Solutions Manual for Introduction to Quantum Mechanics (2nd edition)

By David J. Griffiths

Options guide for advent to Quantum Mechanics

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Likewise sin4 θ four pe 2 π = zero, so zero = zero. 1 = − e (ψ1 ± ψ3 )2 (r)r2 sin θ dr dθ dφ 2 2 1 1 1 r r ˆ 2 sin θ dr dθ dφ =− e 1− ± cos θ e−r/a r(sin θ cos φ ˆi + sin θ sin φ ˆj + cos θ k)r 2 2 2πa 4a 2a 2a 2 e okˆ 1 r r =− 2π 1 − r3 e−r/a cos θ sin θ dr dθ. ± cos θ 2 2πa 4a2 2a 2a ± π zero cos θ sin θ dθ = pe ± π zero cos3 θ sin θ dθ = zero, so merely the cross-term survives: 1 r e ˆ okay ± 1− r cos θ r3 e−r/a cos θ sin θ dr dθ 8a3 a 2a π ∞ r e ˆ e ˆ 2 1 2 1− =∓ cos θ sin θ dθ okay ok r4 e−r/a dr = ∓ four! a5 − five!

Cos θ cos θ sin θ dθ = zero     zeroπ    2 4|Hs |4 = {. . . } sin θ cos θ sin θ dθ = zero     All matrix components of Hs are 0 zero   2π  iφ other than 1|Hs |3 and 3|Hs |1 1|Hs |2 = {. . . } e dφ = zero (which are advanced conjugates,  zero  2π   so just one has to be evaluated). −iφ   1|Hs |4 = {. . . } e dφ = zero    zero  2π     2|Hs |3 = {. . . } e−iφ dφ = zero    zero   2π   −2iφ  2|Hs |4 = {. . . } e dφ = zero    zero   2π   −iφ   3|Hs |4 = {. . . } e dφ = zero zero 1 1 1 1 √ 2πa 2a π2πa 4a2 r e−r/2a re−r/2a cos θ(r cos θ)r2 sin θ dr dθ dφ 2a ∞ eEext r 2 = 1− (2π) cos θ sin θ dθ e−r/a r4 dr three 2πa8a 2a zero zero ∞ ∞ eEext 2 1 eEext 1 four −r/a = four!

This fabric is safe less than all copyright legislation as they at the moment exist. No component of this fabric should be reproduced, in any shape or whatsoever, with out permission in writing from the writer. one hundred twenty bankruptcy four. QUANTUM MECHANICS IN 3 DIMENSIONS + s2 (s2 + 1) − (m − 12 )(m − 32 )|s2 m − | 12 − i 2 + s2 (s2 + 1) − (m − 12 )(m + 12 )|s2 m + 1 2 2i − s2 (s2 + 1) − (m − 12 )(m − 32 )|s2 m − +B three 2 1 |2 four +2 1 1 2|2 2 2 + −i 2 | 12 − 1 2 |s2 m + 1 2 2 + 2 + = 2 A 2 three four 1 2 |s2 m + three 2 1 2 + s2 (s2 + 1) − m − s(s + 1)|s m = 2 s(s + 1) A| 12 | 12 − |s2 m − + B| 12 − 1 2 1 2 1 2 (m + 12 )|s2 m + | 12 1 four s2 (s2 + 1) − m2 + +A 1 2 − 2 + s2 (s2 + 1) − (m + 12 )(m − 12 )|s2 m − 1 2 |s2 m − | 12 − 1 four |s2 m + 1 2 + m +B s2 (s2 + 1) − m2 + 1 four  = s(s + 1)A,   B s2 (s2 + 1) + 1 four − m +A s2 (s2 + 1) − m2 + 1 four = s(s + 1)B,  1 four + m +B s2 (s2 + 1) − m2 + 1 four  = zero,   B s2 (s2 + 1) − s(s + 1) + 1 four − m +A s2 (s2 + 1) − m2 + 1 four = zero,  1 2 .

Fifty six 2 The Schr¨ odinger equation says − 2m ψ = Eψ, or, with the right kind energies (Eq. 2. 27) and a = 1, ψ + (nπ)2 ψ = zero. I’ll commence with a “guess” utilizing nine in preference to π 2 (that is, I’ll use nine for the floor kingdom, 36 for the first excited country, eighty one for the subsequent, and finally 144). Then I’ll tweak the parameter till the graph crosses the axis correct at x = 1. the consequences (see less than) are, to five significant digits: nine. 8696, 39. 478, 88. 826, 157. ninety one. (The real energies are those numbers elevated by means of 2 /2ma2 .

Zero ≤ x ≤ a); ψ(−x) = ψ(x). 2π threeπ ka -h2k mα tan(ka) From the graph, the allowed energies are just a little above ka = nπ (n = 1, three, five, . . . ) so 2 En n2 π 2 2 (n = 1, three, five, . . . ). 2m(2a)2 those energies are a bit of better than the corresponding energies for the infinite sq. good (Eq. 2. 27, with a → 2a). As α → zero, the directly line (− 2 k/mα) will get steeper and steeper, and the intersections catch up with to nπ/2; the energies then decrease to these of the standard infinite good. As α → ∞, the immediately line techniques 2 2 2 π horizontal, and the intersections are at nπ (n = 1, 2, three, .

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