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Page 276 ï~~Algorithmic Conr Quantum Mechanics & the Reginald Bai Northwestern Compt School of Music, Northwe, Evanston, IL 6020 (708) 491-3895/ Abstract The Schrddinger equation, the basic equation c calculation of the wavefunction at some later t time. Wave mechanics explains the quantizat waves. Thus, an analogy may be drawn betwe wave-mechanical system (as exists in an atom) produced by traditional musical instruments, the discrete frequencies are related by integral frequenc waves. Cr Consequently, there are discrete or In the quantum-mechanical system, an forms the boundary conditions for the system-- ^C Â~rnna an eat nrn A an arni ac+fliat Ora wal n faA inr,+
Page 277 ï~~hi hp 9 We define the wavenumber k = 2rr/A P/I 2n E/ta. Thus, F p 2/(2m) and k f2nm A particle which travels in the positive x-dii travels in the x-direction and can be described b harmonic waves sin(t ei Wt -kx) kx) Evaluating complex exponentials using the identit) COSc c + e" 'o)/2, sin a (el a -e-l9/21, can be described as a linear combination of real was Accordingly, e-1 w t -kr) cos (cut -kx) i e+ i (w t - x) - cos(wt -kx) Since the probability for finding the particle mu system, the superposition of sin (Cot a possible wavefunction because at t; 0 tvan a Oitvani
Page 278 ï~~because the wavefunction must be identically ( conditions 't(x,t) O atx = 0, and '(xt) _0 note that the boundary conditions for the quanti identical to that of a vibrating string fixed at bc the boundary conditions Ae-1 E t/h + Be'i Et/lr Aei(Et/h- 2mEL/h) + Bei(Et/h which implies B =-A and Ae'iEt/h(el which reduces to 2iA sinY Due to the sine function the argument f"2mELt thus we can describe the quantization of ener 3 En n~ri4 2mL These quantized eneri distribution is es are referred to as ene
Page 279 ï~~+4? The polar angle s" is referred to as the argumnei equation8 -argz z Once the state equation has been calculated anm has occurred, a predetermined origin is selecte musical analogy of the Schrddinger equation, center ofbalance around which pitches are gen scale. e real component of the state equatic radial displacement r from the onigin in the ga chromatic0 inevlwhich, when added to the cq the nt musical event. The imaginary part, wi in the graphic analogy, determines the interval result, the movement through the pitch space x particle is in a given energy level. One pitch s consequently, the density of the gesture is dire analogous graph.