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SimpHarOsc

# SimpHarOsc - Main Page 1 Schrödinger’s Equation for the...

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Unformatted text preview: Main Page 1 Schrödinger’s Equation for the Simple Harmonic Oscillator Michael Fowler The table of values and calculation are on Sheet 2. which adjusts the energy to minimize the value of the wave function at the extreme right, thus locating bound states. For a guide to using this spreadsheet, see Sheet 3. Energy of particle: E= 8.999744 Strength of Potential D = 1 Begin to integrate from x = Initial wavefunction value f(0): 5 Initial derivative value f'(0): Numerical step size dx: 0.01 Wavefunction at far right: 4.9E-007 the wavefunction changes from oscillatory to exponential. This spreadsheet integrates f"(x) = (V(x)-E)f(x) with V(x) a simple harmonic oscillator. The " Eigenvalue " button activates the Macro of that name, NOTE: We plot the potential V(x) - E , where E is the energy of the wavefunction. At the point where this goes from negative to positive, 1 2 3 4 5 6 7-10-5 5 10 SHO Wavefunction Wavefunction Potential - E T a b le Pa ge 2 N U M E R IC A L SO L U T IO N O F SC H R O D IN G E R 'S E Q U A T IO N T he w a v e func tio n a nd its s e c o nd d e riv a tiv e a re fo und a t inte rv a ls n*d e lta _x , w he re n is a n inte g e r, a nd the firs t d e riv a tiv e is fo und a t the "le a p fro g " p o ints , (n + 0 .5 ) *d e lta _x . S o the num e ric a l s te p find ing the c ha ng e in the w a v e func tio n o v e r a n inte rv a l d e lta _x a s um e s it is d e lta _x m ultip lie d b y the v a lue o f the d e riv a tiv e a t the m id p o int o f the inte rv a l. T his is m uc h m o re a c ura te tha n us ing the v a lue o f the d e riv a tiv e a t the b e g in ing o f the inte rv a l. S im ila rly , to find the c ha ng e in the d e riv a tiv e o v e r a n inte rv a l the m e tho d us e d the v a lue o f the s e c o nd d e riv a tiv e in the m id le o f tha t inte rv a l. Position StepPotential Wavefunction 2nd_derivderiv at 1/2Potential - E v(x)-E 5 -4 .9 87-0.2 49-8.9 974 0.01 0.0 01 4.9 7 50 6-4 .978-0.6747-8.9 964 0.02 0.0 04 4.9 10 23-4 .9157-1.12393-8.9 934 0.03 0.0 09 4.97976302 -4 .8121-1.57205-8.9 8 4 0.04 0.0 16 4.96404251 -4 .6 72-2.01872-8.9 814 0.05 0.0 25 4.94385 27 -4 .481-2.46353-8.9 724 0.06 0.0 36 4.919219 3-4 .254-2.90607-8.9 614 0.07 0.0 49 4.89015918 -43.9862 -3.34594-8.9 484 0.08 0.0 64 4.856 9 82-43.678-3.78272 -8.9 3 4 0.09 0.0 81 4.818 726-43.3296-4.21601-8.9 164 0.1 0.01 4.7 671254 -42.9414 -4.64543-8.98974 0.1 0.0121 4.73025827 -42.5139-5.07057-8.98764 0.12 0.014 4.6795 262 -42.0474-5.49104-8.98534 0.13 0.0169 4.624642 3 -41.5424-5.90646-8.98284 0.14 0.0196 4.565 7 6 -40.9 95-6.31646-8.98014 0.15 0.02 5 4.50241301 -40.4193-6.72065-8.97 24 0.16 0.0256 4.43520649-39.802-7.1 867-8.97414 0.17 0.0289 4.3640197-39.1489-7.51016-8.97084 0.18 0.0324 4.28 91814 -38.4602-7.89476-8.96734 0.19 0.0361 4.209 70496 -37.7367-8.27213-8.96364 0.2 0.04 4.12724918 -36.9791-8.64192-8.95974 0.21 0.04 1 4.040829 6 -36.18 2-9.0 38 -8.95 64 0.2 0.0484 3.95079191 -35.3649 -9.35745-8.95134 0.23 0.0529 3.85721738 -34.509-9.7025-8.94684-8....
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SimpHarOsc - Main Page 1 Schrödinger’s Equation for the...

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