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ฟิสิกส์โอลิมปิก วิทยาศาสตร์โอลิมปิก ข้อสอบแข่งขัน ข้อสอบชิงทุน => GRE - Physics => Topic started by: conantee on January 04, 2010, 04:39:39 AM

Title: GR8677.057
Post by: conantee on January 04, 2010, 04:39:39 AM
57. Which of the following is an eigenfunction of the linear momentum operator  -i \hbar \frac{\partial}{\partial x} with a positive eigenvalue  \hbar k; i.e., an eigenfunction that describes a particle that is moving in free space in the direction of positive x with a precise value of linear momentum?
(A)  \cos kx
(B)  \sin kx
(C) e^{-ikx}
(D) e^{ikx}
(E) e^{-kx}

Title: Re: GR8677.057
Post by: conantee on February 01, 2010, 08:17:15 AM
Let's remind ourselves what the eigenfunction of an operator Q is...
f is eigenfunction of an operation Q if Qf=cf where c is constant.

In this case, we seek for f s.t. -i \hbar \frac{\partial}{\partial x}f(x) = \hbar k f(x).

Looking at choices we have, (A) and (B) are wrong since their derivatives are not in the form of themselves (derivative of sine is cosine, not sine itself, for example).
(E) is also wrong, since its derivative does not give i to cancel with i next to the derivative.

Now, just try plug (C) and (D) to check which one is a solution. We found that the answer is (D).

The correct answer is (D). 59 of 100 got the correct one.