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The particle in an infinite potential well in QM is usually solved by easily solving Schrodinger differential equation. On the other hand particle in the harmonic oscillator oscillator potential can be solved elegantly algebraically using the creation and annihilation operators to find its spectrum.

Is it possible to do the particle in a box problem using creation and annihilation operator and how?

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See, e.g., the following article: "The hidden symmetry for a quantum system with an infinitely deep square-well potential", Shi-Hai Dong, Zhong-Qi Ma, p.520, Am. J. Phys. v.70, No.5, May 2002. EDIT: For the infinitely deep square well, the authors establish the creation and annihilation operators and obtain the matrix elements of the physical quantities from the ladder operators.

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Could you explain the relevant part of the article? Remember that if an answer doesn't contain anything but an external reference, it's not really an answer... –  David Z Feb 18 '12 at 6:04
    
Thank you for the comment. I have added some details. –  akhmeteli Feb 18 '12 at 7:50
    
And the way is not particulary elegant, but it is a good exercise. –  Angel Joaniquet Tukiainen May 1 '13 at 20:54

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