1. Consider a 1D particle of mass m subject to the harmonic potential given by:V(x) = mw²x² (b) Suppose now that the particle remains in the first excited state with ₁(r). Usingthe forms of the position and momentum operators defined in terms of the raisingand lowering operators, derive an expression for the uncertainty product ArAp. Howdoes your answer compare to the uncertainty product associated with the ground statewavefunction?For this question and the next you may find the following relations useful:and:p=iôn(2)=ħ2mw10,[â, a¹] = 1(a¹ + â)âyn(x) = √n+1+1(x)Fmw-(ât - â)√√n-1(2), if n>0if n = 0(2a)(2b)(3a)(3b)(3c)

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Answered: nd: H h 2mw (a¹ + a) (ât - â) Fmw p=i√…

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nd: H h 2mw (a¹ + a) (ât - â) Fmw p=i√ la ati a¹n(x) = √n+1vn+1(x) √√√n-1(2), if n>0 Ôn(2) = {√ if n = 0

Modern Physics

3rd Edition

ISBN:9781111794378

Author:Raymond A. Serway, Clement J. Moses, Curt A. Moyer

Publisher:Raymond A. Serway, Clement J. Moses, Curt A. Moyer

Chapter2: Relativity Ii

Section: Chapter Questions

Problem 18P

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Question

1. Consider a 1D particle of mass m subject to the harmonic potential given by:
V(x) = mw²x²

(b) Suppose now that the particle remains in the first excited state with ₁(r). Using
the forms of the position and momentum operators defined in terms of the raising
and lowering operators, derive an expression for the uncertainty product ArAp. How
does your answer compare to the uncertainty product associated with the ground state
wavefunction?
For this question and the next you may find the following relations useful:
and:
p=i
ôn(2)=
ħ
2mw
10,
[â, a¹] = 1
(a¹ + â)
âyn(x) = √n+1+1(x)
Fmw
-(ât - â)
√√n-1(2), if n>0
if n = 0
(2a)
(2b)
(3a)
(3b)
(3c)

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