Question

Physics 51A Spring 2017 May 31, 2017 Week 9

As shown in Example 7.2, the expectation value of a measurable quantity can be determined by ’sandwiching’ the quantity in between ψ ∗ and ψ and integrating over all space where ψ is nonzero: < x >= Z a 0 ψ ∗xψdx (1) (For real valued ψ, such as sin(x), ψ = ψ ∗ ). Given a particle in a box spanning (0 < x < a) with ψ = q2 a sin nπx a , determine the expectation value for the operators ˆp and ˆp 2 , where ˆp = −ih¯ d dx . ˆp is actually the momentum operator. Appendix B may provide help with the integrals.

Physics 51A Spring 2017 May 31, 2017 Week 9 As shown in Example 7.2, the expectation value of a measurable quantity can be determined by "sandwiching' the quantity in between and b and integrating over all space where v is nonzero: (1) (For real valued y, such as sin Given a particle in a box spanning (0 z a) with t V sin Caz), determine the expectation value for the operators p and p2, where p ih p is actually the momentum operator. Appendix B may provide help with the integrals.

Answer 1

Psi = squareroot 2 /a sin m phi x/a expectation value for p^^= -ih d/dx is < p_x > integral^a_0 [squareroot 2 /a sin n phi x /a]^* [- I h d /dx] (squareroot 2 /a sin n x /a) dx = -I h2 /a inrgeal^a_0 sin nphi x /a d /dx (sin n phi /a) dx = -ih2 /a integral^a_0 sin n phi x /a n phi /a cos n phi x /a dx = (- in 2 /a) (n phi /a) integral^a_0 sin n phi x /a cos n phi x /a dx = (- 2 ih /a) (n phi /a) integral^a_0 1 /2 sin 2 n phi x /a dx = (- in /a) (na /a) [- cos 2n phi x /a]^a_0 = (in /a) (n phi /a) [cos 2n phi - cos 0] or, < Px > = 0

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