, as is the probability for measuring L 6. page 4, Eq. (7.77), st line: Change lower limit of sum to : P Lz ψ (7.77) page 47, Problem 7.7, centered equation: Change " " to "": P( φ,t) sin φ + π I t page 48, Problem 7.6, centered equation: Change sign to sign: L ± e ±iφ ± θ + i cotθ φ 7/7/6 4

5 page 5, Eq. (8.9): Change prefactors of H ( ρ) and H ρ d R dρ ρ e γρ ( ) ρ γ + γ ρ H ρ terms: + ( γρ ) H ( ρ) + ρ H ( ρ) (8.9) page 7, Eq. (8.97), nd line: Change in st denominator to 4 ; rd line: Change in st denominator to 4, change exponent in e r a to e r a, change 4 in last denominator to. ψ (r,θ,φ,t) ψ (r,θ,φ)e ie t/ + ψ (r,θ,φ)e ie t/ r 4 πa a e r /a e ie t/ + πa e ie t/ r 4 πa a e r /a + z e r /a a r cosθ e r /a ie e t/ 8a (8.97) page 7, Problem 8., both equations: Delete nm subscripts from wave functions: ψ (x, y,z) dx dy π ψ (ρ,φ,z) ρ dρ dφ P z P z page 74, Problem 8.5, centered equation: Add "x, y," to argument of function: V ( x, y,z), < x < L, < y < L, < z < L, otherwise page 75, two lines after Eq. (9.): Delete "to" before "zero". page 77, penultimate paragraph, rd line: Delete "what" near end of line. page 8, Eq. (9.8), st line: On right-hand side of equation, delete left parenthesis and change right parenthesis to right bracket: ae E ωa E ( E ω )( a E ) H a E page, last line: Insert spaces on either side of " p ". page, last paragraph, end of 4 th line: add space to separate "of ampli-". page, Problem 9., part (e): Change "that" to "than". page, Eq. (.47): Change subscript on "H" to "nn": (9.8) E () n H nn n () H n () (.47) 7/7/6 5

6 page, Eq. (.49): Insert and remove minus signs to correct sign error in energy: H µ i B ω S z ω ω ω ω (.49) page, nd line after Eq. (.49): (i) Insert and remove minus signs to correct sign error in energy, (ii) Change "decreasing" to "increasing". New line: ( E ) ( ω, E ) ( ω, E ) ( ω, and E ) 4 ω, labeled in order of increasing page, Eq. (.5): Insert and remove minus signs to correct sign error in energy: H µ i B ω S z ω ω ω ω (.5) page 4, Fig..4: Insert and remove minus signs on m labels: From top, m, m, m, m. page 4, Eq. (.5): Insert and remove minus signs to correct sign error in energy: ( E ) ω ( E ) ω ( E ) ω ( E ) 4 ω (.5) page, Eq. (.8): Insert and remove minus signs to correct sign error in energy: H µ i B ω S z ω ω ω ω (.8) page, nd line after Eq. (.8): Insert and remove minus signs to correct sign error in energy. New energies: ( E ) ( ω, E ) ( ω, E ) ( ω, and E ) 4 ω page, first line: Change subscript on B to : B B ˆx 7/7/6 6

7 page, Eq. (.8): Insert minus signs to correct sign error in energy: H µ i B ω S x ω ω 4 ω 4 ω ω ω (.8) page, Eq. (.85): (i) Sum over m, (ii) Insert minus signs ( E ) H m m E ( ) E ( ) ( ) m ω ( ω ω ) ω 4ω H E ( ) E ( ) ( ) (.85) page, Eq. (.86): (i) Sum over m, (ii) Insert and remove minus signs ( E ) H m m E ( ) E ( ) ( ) m H E ( ) E ( ) ( ) 4 ω ( ω ω ) + ω ω 4ω ω ω + H E ( ) E ( ) ( ) (.86) page 4, Eq. (.87): Remove minus signs: ( E ) ω 4ω ( E ) 4 ω 4ω (.87) 7/7/6 7

8 page 4, Eq. (.89): (i) Sum over m, (ii) Insert minus signs in nd line: m H m E ( ) E ( ) m ( ) m ω ω ω H E ( ) E ( ) ( ) (.89) ω ω page 4, Fig..8: Insert and remove minus signs on m labels: From top, m, m, m, m. page 48, Eq. (.4): (i) st line: Insert "" in denominator of wave function polynomial; (ii) rd line: Change " ± " symbol in wave function amplitude to " ". ψ ψ, ± ψ ± ( r,θ,φ) R ( r)y θ,φ ( r,θ,φ) R ( r)y θ,φ r ( r,θ,φ) R ( r)y ± θ,φ a a e r a 4π r e r a ( a ) a 4π cosθ r e r a ( a ) a 8π e±iφ sinθ (.4) page 49, Eq. (.47), nd line: (i) Insert a term before integral, (ii) Insert "" in denominator of wave function polynomial:: H ψ ( )* r,θ,φ ee r cosθ ψ ee ( a ) 4π π cos θ sinθ dθ π ( r,θ,φ)r sinθ dr dθ dφ r ( a ) 4π a a dφ e r a r 4 dr (.47) page 49, Eq. (.48): (i) st line: Insert "" superscripts in ket and bra; (ii) st line: Remove extraneous square right bracket before sign; (iii) st line: Insert " a " in 4π denominator before integral; (iv) nd line: Change denominator before square brackets to a 4 : H ee ( a ) 4πa π r 4 e r a dr a ee 4!a !a a a eea r 5 e r a dr (.48) 7/7/6 8

