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Unformatted text preview: Differentiating under integral sign Asim Gangopadhyaya 1 Department of Physics, Loyola University Chicago, 6525 N. Sheridan Rd., Chicago IL 60626. 1 e-mail: email@example.com 1 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 = 2 + y . J = Z tan- 1 E- tan- 1 E q E- tan 2 y dy To solve it further, let us define I ( ) = Z tan- 1 E- tan- 1 E q E- tan 2 y dy Thus, J = I (1) This leads to I ( ) =- 1 2 Z tan- 1 E- tan- 1 E tan 2 y p E- tan 2 y !...
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