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1. Consider a particle constrained to two spatial dimensions. Suppose it is in a state represented by the (un-normalized) wavefunction given by: (r, 6) = re-ar² sin o, (1) where a is a real constant and (r, o) are the two-dimensional polar coordinates defined in the Figure below. (r, 4) P m!! = y Ø FIG. 1. Polar coordinate system For Parts (a) and (b) you may find the following hints useful. Hint 1: recall the expression of an infinitesimal area in polar coordinates is dA= drdy = rdrdo. Hint 2: Feel free to make use of the following definite integrals: [ 1²4e-an² d du = (2n-1)!! T an2n+1 = S u²n+1e-au² du: n! 2an+1¹ (3) where n is a positive integer, and the definition of the double factorial of a positive integer m is given by: 2 [mx (m - 2) × (m-4) × ... x 6 x 4 x 2, for even m mx (m2) × (m - 4) × x 5 x 3 x 1, for odd m (4) (a) Find the normalized form of the wavefunction (r, o). Call the normalized wavefunction you find (r, o) and use this for Part (b) below.

1. Consider a particle constrained to two spatial dimensions. Suppose it is in a state represented by the (un-normalized) wavefunction given by: (r, 6) = re-ar² sin o, (1) where a is a real constant and (r, o) are the two-dimensional polar coordinates defined in the Figure below. (r, 4) P m!! = y Ø FIG. 1. Polar coordinate system For Parts (a) and (b) you may find the following hints useful. Hint 1: recall the expression of an infinitesimal area in polar coordinates is dA= drdy = rdrdo. Hint 2: Feel free to make use of the following definite integrals: [ 1²4e-an² d du = (2n-1)!! T an2n+1 = S u²n+1e-au² du: n! 2an+1¹ (3) where n is a positive integer, and the definition of the double factorial of a positive integer m is given by: 2 [mx (m - 2) × (m-4) × ... x 6 x 4 x 2, for even m mx (m2) × (m - 4) × x 5 x 3 x 1, for odd m (4) (a) Find the normalized form of the wavefunction (r, o). Call the normalized wavefunction you find (r, o) and use this for Part (b) below.

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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how do i do a? I sit by finding the constant = 1? Pls help me

1. Consider a particle constrained to two spatial dimensions. Suppose it is in a state represented by the (un-normalized) wavefunction given by: (r, 6) = re-ar² sin o, (1) where a is a real constant and (r, o) are the two-dimensional polar coordinates defined in the Figure below. (r, 4) P m!! = y Ø FIG. 1. Polar coordinate system For Parts (a) and (b) you may find the following hints useful. Hint 1: recall the expression of an infinitesimal area in polar coordinates is dA= drdy = rdrdo. Hint 2: Feel free to make use of the following definite integrals: [ 1²4e-an² d du = (2n-1)!! T an2n+1 = S u²n+1e-au² du: n! 2an+1¹ (3) where n is a positive integer, and the definition of the double factorial of a positive integer m is given by: 2 [mx (m - 2) × (m-4) × ... x 6 x 4 x 2, for even m mx (m2) × (m - 4) × x 5 x 3 x 1, for odd m (4) (a) Find the normalized form of the wavefunction (r, o). Call the normalized wavefunction you find (r, o) and use this for Part (b) below.
Transcribed Image Text:1. Consider a particle constrained to two spatial dimensions. Suppose it is in a state represented by the (un-normalized) wavefunction given by: (r, 6) = re-ar² sin o, (1) where a is a real constant and (r, o) are the two-dimensional polar coordinates defined in the Figure below. (r, 4) P m!! = y Ø FIG. 1. Polar coordinate system For Parts (a) and (b) you may find the following hints useful. Hint 1: recall the expression of an infinitesimal area in polar coordinates is dA= drdy = rdrdo. Hint 2: Feel free to make use of the following definite integrals: [ 1²4e-an² d du = (2n-1)!! T an2n+1 = S u²n+1e-au² du: n! 2an+1¹ (3) where n is a positive integer, and the definition of the double factorial of a positive integer m is given by: 2 [mx (m - 2) × (m-4) × ... x 6 x 4 x 2, for even m mx (m2) × (m - 4) × x 5 x 3 x 1, for odd m (4) (a) Find the normalized form of the wavefunction (r, o). Call the normalized wavefunction you find (r, o) and use this for Part (b) below.
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