9 page 56, Eq. (.4): Insert factor of g e before st term in brackets: H hf µ g e µ B g p µ N 4π g e r I i L r S i I + r 5 page 64, Eq. (.45): Change 4 in top line to : ΔE hf µ g e µ B g p µ N α 4 m e c πa g g e p ( S i r) ( I i r) + 8π S i Iδ ( r) m e m p ev h (9)MHz (.4) (.45) page 7, last paragraph: (i) rd line: Change "coupled" to "uncoupled", (ii) 7 th line: Change "the coupled and the coupled" to " the coupled and the uncoupled". pages 75 & 76, Tables..5: Header symbols j and m should be capitalized J and M to be consistent with text. page 98, Eq. (.6): Add minus signs to two elements of the matrix (second / and second / on diagonal): L z page 99, st line after Eq. (.67): Change "play" to "plays". page 4, Eq. (.86): Change exponent on st n to 4: α 4 mc 4n + 4 n + E fs m m s ( +),,,,,, (.6) (.86) page 48, Problem.4, part (b), st and nd lines: Change "the formula we derived in class" to "Eq. (.46)". page 46, Eq. (.6): Interchange last two wave functions on right hand side: SS ψ S ψ ( x, x ) ϕ ( x )ϕ ( x ) + ϕ ( x )ϕ ( x ) (.6) page 45, Eq. (.56): Interchange last two wave functions on right hand side: ψ AS M ϕ x ψ A x, x ϕ ( x ) ϕ ( x )ϕ ( x ) M (.56) 7/7/6 9

10 page 4, Eq. (.79): Add complex conjugation symbols to two wave functions on st line and remove them from two wave functions on nd line: E s, ψ * * * * ( r )ψ m ( r ) ± ψ m ( r )ψ r 4πε r r (.79) ψ r ψ m ( r ) ± ψ m ( r )ψ ( r ) d r d r page 44, Problem.7: Delete second half of last sentence after comma (no integrals required). page 444, Problem.7: (i) part (b): Change "wave function" to "quantum state vector", (ii) part (d): Change "wave functions" to "quantum state vectors". page 448, Eq. (4.6): Change the subscript on the second energy from "n" to "i": i dc k ( t) e i ( E k E i )t k H t dt e i (4.6) page 45, Figure 4., lower label: Insert "/" so that marked time interval is π ω fi ω. page 46, Eq. (4.69), second line: Change the first "E" symbol to a script font for electric field: R π V πe E g( E f ) ˆε i r A π E ω + A page 467, Problem 4.: Change starting time from "t " to "t ". page 468, Problem 4.: Change "l " to " ". (4.69) pages 48 and 48, Eqs. (5.8) and (5.9): An overall phase (which is physically meaningless) has been omitted from Eqs. (5.8) and (5.9). page 48, Fig. 5.: Each wave function is defined only over the range.5a x 8.5a (given that we are using periodic boundary conditions). page 485, Eq. (5.4): Insert absolute value symbols: g( E) g k ( k) dk de L dk π de page 485, Eq. (5.44): Insert minus sign: de dk (5.4) βasin ( ka ) (5.44) 7/7/6

11 page 485, Eq. (5.45): Insert absolute value symbol: g( E) L πβasin ka (5.45) page 486, last line: Change "-.6 mev" to "-.5 mev". page 488, middle of last paragraph before Section 5.8.: Change "- mev" to "- mev". page 49, Eq. (5.58): Change sign of κ term in exponents (4 times): e ika ψ I ( b) ψ II ( b + a) Ae iqb + Be iqb e ika Ce e ika ψ I '( b) ψ II '( b + a) q Ae iqb Be iqb + De iκ a b iκ a b κe ika iκ a b Ce De page 49, Eq. (5.59): Change sign of κ term in exponents (4 times): e iqb e iqb e ika iκ a b e e ika e q q κ κ iκ a b qe iqb qe iqb κe ika iκ a b e κe ika iκ a b e A B C D iκ a b (5.58) (5.59) page 49, Eq. (5.6): Add absolute value symbols to κ terms ( more times): cos( qb)cosh κ a b ( ) q κ q κ sinh κ a b sin qb page 5, Eq. (6.47): Insert prefactor before last term: ( ) cos( ka) (5.6) ψ U CNOT ψ (6.47) page 546, First line after Eq. (D.6): Insert "we" after "if". 7/7/6