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

By David J. Griffiths

Recommendations guide for advent to Quantum Mechanics

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29, the wave functionality is D sin lx, F e−κx , ψ(x) = l ≡ 2m(E + V0 )/ √ κ ≡ −2mE/ (0 < x < a), (x > a), and the boundary stipulations at x = a yield − cot z = with √ z0 = 2mV0 a= (z0 /z)2 − 1 2m(32 2 /ma2 ) a = eight. pertaining to the figure (Problem 2. 29), and noting that (5/2)π = 7. eighty five < z0 < threeπ = nine. forty two, we see that there are 3 sure states. (b) allow a I1 ≡ a |ψ|2 dx = |D|2 zero zero ∞ I2 ≡ sin2 lx dx = |D|2 ∞ |ψ|2 dx = |F |2 a x 1 − sin lx cos lx 2 2l e−2κx dx = |F |2 − a −2κx ∞ 2κ a e = |F |2 a = |D|2 zero −2κa e 2κ a 1 − sin lz cos los angeles ; 2 2l . 2 yet continuity at x = a ⇒ F e−κa = D sin l. a., so I2 = |D|2 sin2κla . Normalizing: a κ 1 sin2 l. a. 1 − sin l. a. cos l. a. + = |D|2 κa − sin l. a. cos l. a. + sin2 l. a. 2 2l 2κ 2κ l yet (referring back to challenge 2. 29) κ/l = − cot l. a., so 1 (1 + κa) = |D|2 κa + cot l. a. sin l. a. cos los angeles + sin2 los angeles = |D|2 . 2κ 2κ 1 = I1 + I2 = |D|2 So |D|2 = 2κ/(1 + κa), and the likelihood of finding the particle open air the good is P = I2 = 2κ sin2 los angeles sin2 l. a. = . 1 + κa 2κ 1 + κa we will convey this interms of z ≡ l. a. and z0 : κa = sin2 l. a. = sin2 z = z02 − z 2 (page 80), 1 1 = = 1 + (z0 /z)2 − 1 1 + cot2 z z z0 2 ⇒P = z2 z02 (1 + z02 − z 2 ) . up to now, this is often right for any sure country. within the current case z0 = eight and z is the 3rd technique to − cot z = (8/z)2 − 1, which happens someplace within the period 7. eighty five < z < eight. Mathematica supplies z = 7. 9573 and P = zero. 54204. c 2005 Pearson schooling, Inc. , higher Saddle River, NJ. All rights reserved. This fabric is secure lower than all copyright legislation as they at present exist. No component of this fabric might be reproduced, in any shape or whatsoever, with out permission in writing from the writer. ¨ bankruptcy 2. THE TIME-INDEPENDENT SCHRODINGER EQUATION forty four challenge 2. forty-one (a) within the normal notation ξ ≡ mω/ x, α ≡ (mω/π )1/4 , Ψ(x, zero) = A(1 − 2ξ)2 e−ξ 2 /2 = A(1 − fourξ + fourξ 2 )e−ξ 2 /2 . it may be expressed as a linear mix of the first 3 desk bound states (Eq. 2. fifty nine and a couple of. sixty two, and challenge 2. 10): √ 2 2 2 α ψ0 (x) = αe−ξ /2 , ψ1 (x) = 2 αξe−ξ /2 , ψ2 (x) = √ (2ξ 2 − 1)e−ξ /2 . 2 √ √ 2 2 1 −ξ /2 So Ψ(x, zero) = c0 ψ0 + c1 ψ1 + c2 ψ2 = α(c0 + 2ξc1 + 2ξ c2 − √2 c2 )e with (equating like powers)  √ √  ⇒ c2 = 2 2A/α, α√2c2 = 4A √ α 2c1 = −4A ⇒ c1 = −2 2A/α,  √ √  α(c0 − c2 / 2) = A ⇒ c0 = (A/α) + c2 / 2 = (1 + 2)A/α = 3A/α. Normalizing: 1 = |c0 |2 + |c1 |2 + |c2 |2 = (8 + eight + 9)(A/α)2 = 25(A/α)2 ⇒ A = α/5. √ √ three 2 2 2 2 c0 = , c1 = − , c2 = . five five five H = 1 nine |cn |2 (n + ) ω = 2 25 1 eight ω + 2 25 three eight ω + 2 25 five ω 2 = seventy three ω (9 + 24 + forty) = ω. 50 50 (b) √ √ √ √ three 2 2 2 2 −iωt/2 2 2 −3iωt/2 2 2 −5iωt/2 −iωt/2 three −iωt Ψ(x, t) = ψ0 e ψ1 e ψ2 e ψ0 − ψ1 e ψ2 e−2iωt . − + =e + five five five five five five to alter the signal of the center time period we'd like e−iωT = −1 (then e−2iωT = 1); obviously ωT = π, or T = π/ω. c 2005 Pearson schooling, Inc. , higher Saddle River, NJ. All rights reserved. This fabric is safe below all copyright legislation as they at present exist. No section of this fabric will be reproduced, in any shape or whatsoever, with out permission in writing from the writer.

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