ME501HW1SolnF2011

# ME501HW1SolnF2011 - 1 ME 501 Homework#1 Due Friday Prof...

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1 ME 501 Homework #1 Due Friday, September 16, 2011 Prof. Lucht (E-mail address: [email protected] ) Note: A useful table of integrals in posted on the class website. 1. Laurendeau, Problem 3.2, p. 147.

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4 2. A particle of mass m is constrained to move along the x-axis in the region 0 x L . The normalized wavefunction for the particle is given by (,) () () s i n e x p , xt xTt L nx L i t xL x x L hn mL  F H I K F H I K   2 0 00 8 11 1 2 1 2 2 The uncertainty B in the determination of the value of a dynamical variable B is given by the square root of the variance, BB B  2 2 . For a particle with quantum number 1 3 n , determine the value of the product x p x . Is this result consistent with Heisenberg's uncertainty principle? Solution:          22 2* 2 * 0 ** * 2 1 0 ,, sin exp exp sin 2 sin xxx L L p p p xx t x x t d x x t xT t t t ii LL d x             From the table of integrals we have: 32 1 sin sin 2 cos 2 64 8 4 x x ax dx ax ax aa a Evaluating the integral for 1 n a L we obtain:
5 32 2 1 3 2 21 sin 64 8 nx xx x La a L      1 2 0 33 22 1 2 11 2 cos 4 1 cos 2 0.3277 63 3 1 8 42 L x aL LL nL L L nn         ** 2 1 00 2 ,, s i n x xt x xt dx x dx   Again from the table of integrals we have 2 2 2 1 sin sin 2 cos 2 44 8 x ax dx ax ax aa 2 1 2 2 sin x LaL 1 2 0 1 2 1 2 1 cos 8 2 cos 2 1 0.5000 8 L L L n 2 2 0.3277 0.5 0.279 x L L L  The expectation value of the x-momentum is zero. The particle is just as likely to be moving in the +x and –x direactions:   * 1 0 2 1 0 1 2 s i n s i n 2 sin cos 1 sin cos sin 2 2 sin cos sin 2 x L L i p x t i x t dx dx xL L x L x n x i dx L L ax ax dx ax a L dx  0 0 0 L x p The expectation value of 2 x p is nonzero

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6  2 22 2* 11 2 00 2 2 2 0 0 1 0 2 2 ,, s i n s i n 2 sin 1 sin sin 2 24 2 sin sin 2 LL x L L L x nx p x t i x t dx dx xL L x L nn x dx L x ax dx ax a L dx Ln L p            2 2 2 L L For 1 3 n : 2 2 2 2 2 2 3 88.83 9.43 0.279 9.43 2.63 0.419 0.0796 4 x xx x x p pp p L h x pL h h L      The result is consistent with the Heisenberg uncertainty principle.
7 3. A particle confined along a line between x = 0 and x = L is in a "mixed" quantum state, i.e., a linear superposition of the eigenstates A and B with quantum numbers 1 3 A n and 1 5 B n , respectively.

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ME501HW1SolnF2011 - 1 ME 501 Homework#1 Due Friday Prof...

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