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Diff_Under_Integration

# Diff_Under_Integration - Dierentiating under integral sign...

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Differentiating under integral sign Asim Gangopadhyaya 1 Department of Physics, Loyola University Chicago, 6525 N. Sheridan Rd., Chicago IL 60626. 1 e-mail: [email protected] 1

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In Supersymmetric quantum mechanics one often calculates a quantity Z x 2 x 1 p E - W 2 ( x ) dx , where x 1 and x 2 are points where the argument under the radical sign vanishes, i.e., they are “classical” turning points. It has been claimed that this quantity is and integral multiple of ¯ h if the energy E is equal to an eigenenergy of the system. We want to verify this assertion for the case of a particle in a flat and infinitely deep well, i.e., W ( x ) = cot x . Let J = Z cot - 1 E - cot - 1 E p E - cot 2 x dx Substitution # 1. x = π 2 + y . J = Z tan - 1 E - tan - 1 E q E - tan 2 y dy To solve it further, let us define I ( α ) = Z tan - 1 E - tan - 1 E q E - α · tan 2 y dy Thus, J = I (1) This leads to ∂I ( α ) ∂α = - 1 2 Z tan - 1 E - tan - 1 E ˆ tan 2 y p E - α · tan 2 y !
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Diff_Under_Integration - Dierentiating under integral sign...

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